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<div class="ChapSects"><a href="chap62.html#X7DDBF6F47A2E021C">62 <span class="Heading">Algebras</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap62.html#X830EDB5F85645FFB">62.1 <span class="Heading">InfoAlgebra (Info Class)</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X8665F459841AAD53">62.1-1 InfoAlgebra</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap62.html#X8686DEBA85D3F3B6">62.2 <span class="Heading">Constructing Algebras by Generators</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7B213851791A594B">62.2-1 Algebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X80FE16EA84EE56CD">62.2-2 AlgebraWithOne</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap62.html#X7A7B00127DC9DD40">62.3 <span class="Heading">Constructing Algebras as Free Algebras</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X83484C917D8F7A1A">62.3-1 FreeAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7FBD04B07B85623D">62.3-2 FreeAlgebraWithOne</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X87835FFE79D2E068">62.3-3 FreeAssociativeAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X845A777584A7D711">62.3-4 FreeAssociativeAlgebraWithOne</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap62.html#X7E8F45547CC07CE5">62.4 <span class="Heading">Constructing Algebras by Structure Constants</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7CC58DFD816E6B65">62.4-1 AlgebraByStructureConstants</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X804ADF0280F67CDC">62.4-2 StructureConstantsTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7F1203A1793411DF">62.4-3 EmptySCTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X817BD086876EC1C4">62.4-4 SetEntrySCTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7F333822780B6731">62.4-5 GapInputSCTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7C23ED85814C0371">62.4-6 TestJacobi</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X78B633CE7A5B9F9A">62.4-7 IdentityFromSCTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7F2A71608602635D">62.4-8 QuotientFromSCTable</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap62.html#X79B7C3078112E7E1">62.5 <span class="Heading">Some Special Algebras</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X83DF4BCC7CE494FC">62.5-1 QuaternionAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7B807702782F56FF">62.5-2 ComplexificationQuat</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X78C88A38853A8443">62.5-3 OctaveAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7D88E42B7DE087B0">62.5-4 FullMatrixAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X78B8BA77869DAA13">62.5-5 NullAlgebra</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap62.html#X7DF5989886BE611E">62.6 <span class="Heading">Subalgebras</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X8396643D7A49EEAD">62.6-1 Subalgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7C6B08657BD836C3">62.6-2 SubalgebraNC</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X83ECF489846F00B0">62.6-3 SubalgebraWithOne</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7A11B177868E76AA">62.6-4 SubalgebraWithOneNC</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7FDD953A84CFC3D2">62.6-5 TrivialSubalgebra</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap62.html#X83629803819C4A6F">62.7 <span class="Heading">Ideals</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap62.html#X7DC95D6982C9D7B0">62.8 <span class="Heading">Categories and Properties of Algebras</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7FEDFAA383AB20D2">62.8-1 IsFLMLOR</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X85C1E13A877DF2C8">62.8-2 IsFLMLORWithOne</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X801ED693808F6C84">62.8-3 IsAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X80B21AC27DE6D068">62.8-4 IsAlgebraWithOne</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X839BAC687B4E1A1D">62.8-5 IsLieAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X877DF13387831A6A">62.8-6 IsSimpleAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7C5AECE87D79D075">62.8-7 IsFiniteDimensional</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X82B3A9077D0CB453">62.8-8 IsQuaternion</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap62.html#X7E9273E47CF38CF1">62.9 <span class="Heading">Attributes and Operations for Algebras</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X83B055F37EBF2438">62.9-1 GeneratorsOfAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7FA408307A5A420E">62.9-2 GeneratorsOfAlgebraWithOne</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7D309FD37D94B196">62.9-3 ProductSpace</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X875CD2B37EE9A8A2">62.9-4 PowerSubalgebraSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X788F4E6184E5C863">62.9-5 AdjointBasis</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X800A410B8536E6DD">62.9-6 IndicesOfAdjointBasis</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7BA35CB28062D407">62.9-7 AsAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X878323367D0B68EB">62.9-8 AsAlgebraWithOne</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7A922D26805AFF99">62.9-9 AsSubalgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7B964BC37A975E48">62.9-10 AsSubalgebraWithOne</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7C280DAC7F840B60">62.9-11 MutableBasisOfClosureUnderAction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7BA1739D7F8B3A2B">62.9-12 MutableBasisOfNonassociativeAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X8467B687823371F9">62.9-13 MutableBasisOfIdealInNonassociativeAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7C591B7C7DEA7EEB">62.9-14 DirectSumOfAlgebras</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7D0EB1437D3D9495">62.9-15 FullMatrixAlgebraCentralizer</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X850C29907A509533">62.9-16 RadicalOfAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X82571785846CF05C">62.9-17 CentralIdempotentsOfAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7CFB230582C26DAA">62.9-18 DirectSumDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X85C58364833E014C">62.9-19 LeviMalcevDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7DCA2568870A2D34">62.9-20 Grading</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap62.html#X7E94B857847F95C1">62.10 <span class="Heading">Homomorphisms of Algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X83CE798C7D39E368">62.10-1 AlgebraGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7A7F97ED8608C882">62.10-2 AlgebraHomomorphismByImages</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X8326D1BD79725462">62.10-3 AlgebraHomomorphismByImagesNC</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X8057E55B864567AD">62.10-4 AlgebraWithOneGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X866F32B5846E5857">62.10-5 AlgebraWithOneHomomorphismByImages</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X80BF4D6A7FDC959A">62.10-6 AlgebraWithOneHomomorphismByImagesNC</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X8712E5C1861CC32C">62.10-7 NaturalHomomorphismByIdeal</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X8705A9C68102FEA3">62.10-8 OperationAlgebraHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7B249E8E86D895F0">62.10-9 NiceAlgebraMonomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X79D770777D873F80">62.10-10 IsomorphismFpAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7FB760F9813B0789">62.10-11 IsomorphismMatrixAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7F8D3DF2863EC50D">62.10-12 IsomorphismSCAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7F34244B81979696">62.10-13 RepresentativeLinearOperation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap62.html#X818DE6C57D1A4B33">62.11 <span class="Heading">Representations of Algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X8055B87F7ADBD66B">62.11-1 LeftAlgebraModuleByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X8026B99B7955A355">62.11-2 RightAlgebraModuleByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7F28A47E876427E0">62.11-3 BiAlgebraModuleByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X852524F581613359">62.11-4 LeftAlgebraModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X8222F2B67D753036">62.11-5 RightAlgebraModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X84517770868DDA02">62.11-6 BiAlgebraModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X79AAB50D83A14A43">62.11-7 GeneratorsOfAlgebraModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X82B708BD84F3DAB1">62.11-8 IsAlgebraModuleElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X80E786467F9163F9">62.11-9 IsLeftAlgebraModuleElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X863756787E2B6E75">62.11-10 IsRightAlgebraModuleElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X85654EF07F708AC3">62.11-11 LeftActingAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X826298B37E1B1520">62.11-12 RightActingAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X8308408D86CFC3C9">62.11-13 ActingAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7C325A507EC9BA18">62.11-14 IsBasisOfAlgebraModuleElementSpace</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X789863037B0E35D2">62.11-15 MatrixOfAction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X8742A7D27F26AFAB">62.11-16 SubAlgebraModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X86E0515987192F0E">62.11-17 LeftModuleByHomomorphismToMatAlg</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7EE41297867E41A8">62.11-18 RightModuleByHomomorphismToMatAlg</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X8729F0A678A4A09C">62.11-19 AdjointModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X84813BCD80BDF3C4">62.11-20 FaithfulModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7E16630185CE2C10">62.11-21 ModuleByRestriction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7885AAC87FDCF649">62.11-22 NaturalHomomorphismBySubAlgebraModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X85D0F3758551DADC">62.11-23 DirectSumOfAlgebraModules</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap62.html#X7D7A6486803B15CE">62.11-24 TranslatorSubalgebra</a></span>
</div></div>
</div>
<h3>62 <span class="Heading">Algebras</span></h3>
<p>An algebra is a vector space equipped with a bilinear map (multiplication). This chapter describes the functions in <strong class="pkg">GAP</strong> that deal with general algebras and associative algebras.</p>
<p>Algebras in <strong class="pkg">GAP</strong> are vector spaces in a natural way. So all the functionality for vector spaces (see Chapter <a href="chap61.html#X7DAD6700787EC845"><span class="RefLink">61</span></a>) is also applicable to algebras.</p>
<p><a id="X830EDB5F85645FFB" name="X830EDB5F85645FFB"></a></p>
<h4>62.1 <span class="Heading">InfoAlgebra (Info Class)</span></h4>
<p><a id="X8665F459841AAD53" name="X8665F459841AAD53"></a></p>
<h5>62.1-1 InfoAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InfoAlgebra</code></td><td class="tdright">( info class )</td></tr></table></div>
<p>is the info class for the functions dealing with algebras (see <a href="chap7.html#X7A9C902479CB6F7C"><span class="RefLink">7.4</span></a>).</p>
<p><a id="X8686DEBA85D3F3B6" name="X8686DEBA85D3F3B6"></a></p>
<h4>62.2 <span class="Heading">Constructing Algebras by Generators</span></h4>
<p><a id="X7B213851791A594B" name="X7B213851791A594B"></a></p>
<h5>62.2-1 Algebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Algebra</code>( <var class="Arg">F</var>, <var class="Arg">gens</var>[, <var class="Arg">zero</var>][, <var class="Arg">"basis"</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="code">Algebra( <var class="Arg">F</var>, <var class="Arg">gens</var> )</code> is the algebra over the division ring <var class="Arg">F</var>, generated by the vectors in the list <var class="Arg">gens</var>.</p>
<p>If there are three arguments, a division ring <var class="Arg">F</var> and a list <var class="Arg">gens</var> and an element <var class="Arg">zero</var>, then <code class="code">Algebra( <var class="Arg">F</var>, <var class="Arg">gens</var>, <var class="Arg">zero</var> )</code> is the <var class="Arg">F</var>-algebra generated by <var class="Arg">gens</var>, with zero element <var class="Arg">zero</var>.</p>
<p>If the last argument is the string <code class="code">"basis"</code> then the vectors in <var class="Arg">gens</var> are known to form a basis of the algebra (as an <var class="Arg">F</var>-vector space).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= [ [ 0, 1, 2 ], [ 0, 0, 3], [ 0, 0, 0 ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Algebra( Rationals, [ m ] );</span>
<algebra over Rationals, with 1 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( A );</span>
2
</pre></div>
<p><a id="X80FE16EA84EE56CD" name="X80FE16EA84EE56CD"></a></p>
<h5>62.2-2 AlgebraWithOne</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlgebraWithOne</code>( <var class="Arg">F</var>, <var class="Arg">gens</var>[, <var class="Arg">zero</var>][, <var class="Arg">"basis"</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="code">AlgebraWithOne( <var class="Arg">F</var>, <var class="Arg">gens</var> )</code> is the algebra-with-one over the division ring <var class="Arg">F</var>, generated by the vectors in the list <var class="Arg">gens</var>.</p>
<p>If there are three arguments, a division ring <var class="Arg">F</var> and a list <var class="Arg">gens</var> and an element <var class="Arg">zero</var>, then <code class="code">AlgebraWithOne( <var class="Arg">F</var>, <var class="Arg">gens</var>, <var class="Arg">zero</var> )</code> is the <var class="Arg">F</var>-algebra-with-one generated by <var class="Arg">gens</var>, with zero element <var class="Arg">zero</var>.</p>
<p>If the last argument is the string <code class="code">"basis"</code> then the vectors in <var class="Arg">gens</var> are known to form a basis of the algebra (as an <var class="Arg">F</var>-vector space).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= [ [ 0, 1, 2 ], [ 0, 0, 3], [ 0, 0, 0 ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= AlgebraWithOne( Rationals, [ m ] );</span>
<algebra-with-one over Rationals, with 1 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( A );</span>
3
<span class="GAPprompt">gap></span> <span class="GAPinput">One(A);</span>
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]
</pre></div>
<p><a id="X7A7B00127DC9DD40" name="X7A7B00127DC9DD40"></a></p>
<h4>62.3 <span class="Heading">Constructing Algebras as Free Algebras</span></h4>
<p><a id="X83484C917D8F7A1A" name="X83484C917D8F7A1A"></a></p>
<h5>62.3-1 FreeAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeAlgebra</code>( <var class="Arg">R</var>, <var class="Arg">rank</var>[, <var class="Arg">name</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeAlgebra</code>( <var class="Arg">R</var>, <var class="Arg">name1</var>, <var class="Arg">name2</var>, <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is a free (nonassociative) algebra of rank <var class="Arg">rank</var> over the division ring <var class="Arg">R</var>. Here <var class="Arg">name</var>, and <var class="Arg">name1</var>, <var class="Arg">name2</var>, ... are optional strings that can be used to provide names for the generators.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= FreeAlgebra( Rationals, "a", "b" );</span>
<algebra over Rationals, with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= GeneratorsOfAlgebra( A );</span>
[ (1)*a, (1)*b ]
<span class="GAPprompt">gap></span> <span class="GAPinput">(g[1]*g[2])*((g[2]*g[1])*g[1]);</span>
(1)*((a*b)*((b*a)*a))
</pre></div>
<p><a id="X7FBD04B07B85623D" name="X7FBD04B07B85623D"></a></p>
<h5>62.3-2 FreeAlgebraWithOne</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeAlgebraWithOne</code>( <var class="Arg">R</var>, <var class="Arg">rank</var>[, <var class="Arg">name</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeAlgebraWithOne</code>( <var class="Arg">R</var>, <var class="Arg">name1</var>, <var class="Arg">name2</var>, <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is a free (nonassociative) algebra-with-one of rank <var class="Arg">rank</var> over the division ring <var class="Arg">R</var>. Here <var class="Arg">name</var>, and <var class="Arg">name1</var>, <var class="Arg">name2</var>, ... are optional strings that can be used to provide names for the generators.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= FreeAlgebraWithOne( Rationals, 4, "q" );</span>
<algebra-with-one over Rationals, with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfAlgebra( A );</span>
[ (1)*<identity ...>, (1)*q.1, (1)*q.2, (1)*q.3, (1)*q.4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">One( A );</span>
(1)*<identity ...>
</pre></div>
<p><a id="X87835FFE79D2E068" name="X87835FFE79D2E068"></a></p>
<h5>62.3-3 FreeAssociativeAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeAssociativeAlgebra</code>( <var class="Arg">R</var>, <var class="Arg">rank</var>[, <var class="Arg">name</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeAssociativeAlgebra</code>( <var class="Arg">R</var>, <var class="Arg">name1</var>, <var class="Arg">name2</var>, <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is a free associative algebra of rank <var class="Arg">rank</var> over the division ring <var class="Arg">R</var>. Here <var class="Arg">name</var>, and <var class="Arg">name1</var>, <var class="Arg">name2</var>, ... are optional strings that can be used to provide names for the generators.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= FreeAssociativeAlgebra( GF( 5 ), 4, "a" );</span>
<algebra over GF(5), with 4 generators>
</pre></div>
<p><a id="X845A777584A7D711" name="X845A777584A7D711"></a></p>
<h5>62.3-4 FreeAssociativeAlgebraWithOne</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeAssociativeAlgebraWithOne</code>( <var class="Arg">R</var>, <var class="Arg">rank</var>[, <var class="Arg">name</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeAssociativeAlgebraWithOne</code>( <var class="Arg">R</var>, <var class="Arg">name1</var>, <var class="Arg">name2</var>, <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is a free associative algebra-with-one of rank <var class="Arg">rank</var> over the division ring <var class="Arg">R</var>. Here <var class="Arg">name</var>, and <var class="Arg">name1</var>, <var class="Arg">name2</var>, ... are optional strings that can be used to provide names for the generators.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= FreeAssociativeAlgebraWithOne( Rationals, "a", "b", "c" );</span>
<algebra-with-one over Rationals, with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfAlgebra( A );</span>
[ (1)*<identity ...>, (1)*a, (1)*b, (1)*c ]
<span class="GAPprompt">gap></span> <span class="GAPinput">One( A );</span>
(1)*<identity ...>
</pre></div>
<p><a id="X7E8F45547CC07CE5" name="X7E8F45547CC07CE5"></a></p>
<h4>62.4 <span class="Heading">Constructing Algebras by Structure Constants</span></h4>
<p>For an introduction into structure constants and how they are handled by <strong class="pkg">GAP</strong>, we refer to Section <a href="../../doc/tut/chap6.html#X7DDBF6F47A2E021C"><span class="RefLink">Tutorial: Algebras</span></a> of the user's tutorial.</p>
<p><a id="X7CC58DFD816E6B65" name="X7CC58DFD816E6B65"></a></p>
<h5>62.4-1 AlgebraByStructureConstants</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlgebraByStructureConstants</code>( <var class="Arg">R</var>, <var class="Arg">sctable</var>[, <var class="Arg">nameinfo</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a free left module <span class="SimpleMath">A</span> over the division ring <var class="Arg">R</var>, with multiplication defined by the structure constants table <var class="Arg">sctable</var>. The optional argument <var class="Arg">nameinfo</var> can be used to prescribe names for the elements of the canonical basis of <span class="SimpleMath">A</span>; it can be either a string <var class="Arg">name</var> (then <var class="Arg">name</var><code class="code">1</code>, <var class="Arg">name</var><code class="code">2</code> etc. are chosen) or a list of strings which are then chosen. The vectors of the canonical basis of <span class="SimpleMath">A</span> correspond to the vectors of the basis given by <var class="Arg">sctable</var>.</p>
<p>It is <em>not</em> checked whether the coefficients in <var class="Arg">sctable</var> are really elements in <var class="Arg">R</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">T:= EmptySCTable( 2, 0 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( T, 1, 1, [ 1/2, 1, 2/3, 2 ] );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= AlgebraByStructureConstants( Rationals, T );</span>
<algebra of dimension 2 over Rationals>
<span class="GAPprompt">gap></span> <span class="GAPinput">b:= BasisVectors( Basis( A ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">b[1]^2;</span>
(1/2)*v.1+(2/3)*v.2
<span class="GAPprompt">gap></span> <span class="GAPinput">b[1]*b[2];</span>
0*v.1
</pre></div>
<p><a id="X804ADF0280F67CDC" name="X804ADF0280F67CDC"></a></p>
<h5>62.4-2 StructureConstantsTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StructureConstantsTable</code>( <var class="Arg">B</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Let <var class="Arg">B</var> be a basis of a free left module <span class="SimpleMath">R</span>, say, that is also a ring. In this case <code class="func">StructureConstantsTable</code> returns a structure constants table <span class="SimpleMath">T</span> in sparse representation, as used for structure constants algebras (see Section <a href="../../doc/tut/chap6.html#X7DDBF6F47A2E021C"><span class="RefLink">Tutorial: Algebras</span></a> of the <strong class="pkg">GAP</strong> User's Tutorial).</p>
<p>If <var class="Arg">B</var> has length <span class="SimpleMath">n</span> then <span class="SimpleMath">T</span> is a list of length <span class="SimpleMath">n+2</span>. The first <span class="SimpleMath">n</span> entries of <span class="SimpleMath">T</span> are lists of length <span class="SimpleMath">n</span>. <span class="SimpleMath">T[ n+1 ]</span> is one of <span class="SimpleMath">1</span>, <span class="SimpleMath">-1</span>, or <span class="SimpleMath">0</span>; in the case of <span class="SimpleMath">1</span> the table is known to be symmetric, in the case of <span class="SimpleMath">-1</span> it is known to be antisymmetric, and <span class="SimpleMath">0</span> occurs in all other cases. <span class="SimpleMath">T[ n+2 ]</span> is the zero element of the coefficient domain.</p>
<p>The coefficients w.r.t. <var class="Arg">B</var> of the product of the <span class="SimpleMath">i</span>-th and <span class="SimpleMath">j</span>-th basis vector of <var class="Arg">B</var> are stored in <span class="SimpleMath">T[i][j]</span> as a list of length <span class="SimpleMath">2</span>; its first entry is the list of positions of nonzero coefficients, the second entry is the list of these coefficients themselves.</p>
<p>The multiplication in an algebra <span class="SimpleMath">A</span> with vector space basis <var class="Arg">B</var> with basis vectors <span class="SimpleMath">[ v_1, ..., v_n ]</span> is determined by the so-called structure matrices <span class="SimpleMath">M_k = [ m_ijk ]_ij</span>, <span class="SimpleMath">1 ≤ k ≤ n</span>. The <span class="SimpleMath">M_k</span> are defined by <span class="SimpleMath">v_i v_j = ∑_k m_ijk v_k</span>. Let <span class="SimpleMath">a = [ a_1, ..., a_n ]</span> and <span class="SimpleMath">b = [ b_1, ..., b_n ]</span>. Then</p>
<p class="pcenter">( ∑_i a_i v_i ) ( ∑_j b_j v_j ) = ∑_{i,j} a_i b_j ( v_i v_j ) = ∑_k ( ∑_j ( ∑_i a_i m_ijk ) b_j ) v_k = ∑_k ( a M_k b^tr ) v_k.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= QuaternionAlgebra( Rationals );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureConstantsTable( Basis( A ) );</span>
[ [ [ [ 1 ], [ 1 ] ], [ [ 2 ], [ 1 ] ], [ [ 3 ], [ 1 ] ],
[ [ 4 ], [ 1 ] ] ],
[ [ [ 2 ], [ 1 ] ], [ [ 1 ], [ -1 ] ], [ [ 4 ], [ 1 ] ],
[ [ 3 ], [ -1 ] ] ],
[ [ [ 3 ], [ 1 ] ], [ [ 4 ], [ -1 ] ], [ [ 1 ], [ -1 ] ],
[ [ 2 ], [ 1 ] ] ],
[ [ [ 4 ], [ 1 ] ], [ [ 3 ], [ 1 ] ], [ [ 2 ], [ -1 ] ],
[ [ 1 ], [ -1 ] ] ], 0, 0 ]
</pre></div>
<p><a id="X7F1203A1793411DF" name="X7F1203A1793411DF"></a></p>
<h5>62.4-3 EmptySCTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EmptySCTable</code>( <var class="Arg">dim</var>, <var class="Arg">zero</var>[, <var class="Arg">flag</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">EmptySCTable</code> returns a structure constants table for an algebra of dimension <var class="Arg">dim</var>, describing trivial multiplication. <var class="Arg">zero</var> must be the zero of the coefficients domain. If the multiplication is known to be (anti)commutative then this can be indicated by the optional third argument <var class="Arg">flag</var>, which must be one of the strings <code class="code">"symmetric"</code>, <code class="code">"antisymmetric"</code>.</p>
<p>For filling up the structure constants table, see <code class="func">SetEntrySCTable</code> (<a href="chap62.html#X817BD086876EC1C4"><span class="RefLink">62.4-4</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">EmptySCTable( 2, Zero( GF(5) ), "antisymmetric" );</span>
[ [ [ [ ], [ ] ], [ [ ], [ ] ] ],
[ [ [ ], [ ] ], [ [ ], [ ] ] ], -1, 0*Z(5) ]
</pre></div>
<p><a id="X817BD086876EC1C4" name="X817BD086876EC1C4"></a></p>
<h5>62.4-4 SetEntrySCTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SetEntrySCTable</code>( <var class="Arg">T</var>, <var class="Arg">i</var>, <var class="Arg">j</var>, <var class="Arg">list</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>sets the entry of the structure constants table <var class="Arg">T</var> that describes the product of the <var class="Arg">i</var>-th basis element with the <var class="Arg">j</var>-th basis element to the value given by the list <var class="Arg">list</var>.</p>
<p>If <var class="Arg">T</var> is known to be antisymmetric or symmetric then also the value <code class="code"><var class="Arg">T</var>[<var class="Arg">j</var>][<var class="Arg">i</var>]</code> is set.</p>
<p><var class="Arg">list</var> must be of the form <span class="SimpleMath">[ c_ij^{k_1}, k_1, c_ij^{k_2}, k_2, ... ]</span>.</p>
<p>The entries at the odd positions of <var class="Arg">list</var> must be compatible with the zero element stored in <var class="Arg">T</var>. For convenience, these entries may also be rational numbers that are automatically replaced by the corresponding elements in the appropriate prime field in finite characteristic if necessary.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">T:= EmptySCTable( 2, 0 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( T, 1, 1, [ 1/2, 1, 2/3, 2 ] );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">T;</span>
[ [ [ [ 1, 2 ], [ 1/2, 2/3 ] ], [ [ ], [ ] ] ],
[ [ [ ], [ ] ], [ [ ], [ ] ] ], 0, 0 ]
</pre></div>
<p><a id="X7F333822780B6731" name="X7F333822780B6731"></a></p>
<h5>62.4-5 GapInputSCTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GapInputSCTable</code>( <var class="Arg">T</var>, <var class="Arg">varname</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is a string that describes the structure constants table <var class="Arg">T</var> in terms of <code class="func">EmptySCTable</code> (<a href="chap62.html#X7F1203A1793411DF"><span class="RefLink">62.4-3</span></a>) and <code class="func">SetEntrySCTable</code> (<a href="chap62.html#X817BD086876EC1C4"><span class="RefLink">62.4-4</span></a>). The assignments are made to the variable <var class="Arg">varname</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">T:= EmptySCTable( 2, 0 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( T, 1, 2, [ 1, 2 ] );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( T, 2, 1, [ 1, 2 ] );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GapInputSCTable( T, "T" );</span>
"T:= EmptySCTable( 2, 0 );\nSetEntrySCTable( T, 1, 2, [1,2] );\nSetEnt\
rySCTable( T, 2, 1, [1,2] );\n"
</pre></div>
<p><a id="X7C23ED85814C0371" name="X7C23ED85814C0371"></a></p>
<h5>62.4-6 TestJacobi</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TestJacobi</code>( <var class="Arg">T</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>tests whether the structure constants table <var class="Arg">T</var> satisfies the Jacobi identity <span class="SimpleMath">v_i * (v_j * v_k) + v_j * (v_k * v_i) + v_k * (v_i * v_j) = 0</span> for all basis vectors <span class="SimpleMath">v_i</span> of the underlying algebra, where <span class="SimpleMath">i ≤ j ≤ k</span>. (Thus antisymmetry is assumed.)</p>
<p>The function returns <code class="keyw">true</code> if the Jacobi identity is satisfied, and a failing triple <span class="SimpleMath">[ i, j, k ]</span> otherwise.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">T:= EmptySCTable( 2, 0, "antisymmetric" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( T, 1, 2, [ 1, 2 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">TestJacobi( T );</span>
true
</pre></div>
<p><a id="X78B633CE7A5B9F9A" name="X78B633CE7A5B9F9A"></a></p>
<h5>62.4-7 IdentityFromSCTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdentityFromSCTable</code>( <var class="Arg">T</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">T</var> be a structure constants table of an algebra <span class="SimpleMath">A</span> of dimension <span class="SimpleMath">n</span>. <code class="code">IdentityFromSCTable( <var class="Arg">T</var> )</code> is either <code class="keyw">fail</code> or the vector of length <span class="SimpleMath">n</span> that contains the coefficients of the multiplicative identity of <span class="SimpleMath">A</span> with respect to the basis that belongs to <var class="Arg">T</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">T:= EmptySCTable( 2, 0 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( T, 1, 1, [ 1, 1 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( T, 1, 2, [ 1, 2 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( T, 2, 1, [ 1, 2 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IdentityFromSCTable( T );</span>
[ 1, 0 ]
</pre></div>
<p><a id="X7F2A71608602635D" name="X7F2A71608602635D"></a></p>
<h5>62.4-8 QuotientFromSCTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ QuotientFromSCTable</code>( <var class="Arg">T</var>, <var class="Arg">num</var>, <var class="Arg">den</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">T</var> be a structure constants table of an algebra <span class="SimpleMath">A</span> of dimension <span class="SimpleMath">n</span>. <code class="code">QuotientFromSCTable( <var class="Arg">T</var> )</code> is either <code class="keyw">fail</code> or the vector of length <span class="SimpleMath">n</span> that contains the coefficients of the quotient of <var class="Arg">num</var> and <var class="Arg">den</var> with respect to the basis that belongs to <var class="Arg">T</var>.</p>
<p>We solve the equation system <var class="Arg">num</var><span class="SimpleMath">= x *</span> <var class="Arg">den</var>. If no solution exists, <code class="keyw">fail</code> is returned.</p>
<p>In terms of the basis <span class="SimpleMath">B</span> with vectors <span class="SimpleMath">b_1, ..., b_n</span> this means for <span class="SimpleMath"><var class="Arg">num</var> = ∑_{i = 1}^n a_i b_i</span>, <span class="SimpleMath"><var class="Arg">den</var> = ∑_{i = 1}^n c_i b_i</span>, <span class="SimpleMath">x = ∑_{i = 1}^n x_i b_i</span> that <span class="SimpleMath">a_k = ∑_{i,j} c_i x_j c_ijk</span> for all <span class="SimpleMath">k</span>. Here <span class="SimpleMath">c_ijk</span> denotes the structure constants with respect to <span class="SimpleMath">B</span>. This means that (as a vector) <span class="SimpleMath">a = x M</span> with <span class="SimpleMath">M_jk = ∑_{i = 1}^n c_ijk c_i</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">T:= EmptySCTable( 2, 0 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( T, 1, 1, [ 1, 1 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( T, 2, 1, [ 1, 2 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( T, 1, 2, [ 1, 2 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">QuotientFromSCTable( T, [0,1], [1,0] );</span>
[ 0, 1 ]
</pre></div>
<p><a id="X79B7C3078112E7E1" name="X79B7C3078112E7E1"></a></p>
<h4>62.5 <span class="Heading">Some Special Algebras</span></h4>
<p><a id="X83DF4BCC7CE494FC" name="X83DF4BCC7CE494FC"></a></p>
<h5>62.5-1 QuaternionAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ QuaternionAlgebra</code>( <var class="Arg">F</var>[, <var class="Arg">a</var>, <var class="Arg">b</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a quaternion algebra over <var class="Arg">F</var>, with parameters <var class="Arg">a</var> and <var class="Arg">b</var>.</p>
<p>Let <var class="Arg">F</var> be a field or a list of field elements, let <span class="SimpleMath">F</span> be the field generated by <var class="Arg">F</var>, and let <var class="Arg">a</var> and <var class="Arg">b</var> two elements in <span class="SimpleMath">F</span>. <code class="func">QuaternionAlgebra</code> returns a quaternion algebra over <span class="SimpleMath">F</span>, with parameters <var class="Arg">a</var> and <var class="Arg">b</var>, i.e., a four-dimensional associative <span class="SimpleMath">F</span>-algebra with basis <span class="SimpleMath">(e,i,j,k)</span> and multiplication defined by <span class="SimpleMath">e e = e</span>, <span class="SimpleMath">e i = i e = i</span>, <span class="SimpleMath">e j = j e = j</span>, <span class="SimpleMath">e k = k e = k</span>, <span class="SimpleMath">i i = <var class="Arg">a</var> e</span>, <span class="SimpleMath">i j = - j i = k</span>, <span class="SimpleMath">i k = - k i = <var class="Arg">a</var> j</span>, <span class="SimpleMath">j j = <var class="Arg">b</var> e</span>, <span class="SimpleMath">j k = - k j = <var class="Arg">b</var> i</span>, <span class="SimpleMath">k k = - <var class="Arg">a</var> <var class="Arg">b</var> e</span>. The default value for both <var class="Arg">a</var> and <var class="Arg">b</var> is <span class="SimpleMath">-1 ∈ F</span>.</p>
<p>The <code class="func">GeneratorsOfAlgebra</code> (<a href="chap62.html#X83B055F37EBF2438"><span class="RefLink">62.9-1</span></a>) and <code class="func">CanonicalBasis</code> (<a href="chap61.html#X7C8EBFF5805F8C51"><span class="RefLink">61.5-3</span></a>) value of an algebra constructed with <code class="func">QuaternionAlgebra</code> is the list <span class="SimpleMath">[ e, i, j, k ]</span>.</p>
<p>Two quaternion algebras with the same parameters <var class="Arg">a</var>, <var class="Arg">b</var> lie in the same family, so it makes sense to consider their intersection or to ask whether they are contained in each other. (This is due to the fact that the results of <code class="func">QuaternionAlgebra</code> are cached, in the global variable <code class="code">QuaternionAlgebraData</code>.)</p>
<p>The embedding of the field <code class="func">GaussianRationals</code> (<a href="chap60.html#X82F53C65802FF551"><span class="RefLink">60.1-3</span></a>) into a quaternion algebra <span class="SimpleMath">A</span> over <code class="func">Rationals</code> (<a href="chap17.html#X7B6029D18570C08A"><span class="RefLink">17.1-1</span></a>) is not uniquely determined. One can specify one embedding as a vector space homomorphism that maps <code class="code">1</code> to the first algebra generator of <span class="SimpleMath">A</span>, and <code class="code">E(4)</code> to one of the others.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">QuaternionAlgebra( Rationals );</span>
<algebra-with-one of dimension 4 over Rationals>
</pre></div>
<p><a id="X7B807702782F56FF" name="X7B807702782F56FF"></a></p>
<h5>62.5-2 ComplexificationQuat</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ComplexificationQuat</code>( <var class="Arg">vector</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ComplexificationQuat</code>( <var class="Arg">matrix</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <span class="SimpleMath">A = e F ⊕ i F ⊕ j F ⊕ k F</span> be a quaternion algebra over the field <span class="SimpleMath">F</span> of cyclotomics, with basis <span class="SimpleMath">(e,i,j,k)</span>.</p>
<p>If <span class="SimpleMath">v = v_1 + v_2 j</span> is a row vector over <span class="SimpleMath">A</span> with <span class="SimpleMath">v_1 = e w_1 + i w_2</span> and <span class="SimpleMath">v_2 = e w_3 + i w_4</span> then <code class="func">ComplexificationQuat</code> called with argument <span class="SimpleMath">v</span> returns the concatenation of <span class="SimpleMath">w_1 +</span><code class="code">E(4)</code><span class="SimpleMath">w_2</span> and <span class="SimpleMath">w_3 +</span><code class="code">E(4)</code><span class="SimpleMath">w_4</span>.</p>
<p>If <span class="SimpleMath">M = M_1 + M_2 j</span> is a matrix over <span class="SimpleMath">A</span> with <span class="SimpleMath">M_1 = e N_1 + i N_2</span> and <span class="SimpleMath">M_2 = e N_3 + i N_4</span> then <code class="func">ComplexificationQuat</code> called with argument <span class="SimpleMath">M</span> returns the block matrix <span class="SimpleMath">A</span> over <span class="SimpleMath">e F ⊕ i F</span> such that <span class="SimpleMath">A(1,1) = N_1 +</span><code class="code">E(4)</code><span class="SimpleMath">N_2</span>, <span class="SimpleMath">A(2,2) = N_1 -</span><code class="code">E(4)</code><span class="SimpleMath">N_2</span>, <span class="SimpleMath">A(1,2) = N_3 +</span><code class="code">E(4)</code><span class="SimpleMath">N_4</span>, and <span class="SimpleMath">A(2,1) = - N_3 +</span><code class="code">E(4)</code><span class="SimpleMath">N_4</span>.</p>
<p>Then <code class="code">ComplexificationQuat(<var class="Arg">v</var>) * ComplexificationQuat(<var class="Arg">M</var>)= ComplexificationQuat(<var class="Arg">v</var> * <var class="Arg">M</var>)</code>, since</p>
<p class="pcenter">v M = v_1 M_1 + v_2 j M_1 + v_1 M_2 j + v_2 j M_2 j = ( v_1 M_1 - v_2 overline{M_2} ) + ( v_1 M_2 + v_2 overline{M_1} ) j.</p>
<p><a id="X78C88A38853A8443" name="X78C88A38853A8443"></a></p>
<h5>62.5-3 OctaveAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OctaveAlgebra</code>( <var class="Arg">F</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The algebra of octonions over <var class="Arg">F</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">OctaveAlgebra( Rationals );</span>
<algebra of dimension 8 over Rationals>
</pre></div>
<p><a id="X7D88E42B7DE087B0" name="X7D88E42B7DE087B0"></a></p>
<h5>62.5-4 FullMatrixAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FullMatrixAlgebra</code>( <var class="Arg">R</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MatrixAlgebra</code>( <var class="Arg">R</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MatAlgebra</code>( <var class="Arg">R</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is the full matrix algebra of <span class="SimpleMath"><var class="Arg">n</var> × <var class="Arg">n</var></span> matrices over the ring <var class="Arg">R</var>, for a nonnegative integer <var class="Arg">n</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:=FullMatrixAlgebra( Rationals, 20 );</span>
( Rationals^[ 20, 20 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( A );</span>
400
</pre></div>
<p><a id="X78B8BA77869DAA13" name="X78B8BA77869DAA13"></a></p>
<h5>62.5-5 NullAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NullAlgebra</code>( <var class="Arg">R</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The zero-dimensional algebra over <var class="Arg">R</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= NullAlgebra( Rationals );</span>
<algebra over Rationals>
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( A );</span>
0
</pre></div>
<p><a id="X7DF5989886BE611E" name="X7DF5989886BE611E"></a></p>
<h4>62.6 <span class="Heading">Subalgebras</span></h4>
<p><a id="X8396643D7A49EEAD" name="X8396643D7A49EEAD"></a></p>
<h5>62.6-1 Subalgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Subalgebra</code>( <var class="Arg">A</var>, <var class="Arg">gens</var>[, <var class="Arg">"basis"</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>is the <span class="SimpleMath">F</span>-algebra generated by <var class="Arg">gens</var>, with parent algebra <var class="Arg">A</var>, where <span class="SimpleMath">F</span> is the left acting domain of <var class="Arg">A</var>.</p>
<p><em>Note</em> that being a subalgebra of <var class="Arg">A</var> means to be an algebra, to be contained in <var class="Arg">A</var>, <em>and</em> to have the same left acting domain as <var class="Arg">A</var>.</p>
<p>An optional argument <code class="code">"basis"</code> may be added if it is known that the generators already form a basis of the algebra. Then it is <em>not</em> checked whether <var class="Arg">gens</var> really are linearly independent and whether all elements in <var class="Arg">gens</var> lie in <var class="Arg">A</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= [ [ 0, 1, 2 ], [ 0, 0, 3], [ 0, 0, 0 ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Algebra( Rationals, [ m ] );</span>
<algebra over Rationals, with 1 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Subalgebra( A, [ m^2 ] );</span>
<algebra over Rationals, with 1 generators>
</pre></div>
<p><a id="X7C6B08657BD836C3" name="X7C6B08657BD836C3"></a></p>
<h5>62.6-2 SubalgebraNC</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SubalgebraNC</code>( <var class="Arg">A</var>, <var class="Arg">gens</var>[, <var class="Arg">"basis"</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">SubalgebraNC</code> does the same as <code class="func">Subalgebra</code> (<a href="chap62.html#X8396643D7A49EEAD"><span class="RefLink">62.6-1</span></a>), except that it does not check whether all elements in <var class="Arg">gens</var> lie in <var class="Arg">A</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= RandomMat( 3, 3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Algebra( Rationals, [ m ] );</span>
<algebra over Rationals, with 1 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">SubalgebraNC( A, [ IdentityMat( 3, 3 ) ], "basis" );</span>
<algebra of dimension 1 over Rationals>
</pre></div>
<p><a id="X83ECF489846F00B0" name="X83ECF489846F00B0"></a></p>
<h5>62.6-3 SubalgebraWithOne</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SubalgebraWithOne</code>( <var class="Arg">A</var>, <var class="Arg">gens</var>[, <var class="Arg">"basis"</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>is the algebra-with-one generated by <var class="Arg">gens</var>, with parent algebra <var class="Arg">A</var>.</p>
<p>The optional third argument, the string <code class="code">"basis"</code>, may be added if it is known that the elements from <var class="Arg">gens</var> are linearly independent. Then it is <em>not</em> checked whether <var class="Arg">gens</var> really are linearly independent and whether all elements in <var class="Arg">gens</var> lie in <var class="Arg">A</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= [ [ 0, 1, 2 ], [ 0, 0, 3], [ 0, 0, 0 ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= AlgebraWithOne( Rationals, [ m ] );</span>
<algebra-with-one over Rationals, with 1 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">B1:= SubalgebraWithOne( A, [ m ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B2:= Subalgebra( A, [ m ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( B1 );</span>
3
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( B2 );</span>
2
</pre></div>
<p><a id="X7A11B177868E76AA" name="X7A11B177868E76AA"></a></p>
<h5>62.6-4 SubalgebraWithOneNC</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SubalgebraWithOneNC</code>( <var class="Arg">A</var>, <var class="Arg">gens</var>[, <var class="Arg">"basis"</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">SubalgebraWithOneNC</code> does the same as <code class="func">SubalgebraWithOne</code> (<a href="chap62.html#X83ECF489846F00B0"><span class="RefLink">62.6-3</span></a>), except that it does not check whether all elements in <var class="Arg">gens</var> lie in <var class="Arg">A</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= RandomMat( 3, 3 );; A:= Algebra( Rationals, [ m ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SubalgebraWithOneNC( A, [ m ] );</span>
<algebra-with-one over Rationals, with 1 generators>
</pre></div>
<p><a id="X7FDD953A84CFC3D2" name="X7FDD953A84CFC3D2"></a></p>
<h5>62.6-5 TrivialSubalgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TrivialSubalgebra</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The zero dimensional subalgebra of the algebra <var class="Arg">A</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= QuaternionAlgebra( Rationals );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= TrivialSubalgebra( A );</span>
<algebra over Rationals>
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( B );</span>
0
</pre></div>
<p><a id="X83629803819C4A6F" name="X83629803819C4A6F"></a></p>
<h4>62.7 <span class="Heading">Ideals</span></h4>
<p>For constructing and working with ideals in algebras the same functions are available as for ideals in rings. So for the precise description of these functions we refer to Chapter <a href="chap56.html#X81897F6082CACB59"><span class="RefLink">56</span></a>. Here we give examples demonstrating the use of ideals in algebras. For an introduction into the construction of quotient algebras we refer to Chapter <a href="../../doc/tut/chap6.html#X7DDBF6F47A2E021C"><span class="RefLink">Tutorial: Algebras</span></a> of the user's tutorial.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= [ [ 0, 2, 3 ], [ 0, 0, 4 ], [ 0, 0, 0] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= AlgebraWithOne( Rationals, [ m ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">I:= Ideal( A, [ m ] ); # the two-sided ideal of `A' generated by `m'</span>
<two-sided ideal in <algebra-with-one of dimension 3 over Rationals>,
(1 generators)>
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( I );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfIdeal( I );</span>
[ [ [ 0, 2, 3 ], [ 0, 0, 4 ], [ 0, 0, 0 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">BasisVectors( Basis( I ) );</span>
[ [ [ 0, 1, 3/2 ], [ 0, 0, 2 ], [ 0, 0, 0 ] ],
[ [ 0, 0, 1 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= FullMatrixAlgebra( Rationals, 4 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= NullMat( 4, 4 );; m[1][4]:=1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">I:= LeftIdeal( A, [ m ] );</span>
<left ideal in ( Rationals^[ 4, 4 ] ), (1 generators)>
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( I );</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfLeftIdeal( I );</span>
[ [ [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">mats:= [ [[1,0],[0,0]], [[0,1],[0,0]], [[0,0],[0,1]] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Algebra( Rationals, mats );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"># Form the two-sided ideal for which `mats[2]' is known to be</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"># the unique basis element.</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">I:= Ideal( A, [ mats[2] ], "basis" );</span>
<two-sided ideal in <algebra of dimension 3 over Rationals>,
(dimension 1)>
</pre></div>
<p><a id="X7DC95D6982C9D7B0" name="X7DC95D6982C9D7B0"></a></p>
<h4>62.8 <span class="Heading">Categories and Properties of Algebras</span></h4>
<p><a id="X7FEDFAA383AB20D2" name="X7FEDFAA383AB20D2"></a></p>
<h5>62.8-1 IsFLMLOR</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFLMLOR</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>A FLMLOR ("free left module left operator ring") in <strong class="pkg">GAP</strong> is a ring that is also a free left module.</p>
<p>Note that this means that being a FLMLOR is not a property a ring can get, since a ring is usually not represented as an external left set.</p>
<p>Examples are magma rings (e.g. over the integers) or algebras.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= FullMatrixAlgebra( Rationals, 2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsFLMLOR ( A );</span>
true
</pre></div>
<p><a id="X85C1E13A877DF2C8" name="X85C1E13A877DF2C8"></a></p>
<h5>62.8-2 IsFLMLORWithOne</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFLMLORWithOne</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>A FLMLOR-with-one in <strong class="pkg">GAP</strong> is a ring-with-one that is also a free left module.</p>
<p>Note that this means that being a FLMLOR-with-one is not a property a ring-with-one can get, since a ring-with-one is usually not represented as an external left set.</p>
<p>Examples are magma rings-with-one or algebras-with-one (but also over the integers).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= FullMatrixAlgebra( Rationals, 2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsFLMLORWithOne ( A );</span>
true
</pre></div>
<p><a id="X801ED693808F6C84" name="X801ED693808F6C84"></a></p>
<h5>62.8-3 IsAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAlgebra</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>An algebra in <strong class="pkg">GAP</strong> is a ring that is also a left vector space. Note that this means that being an algebra is not a property a ring can get, since a ring is usually not represented as an external left set.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= MatAlgebra( Rationals, 3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAlgebra( A );</span>
true
</pre></div>
<p><a id="X80B21AC27DE6D068" name="X80B21AC27DE6D068"></a></p>
<h5>62.8-4 IsAlgebraWithOne</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAlgebraWithOne</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>An algebra-with-one in <strong class="pkg">GAP</strong> is a ring-with-one that is also a left vector space. Note that this means that being an algebra-with-one is not a property a ring-with-one can get, since a ring-with-one is usually not represented as an external left set.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= MatAlgebra( Rationals, 3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAlgebraWithOne( A );</span>
true
</pre></div>
<p><a id="X839BAC687B4E1A1D" name="X839BAC687B4E1A1D"></a></p>
<h5>62.8-5 IsLieAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLieAlgebra</code>( <var class="Arg">A</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>An algebra <var class="Arg">A</var> is called Lie algebra if <span class="SimpleMath">a * a = 0</span> for all <span class="SimpleMath">a</span> in <var class="Arg">A</var> and <span class="SimpleMath">( a * ( b * c ) ) + ( b * ( c * a ) ) + ( c * ( a * b ) ) = 0</span> for all <span class="SimpleMath">a, b, c ∈</span><var class="Arg">A</var> (Jacobi identity).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= FullMatrixLieAlgebra( Rationals, 3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsLieAlgebra( A );</span>
true
</pre></div>
<p><a id="X877DF13387831A6A" name="X877DF13387831A6A"></a></p>
<h5>62.8-6 IsSimpleAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSimpleAlgebra</code>( <var class="Arg">A</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>is <code class="keyw">true</code> if the algebra <var class="Arg">A</var> is simple, and <code class="keyw">false</code> otherwise. This function is only implemented for the cases where <var class="Arg">A</var> is an associative or a Lie algebra. And for Lie algebras it is only implemented for the case where the ground field is of characteristic zero.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= FullMatrixLieAlgebra( Rationals, 3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSimpleAlgebra( A );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= MatAlgebra( Rationals, 3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSimpleAlgebra( A );</span>
true
</pre></div>
<p><a id="X7C5AECE87D79D075" name="X7C5AECE87D79D075"></a></p>
<h5>62.8-7 IsFiniteDimensional</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFiniteDimensional</code>( <var class="Arg">matalg</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>returns <code class="keyw">true</code> (always) for a matrix algebra <var class="Arg">matalg</var>, since matrix algebras are always finite dimensional.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= MatAlgebra( Rationals, 3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsFiniteDimensional( A );</span>
true
</pre></div>
<p><a id="X82B3A9077D0CB453" name="X82B3A9077D0CB453"></a></p>
<h5>62.8-8 IsQuaternion</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsQuaternion</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsQuaternionCollection</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsQuaternionCollColl</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p><code class="func">IsQuaternion</code> is the category of elements in an algebra constructed by <code class="func">QuaternionAlgebra</code> (<a href="chap62.html#X83DF4BCC7CE494FC"><span class="RefLink">62.5-1</span></a>). A collection of quaternions lies in the category <code class="func">IsQuaternionCollection</code>. Finally, a collection of quaternion collections (e.g., a matrix of quaternions) lies in the category <code class="func">IsQuaternionCollColl</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= QuaternionAlgebra( Rationals );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">b:= BasisVectors( Basis( A ) );</span>
[ e, i, j, k ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsQuaternion( b[1] );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsQuaternionCollColl( [ [ b[1], b[2] ], [ b[3], b[4] ] ] );</span>
true
</pre></div>
<p><a id="X7E9273E47CF38CF1" name="X7E9273E47CF38CF1"></a></p>
<h4>62.9 <span class="Heading">Attributes and Operations for Algebras</span></h4>
<p><a id="X83B055F37EBF2438" name="X83B055F37EBF2438"></a></p>
<h5>62.9-1 GeneratorsOfAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneratorsOfAlgebra</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of elements that generate <var class="Arg">A</var> as an algebra.</p>
<p>For a free algebra, each generator can also be accessed using the <code class="code">.</code> operator (see <code class="func">GeneratorsOfDomain</code> (<a href="chap31.html#X7E353DD1838AB223"><span class="RefLink">31.9-2</span></a>)).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= [ [ 0, 1, 2 ], [ 0, 0, 3 ], [ 0, 0, 0 ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= AlgebraWithOne( Rationals, [ m ] );</span>
<algebra-with-one over Rationals, with 1 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfAlgebra( A );</span>
[ [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
[ [ 0, 1, 2 ], [ 0, 0, 3 ], [ 0, 0, 0 ] ] ]
</pre></div>
<p><a id="X7FA408307A5A420E" name="X7FA408307A5A420E"></a></p>
<h5>62.9-2 GeneratorsOfAlgebraWithOne</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneratorsOfAlgebraWithOne</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of elements of <var class="Arg">A</var> that generate <var class="Arg">A</var> as an algebra with one.</p>
<p>For a free algebra with one, each generator can also be accessed using the <code class="code">.</code> operator (see <code class="func">GeneratorsOfDomain</code> (<a href="chap31.html#X7E353DD1838AB223"><span class="RefLink">31.9-2</span></a>)).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= [ [ 0, 1, 2 ], [ 0, 0, 3 ], [ 0, 0, 0 ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= AlgebraWithOne( Rationals, [ m ] );</span>
<algebra-with-one over Rationals, with 1 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfAlgebraWithOne( A );</span>
[ [ [ 0, 1, 2 ], [ 0, 0, 3 ], [ 0, 0, 0 ] ] ]
</pre></div>
<p><a id="X7D309FD37D94B196" name="X7D309FD37D94B196"></a></p>
<h5>62.9-3 ProductSpace</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProductSpace</code>( <var class="Arg">U</var>, <var class="Arg">V</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>is the vector space <span class="SimpleMath">⟨ u * v ; u ∈ U, v ∈ V ⟩</span>, where <span class="SimpleMath">U</span> and <span class="SimpleMath">V</span> are subspaces of the same algebra.</p>
<p>If <span class="SimpleMath"><var class="Arg">U</var> = <var class="Arg">V</var></span> is known to be an algebra then the product space is also an algebra, moreover it is an ideal in <var class="Arg">U</var>. If <var class="Arg">U</var> and <var class="Arg">V</var> are known to be ideals in an algebra <span class="SimpleMath">A</span> then the product space is known to be an algebra and an ideal in <span class="SimpleMath">A</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= QuaternionAlgebra( Rationals );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">b:= BasisVectors( Basis( A ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Subalgebra( A, [ b[4] ] );</span>
<algebra over Rationals, with 1 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">ProductSpace( A, B );</span>
<vector space of dimension 4 over Rationals>
</pre></div>
<p><a id="X875CD2B37EE9A8A2" name="X875CD2B37EE9A8A2"></a></p>
<h5>62.9-4 PowerSubalgebraSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PowerSubalgebraSeries</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of subalgebras of <var class="Arg">A</var>, the first term of which is <var class="Arg">A</var>; and every next term is the product space of the previous term with itself.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= QuaternionAlgebra( Rationals );</span>
<algebra-with-one of dimension 4 over Rationals>
<span class="GAPprompt">gap></span> <span class="GAPinput">PowerSubalgebraSeries( A );</span>
[ <algebra-with-one of dimension 4 over Rationals> ]
</pre></div>
<p><a id="X788F4E6184E5C863" name="X788F4E6184E5C863"></a></p>
<h5>62.9-5 AdjointBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AdjointBasis</code>( <var class="Arg">B</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The <em>adjoint map</em> <span class="SimpleMath">ad(x)</span> of an element <span class="SimpleMath">x</span> in an <span class="SimpleMath">F</span>-algebra <span class="SimpleMath">A</span>, say, is the left multiplication by <span class="SimpleMath">x</span>. This map is <span class="SimpleMath">F</span>-linear and thus, w.r.t. the given basis <var class="Arg">B</var><span class="SimpleMath">= (x_1, x_2, ..., x_n)</span> of <span class="SimpleMath">A</span>, <span class="SimpleMath">ad(x)</span> can be represented by a matrix over <span class="SimpleMath">F</span>. Let <span class="SimpleMath">V</span> denote the <span class="SimpleMath">F</span>-vector space of the matrices corresponding to <span class="SimpleMath">ad(x)</span>, for <span class="SimpleMath">x ∈ A</span>. Then <code class="func">AdjointBasis</code> returns the basis of <span class="SimpleMath">V</span> that consists of the matrices for <span class="SimpleMath">ad(x_1), ..., ad(x_n)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= QuaternionAlgebra( Rationals );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">AdjointBasis( Basis( A ) );</span>
Basis( <vector space over Rationals, with 4 generators>,
[ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ],
[ [ 0, -1, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, -1 ], [ 0, 0, 1, 0 ] ]
,
[ [ 0, 0, -1, 0 ], [ 0, 0, 0, 1 ], [ 1, 0, 0, 0 ], [ 0, -1, 0, 0 ] ]
,
[ [ 0, 0, 0, -1 ], [ 0, 0, -1, 0 ], [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ]
] ] )
</pre></div>
<p><a id="X800A410B8536E6DD" name="X800A410B8536E6DD"></a></p>
<h5>62.9-6 IndicesOfAdjointBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IndicesOfAdjointBasis</code>( <var class="Arg">B</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Let <var class="Arg">A</var> be an algebra and let <var class="Arg">B</var> be the basis that is output by <code class="code">AdjointBasis( Basis( <var class="Arg">A</var> ) )</code>. This function returns a list of indices. If <span class="SimpleMath">i</span> is an index belonging to this list, then <span class="SimpleMath">ad x_i</span> is a basis vector of the matrix space spanned by <span class="SimpleMath">ad A</span>, where <span class="SimpleMath">x_i</span> is the <span class="SimpleMath">i</span>-th basis vector of the basis <var class="Arg">B</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:= FullMatrixLieAlgebra( Rationals, 3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= AdjointBasis( Basis( L ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IndicesOfAdjointBasis( B );</span>
[ 1, 2, 3, 4, 5, 6, 7, 8 ]
</pre></div>
<p><a id="X7BA35CB28062D407" name="X7BA35CB28062D407"></a></p>
<h5>62.9-7 AsAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsAlgebra</code>( <var class="Arg">F</var>, <var class="Arg">A</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns the algebra over <var class="Arg">F</var> generated by <var class="Arg">A</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= VectorSpace( Rationals, [ IdentityMat( 2 ) ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">AsAlgebra( Rationals, V );</span>
<algebra of dimension 1 over Rationals>
</pre></div>
<p><a id="X878323367D0B68EB" name="X878323367D0B68EB"></a></p>
<h5>62.9-8 AsAlgebraWithOne</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsAlgebraWithOne</code>( <var class="Arg">F</var>, <var class="Arg">A</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>If the algebra <var class="Arg">A</var> has an identity, then it can be viewed as an algebra with one over <var class="Arg">F</var>. This function returns this algebra with one.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= VectorSpace( Rationals, [ IdentityMat( 2 ) ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= AsAlgebra( Rationals, V );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">AsAlgebraWithOne( Rationals, A );</span>
<algebra-with-one over Rationals, with 1 generators>
</pre></div>
<p><a id="X7A922D26805AFF99" name="X7A922D26805AFF99"></a></p>
<h5>62.9-9 AsSubalgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsSubalgebra</code>( <var class="Arg">A</var>, <var class="Arg">B</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>If all elements of the algebra <var class="Arg">B</var> happen to be contained in the algebra <var class="Arg">A</var>, then <var class="Arg">B</var> can be viewed as a subalgebra of <var class="Arg">A</var>. This function returns this subalgebra.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= FullMatrixAlgebra( Rationals, 2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= VectorSpace( Rationals, [ IdentityMat( 2 ) ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= AsAlgebra( Rationals, V );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BA:= AsSubalgebra( A, B );</span>
<algebra of dimension 1 over Rationals>
</pre></div>
<p><a id="X7B964BC37A975E48" name="X7B964BC37A975E48"></a></p>
<h5>62.9-10 AsSubalgebraWithOne</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsSubalgebraWithOne</code>( <var class="Arg">A</var>, <var class="Arg">B</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>If <var class="Arg">B</var> is an algebra with one, all elements of which happen to be contained in the algebra with one <var class="Arg">A</var>, then <var class="Arg">B</var> can be viewed as a subalgebra with one of <var class="Arg">A</var>. This function returns this subalgebra with one.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= FullMatrixAlgebra( Rationals, 2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= VectorSpace( Rationals, [ IdentityMat( 2 ) ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= AsAlgebra( Rationals, V );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:= AsAlgebraWithOne( Rationals, B );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">AC:= AsSubalgebraWithOne( A, C );</span>
<algebra-with-one over Rationals, with 1 generators>
</pre></div>
<p><a id="X7C280DAC7F840B60" name="X7C280DAC7F840B60"></a></p>
<h5>62.9-11 MutableBasisOfClosureUnderAction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MutableBasisOfClosureUnderAction</code>( <var class="Arg">F</var>, <var class="Arg">Agens</var>, <var class="Arg">from</var>, <var class="Arg">init</var>, <var class="Arg">opr</var>, <var class="Arg">zero</var>, <var class="Arg">maxdim</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">F</var> be a ring, <var class="Arg">Agens</var> a list of generators for an <var class="Arg">F</var>-algebra <span class="SimpleMath">A</span>, and <var class="Arg">from</var> one of <code class="code">"left"</code>, <code class="code">"right"</code>, <code class="code">"both"</code>; this means that elements of <span class="SimpleMath">A</span> act via multiplication from the respective side(s). <var class="Arg">init</var> must be a list of initial generating vectors, and <var class="Arg">opr</var> the operation (a function of two arguments).</p>
<p><code class="func">MutableBasisOfClosureUnderAction</code> returns a mutable basis of the <var class="Arg">F</var>-free left module generated by the vectors in <var class="Arg">init</var> and their images under the action of <var class="Arg">Agens</var> from the respective side(s).</p>
<p><var class="Arg">zero</var> is the zero element of the desired module. <var class="Arg">maxdim</var> is an upper bound for the dimension of the closure; if no such upper bound is known then the value of <var class="Arg">maxdim</var> must be <code class="func">infinity</code> (<a href="chap18.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>).</p>
<p><code class="func">MutableBasisOfClosureUnderAction</code> can be used to compute a basis of an <em>associative</em> algebra generated by the elements in <var class="Arg">Agens</var>. In this case <var class="Arg">from</var> may be <code class="code">"left"</code> or <code class="code">"right"</code>, <var class="Arg">opr</var> is the multiplication <code class="code">*</code>, and <var class="Arg">init</var> is a list containing either the identity of the algebra or a list of algebra generators. (Note that if the algebra has an identity then it is in general not sufficient to take algebra-with-one generators as <var class="Arg">init</var>, whereas of course <var class="Arg">Agens</var> need not contain the identity.)</p>
<p>(Note that bases of <em>not</em> necessarily associative algebras can be computed using <code class="func">MutableBasisOfNonassociativeAlgebra</code> (<a href="chap62.html#X7BA1739D7F8B3A2B"><span class="RefLink">62.9-12</span></a>).)</p>
<p>Other applications of <code class="func">MutableBasisOfClosureUnderAction</code> are the computations of bases for (left/ right/ two-sided) ideals <span class="SimpleMath">I</span> in an <em>associative</em> algebra <span class="SimpleMath">A</span> from ideal generators of <span class="SimpleMath">I</span>; in these cases <var class="Arg">Agens</var> is a list of algebra generators of <span class="SimpleMath">A</span>, <var class="Arg">from</var> denotes the appropriate side(s), <var class="Arg">init</var> is a list of ideal generators of <span class="SimpleMath">I</span>, and <var class="Arg">opr</var> is again <code class="code">*</code>.</p>
<p>(Note that bases of ideals in <em>not</em> necessarily associative algebras can be computed using <code class="func">MutableBasisOfIdealInNonassociativeAlgebra</code> (<a href="chap62.html#X8467B687823371F9"><span class="RefLink">62.9-13</span></a>).)</p>
<p>Finally, bases of right <span class="SimpleMath">A</span>-modules also can be computed using <code class="func">MutableBasisOfClosureUnderAction</code>. The only difference to the ideal case is that <var class="Arg">init</var> is now a list of right module generators, and <var class="Arg">opr</var> is the operation of the module.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= QuaternionAlgebra( Rationals );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= GeneratorsOfAlgebra( A );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= MutableBasisOfClosureUnderAction( Rationals, </span>
<span class="GAPprompt">></span> <span class="GAPinput"> g, "left", [ g[1] ], \*, Zero(A), 4 );</span>
<mutable basis over Rationals, 4 vectors>
<span class="GAPprompt">gap></span> <span class="GAPinput">BasisVectors( B );</span>
[ e, i, j, k ]
</pre></div>
<p><a id="X7BA1739D7F8B3A2B" name="X7BA1739D7F8B3A2B"></a></p>
<h5>62.9-12 MutableBasisOfNonassociativeAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MutableBasisOfNonassociativeAlgebra</code>( <var class="Arg">F</var>, <var class="Arg">Agens</var>, <var class="Arg">zero</var>, <var class="Arg">maxdim</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is a mutable basis of the (not necessarily associative) <var class="Arg">F</var>-algebra that is generated by <var class="Arg">Agens</var>, has zero element <var class="Arg">zero</var>, and has dimension at most <var class="Arg">maxdim</var>. If no finite bound for the dimension is known then <code class="func">infinity</code> (<a href="chap18.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>) must be the value of <var class="Arg">maxdim</var>.</p>
<p>The difference to <code class="func">MutableBasisOfClosureUnderAction</code> (<a href="chap62.html#X7C280DAC7F840B60"><span class="RefLink">62.9-11</span></a>) is that in general it is not sufficient to multiply just with algebra generators. (For special cases of nonassociative algebras, especially for Lie algebras, multiplying with algebra generators suffices.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:= FullMatrixLieAlgebra( Rationals, 4 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m1:= Random( L );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m2:= Random( L );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">MutableBasisOfNonassociativeAlgebra( Rationals, [ m1, m2 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput">Zero( L ), 16 );</span>
<mutable basis over Rationals, 16 vectors>
</pre></div>
<p><a id="X8467B687823371F9" name="X8467B687823371F9"></a></p>
<h5>62.9-13 MutableBasisOfIdealInNonassociativeAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MutableBasisOfIdealInNonassociativeAlgebra</code>( <var class="Arg">F</var>, <var class="Arg">Vgens</var>, <var class="Arg">Igens</var>, <var class="Arg">zero</var>, <var class="Arg">from</var>, <var class="Arg">maxdim</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is a mutable basis of the ideal generated by <var class="Arg">Igens</var> under the action of the (not necessarily associative) <var class="Arg">F</var>-algebra with vector space generators <var class="Arg">Vgens</var>. The zero element of the ideal is <var class="Arg">zero</var>, <var class="Arg">from</var> is one of <code class="code">"left"</code>, <code class="code">"right"</code>, <code class="code">"both"</code> (with the same meaning as in <code class="func">MutableBasisOfClosureUnderAction</code> (<a href="chap62.html#X7C280DAC7F840B60"><span class="RefLink">62.9-11</span></a>)), and <var class="Arg">maxdim</var> is a known upper bound on the dimension of the ideal; if no finite bound for the dimension is known then <code class="func">infinity</code> (<a href="chap18.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>) must be the value of <var class="Arg">maxdim</var>.</p>
<p>The difference to <code class="func">MutableBasisOfClosureUnderAction</code> (<a href="chap62.html#X7C280DAC7F840B60"><span class="RefLink">62.9-11</span></a>) is that in general it is not sufficient to multiply just with algebra generators. (For special cases of nonassociative algebras, especially for Lie algebras, multiplying with algebra generators suffices.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mats:= [ [[ 1, 0 ], [ 0, -1 ]], [[0,1],[0,0]] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Algebra( Rationals, mats );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">basA:= BasisVectors( Basis( A ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= MutableBasisOfIdealInNonassociativeAlgebra( Rationals, basA,</span>
<span class="GAPprompt">></span> <span class="GAPinput">[ mats[2] ], 0*mats[1], "both", infinity );</span>
<mutable basis over Rationals, 1 vectors>
<span class="GAPprompt">gap></span> <span class="GAPinput">BasisVectors( B );</span>
[ [ [ 0, 1 ], [ 0, 0 ] ] ]
</pre></div>
<p><a id="X7C591B7C7DEA7EEB" name="X7C591B7C7DEA7EEB"></a></p>
<h5>62.9-14 DirectSumOfAlgebras</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DirectSumOfAlgebras</code>( <var class="Arg">A1</var>, <var class="Arg">A2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DirectSumOfAlgebras</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>is the direct sum of the two algebras <var class="Arg">A1</var> and <var class="Arg">A2</var> respectively of the algebras in the list <var class="Arg">list</var>.</p>
<p>If all involved algebras are associative algebras then the result is also known to be associative. If all involved algebras are Lie algebras then the result is also known to be a Lie algebra.</p>
<p>All involved algebras must have the same left acting domain.</p>
<p>The default case is that the result is a structure constants algebra. If all involved algebras are matrix algebras, and either both are Lie algebras or both are associative then the result is again a matrix algebra of the appropriate type.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= QuaternionAlgebra( Rationals );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DirectSumOfAlgebras( [A, A, A] );</span>
<algebra of dimension 12 over Rationals>
</pre></div>
<p><a id="X7D0EB1437D3D9495" name="X7D0EB1437D3D9495"></a></p>
<h5>62.9-15 FullMatrixAlgebraCentralizer</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FullMatrixAlgebraCentralizer</code>( <var class="Arg">F</var>, <var class="Arg">lst</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">lst</var> be a nonempty list of square matrices of the same dimension <span class="SimpleMath">n</span>, say, with entries in the field <var class="Arg">F</var>. <code class="func">FullMatrixAlgebraCentralizer</code> returns the (pointwise) centralizer of all matrices in <var class="Arg">lst</var>, inside the full matrix algebra of <span class="SimpleMath">n × n</span> matrices over <var class="Arg">F</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= QuaternionAlgebra( Rationals );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">b:= Basis( A );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mats:= List( BasisVectors( b ), x -> AdjointMatrix( b, x ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">FullMatrixAlgebraCentralizer( Rationals, mats );</span>
<algebra-with-one of dimension 4 over Rationals>
</pre></div>
<p><a id="X850C29907A509533" name="X850C29907A509533"></a></p>
<h5>62.9-16 RadicalOfAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RadicalOfAlgebra</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is the maximal nilpotent ideal of <var class="Arg">A</var>, where <var class="Arg">A</var> is an associative algebra.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= [ [ 0, 1, 2 ], [ 0, 0, 3 ], [ 0, 0, 0 ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= AlgebraWithOneByGenerators( Rationals, [ m ] );</span>
<algebra-with-one over Rationals, with 1 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">RadicalOfAlgebra( A );</span>
<algebra of dimension 2 over Rationals>
</pre></div>
<p><a id="X82571785846CF05C" name="X82571785846CF05C"></a></p>
<h5>62.9-17 CentralIdempotentsOfAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CentralIdempotentsOfAlgebra</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For an associative algebra <var class="Arg">A</var>, this function returns a list of central primitive idempotents such that their sum is the identity element of <var class="Arg">A</var>. Therefore <var class="Arg">A</var> is required to have an identity.</p>
<p>(This is a synonym of <code class="code">CentralIdempotentsOfSemiring</code>.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= QuaternionAlgebra( Rationals );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= DirectSumOfAlgebras( [A, A, A] );</span>
<algebra of dimension 12 over Rationals>
<span class="GAPprompt">gap></span> <span class="GAPinput">CentralIdempotentsOfAlgebra( B );</span>
[ v.9, v.5, v.1 ]
</pre></div>
<p><a id="X7CFB230582C26DAA" name="X7CFB230582C26DAA"></a></p>
<h5>62.9-18 DirectSumDecomposition</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DirectSumDecomposition</code>( <var class="Arg">L</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This function calculates a list of ideals of the algebra <var class="Arg">L</var> such that <var class="Arg">L</var> is equal to their direct sum. Currently this is only implemented for semisimple associative algebras, and for Lie algebras (semisimple or not).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:= SymmetricGroup( 4 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= GroupRing( Rationals, G );</span>
<algebra-with-one over Rationals, with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">dd:= DirectSumDecomposition( A );</span>
[ <two-sided ideal in
<algebra-with-one of dimension 24 over Rationals>,
(1 generators)>,
<two-sided ideal in
<algebra-with-one of dimension 24 over Rationals>,
(1 generators)>,
<two-sided ideal in
<algebra-with-one of dimension 24 over Rationals>,
(1 generators)>,
<two-sided ideal in
<algebra-with-one of dimension 24 over Rationals>,
(1 generators)>,
<two-sided ideal in
<algebra-with-one of dimension 24 over Rationals>,
(1 generators)> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( dd, Dimension );</span>
[ 1, 1, 4, 9, 9 ]
</pre></div>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:= FullMatrixLieAlgebra( Rationals, 5 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DirectSumDecomposition( L );</span>
[ <two-sided ideal in
<two-sided ideal in <Lie algebra of dimension 25 over Rationals>
, (dimension 1)>, (dimension 1)>,
<two-sided ideal in
<two-sided ideal in <Lie algebra of dimension 25 over Rationals>
, (dimension 24)>, (dimension 24)> ]
</pre></div>
<p><a id="X85C58364833E014C" name="X85C58364833E014C"></a></p>
<h5>62.9-19 LeviMalcevDecomposition</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeviMalcevDecomposition</code>( <var class="Arg">L</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>A Levi-Malcev subalgebra of the algebra <var class="Arg">L</var> is a semisimple subalgebra complementary to the radical of <var class="Arg">L</var>. This function returns a list with two components. The first component is a Levi-Malcev subalgebra, the second the radical. This function is implemented for associative and Lie algebras.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= [ [ 1, 2, 0 ], [ 0, 1, 3 ], [ 0, 0, 1] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Algebra( Rationals, [ m ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">LeviMalcevDecomposition( A );</span>
[ <algebra of dimension 1 over Rationals>,
<algebra of dimension 2 over Rationals> ]
</pre></div>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:= FullMatrixLieAlgebra( Rationals, 5 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">LeviMalcevDecomposition( L );</span>
[ <Lie algebra of dimension 24 over Rationals>,
<two-sided ideal in <Lie algebra of dimension 25 over Rationals>,
(dimension 1)> ]
</pre></div>
<p><a id="X7DCA2568870A2D34" name="X7DCA2568870A2D34"></a></p>
<h5>62.9-20 Grading</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Grading</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Let <span class="SimpleMath">G</span> be an Abelian group and <span class="SimpleMath">A</span> an algebra. Then <span class="SimpleMath">A</span> is said to be graded over <span class="SimpleMath">G</span> if for every <span class="SimpleMath">g ∈ G</span> there is a subspace <span class="SimpleMath">A_g</span> of <span class="SimpleMath">A</span> such that <span class="SimpleMath">A_g ⋅ A_h ⊂ A_{g+h}</span> for <span class="SimpleMath">g, h ∈ G</span>. In <strong class="pkg">GAP</strong> 4 a <em>grading</em> of an algebra is a record containing the following components.</p>
<dl>
<dt><strong class="Mark"><code class="code">source</code></strong></dt>
<dd><p>the Abelian group over which the algebra is graded.</p>
</dd>
<dt><strong class="Mark"><code class="code">hom_components</code></strong></dt>
<dd><p>a function assigning to each element from the source a subspace of the algebra.</p>
</dd>
<dt><strong class="Mark"><code class="code">min_degree</code></strong></dt>
<dd><p>in the case where the algebra is graded over the integers this is the minimum number for which <code class="code">hom_components</code> returns a nonzero subspace.</p>
</dd>
<dt><strong class="Mark"><code class="code">max_degree</code></strong></dt>
<dd><p>is analogous to <code class="code">min_degree</code>.</p>
</dd>
</dl>
<p>We note that there are no methods to compute a grading of an arbitrary algebra; however some algebras get a natural grading when they are constructed (see <code class="func">JenningsLieAlgebra</code> (<a href="chap64.html#X8692ADD581359CA1"><span class="RefLink">64.8-4</span></a>), <code class="func">NilpotentQuotientOfFpLieAlgebra</code> (<a href="chap64.html#X79FD70C487EA9438"><span class="RefLink">64.11-2</span></a>)).</p>
<p>We note also that these components may be not enough to handle the grading efficiently, and another record component may be needed. For instance in a Lie algebra <span class="SimpleMath">L</span> constructed by <code class="func">JenningsLieAlgebra</code> (<a href="chap64.html#X8692ADD581359CA1"><span class="RefLink">64.8-4</span></a>), the length of the of the range <code class="code">[ Grading(L)!.min_degree .. Grading(L)!.max_degree ]</code> may be non-polynomial in the dimension of <span class="SimpleMath">L</span>. To handle efficiently this situation, an optional component can be used:</p>
<dl>
<dt><strong class="Mark"><code class="code">non_zero_hom_components</code></strong></dt>
<dd><p>the subset of <code class="code">source</code> for which <code class="code">hom_components</code> returns a nonzero subspace.</p>
</dd>
</dl>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:= SmallGroup(3^6, 100 );</span>
<pc group of size 729 with 6 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:= JenningsLieAlgebra( G );</span>
<Lie algebra of dimension 6 over GF(3)>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Grading( L );</span>
rec( hom_components := function( d ) ... end, max_degree := 9,
min_degree := 1, source := Integers )
<span class="GAPprompt">gap></span> <span class="GAPinput">g.hom_components( 3 );</span>
<vector space over GF(3), with 1 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">g.hom_components( 14 );</span>
<vector space over GF(3), with 0 generators>
</pre></div>
<p><a id="X7E94B857847F95C1" name="X7E94B857847F95C1"></a></p>
<h4>62.10 <span class="Heading">Homomorphisms of Algebras</span></h4>
<p>Algebra homomorphisms are vector space homomorphisms that preserve the multiplication. So the default methods for vector space homomorphisms work, and in fact there is not much use of the fact that source and range are algebras, except that preimages and images are algebras (or even ideals) in certain cases.</p>
<p><a id="X83CE798C7D39E368" name="X83CE798C7D39E368"></a></p>
<h5>62.10-1 AlgebraGeneralMappingByImages</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlgebraGeneralMappingByImages</code>( <var class="Arg">A</var>, <var class="Arg">B</var>, <var class="Arg">gens</var>, <var class="Arg">imgs</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>is a general mapping from the <span class="SimpleMath">F</span>-algebra <var class="Arg">A</var> to the <span class="SimpleMath">F</span>-algebra <var class="Arg">B</var>. This general mapping is defined by mapping the entries in the list <var class="Arg">gens</var> (elements of <var class="Arg">A</var>) to the entries in the list <var class="Arg">imgs</var> (elements of <var class="Arg">B</var>), and taking the <span class="SimpleMath">F</span>-linear and multiplicative closure.</p>
<p><var class="Arg">gens</var> need not generate <var class="Arg">A</var> as an <span class="SimpleMath">F</span>-algebra, and if the specification does not define a linear and multiplicative mapping then the result will be multivalued. Hence, in general it is not a mapping. For constructing a linear map that is not necessarily multiplicative, we refer to <code class="func">LeftModuleHomomorphismByImages</code> (<a href="chap61.html#X85F5293983E47B5A"><span class="RefLink">61.10-2</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= QuaternionAlgebra( Rationals );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= FullMatrixAlgebra( Rationals, 2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">bA:= BasisVectors( Basis( A ) );; bB:= BasisVectors( Basis( B ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= AlgebraGeneralMappingByImages( A, B, bA, bB );</span>
[ e, i, j, k ] -> [ [ [ 1, 0 ], [ 0, 0 ] ], [ [ 0, 1 ], [ 0, 0 ] ],
[ [ 0, 0 ], [ 1, 0 ] ], [ [ 0, 0 ], [ 0, 1 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Images( f, bA[1] );</span>
<add. coset of <algebra over Rationals, with 16 generators>>
</pre></div>
<p><a id="X7A7F97ED8608C882" name="X7A7F97ED8608C882"></a></p>
<h5>62.10-2 AlgebraHomomorphismByImages</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlgebraHomomorphismByImages</code>( <var class="Arg">A</var>, <var class="Arg">B</var>, <var class="Arg">gens</var>, <var class="Arg">imgs</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">AlgebraHomomorphismByImages</code> returns the algebra homomorphism with source <var class="Arg">A</var> and range <var class="Arg">B</var> that is defined by mapping the list <var class="Arg">gens</var> of generators of <var class="Arg">A</var> to the list <var class="Arg">imgs</var> of images in <var class="Arg">B</var>.</p>
<p>If <var class="Arg">gens</var> does not generate <var class="Arg">A</var> or if the homomorphism does not exist (i.e., if mapping the generators describes only a multi-valued mapping) then <code class="keyw">fail</code> is returned.</p>
<p>One can avoid the checks by calling <code class="func">AlgebraHomomorphismByImagesNC</code> (<a href="chap62.html#X8326D1BD79725462"><span class="RefLink">62.10-3</span></a>), and one can construct multi-valued mappings with <code class="func">AlgebraGeneralMappingByImages</code> (<a href="chap62.html#X83CE798C7D39E368"><span class="RefLink">62.10-1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">T:= EmptySCTable( 2, 0 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( T, 1, 1, [1,1] ); SetEntrySCTable( T, 2, 2, [1,2] );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= AlgebraByStructureConstants( Rationals, T );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m1:= NullMat( 2, 2 );; m1[1][1]:= 1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m2:= NullMat( 2, 2 );; m2[2][2]:= 1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= AlgebraByGenerators( Rationals, [ m1, m2 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">bA:= BasisVectors( Basis( A ) );; bB:= BasisVectors( Basis( B ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= AlgebraHomomorphismByImages( A, B, bA, bB );</span>
[ v.1, v.2 ] -> [ [ [ 1, 0 ], [ 0, 0 ] ], [ [ 0, 0 ], [ 0, 1 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Image( f, bA[1]+bA[2] );</span>
[ [ 1, 0 ], [ 0, 1 ] ]
</pre></div>
<p><a id="X8326D1BD79725462" name="X8326D1BD79725462"></a></p>
<h5>62.10-3 AlgebraHomomorphismByImagesNC</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlgebraHomomorphismByImagesNC</code>( <var class="Arg">A</var>, <var class="Arg">B</var>, <var class="Arg">gens</var>, <var class="Arg">imgs</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">AlgebraHomomorphismByImagesNC</code> is the operation that is called by the function <code class="func">AlgebraHomomorphismByImages</code> (<a href="chap62.html#X7A7F97ED8608C882"><span class="RefLink">62.10-2</span></a>). Its methods may assume that <var class="Arg">gens</var> generates <var class="Arg">A</var> and that the mapping of <var class="Arg">gens</var> to <var class="Arg">imgs</var> defines an algebra homomorphism. Results are unpredictable if these conditions do not hold.</p>
<p>For creating a possibly multi-valued mapping from <var class="Arg">A</var> to <var class="Arg">B</var> that respects addition, multiplication, and scalar multiplication, <code class="func">AlgebraGeneralMappingByImages</code> (<a href="chap62.html#X83CE798C7D39E368"><span class="RefLink">62.10-1</span></a>) can be used.</p>
<p>For the definitions of the algebras <code class="code">A</code> and <code class="code">B</code> in the next example we refer to the previous example.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= AlgebraHomomorphismByImagesNC( A, B, bA, bB );</span>
[ v.1, v.2 ] -> [ [ [ 1, 0 ], [ 0, 0 ] ], [ [ 0, 0 ], [ 0, 1 ] ] ]
</pre></div>
<p><a id="X8057E55B864567AD" name="X8057E55B864567AD"></a></p>
<h5>62.10-4 AlgebraWithOneGeneralMappingByImages</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlgebraWithOneGeneralMappingByImages</code>( <var class="Arg">A</var>, <var class="Arg">B</var>, <var class="Arg">gens</var>, <var class="Arg">imgs</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This function is analogous to <code class="func">AlgebraGeneralMappingByImages</code> (<a href="chap62.html#X83CE798C7D39E368"><span class="RefLink">62.10-1</span></a>); the only difference being that the identity of <var class="Arg">A</var> is automatically mapped to the identity of <var class="Arg">B</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= QuaternionAlgebra( Rationals );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= FullMatrixAlgebra( Rationals, 2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">bA:= BasisVectors( Basis( A ) );; bB:= BasisVectors( Basis( B ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:=AlgebraWithOneGeneralMappingByImages(A,B,bA{[2,3,4]},bB{[1,2,3]});</span>
[ i, j, k, e ] -> [ [ [ 1, 0 ], [ 0, 0 ] ], [ [ 0, 1 ], [ 0, 0 ] ],
[ [ 0, 0 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ]
</pre></div>
<p><a id="X866F32B5846E5857" name="X866F32B5846E5857"></a></p>
<h5>62.10-5 AlgebraWithOneHomomorphismByImages</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlgebraWithOneHomomorphismByImages</code>( <var class="Arg">A</var>, <var class="Arg">B</var>, <var class="Arg">gens</var>, <var class="Arg">imgs</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">AlgebraWithOneHomomorphismByImages</code> returns the algebra-with-one homomorphism with source <var class="Arg">A</var> and range <var class="Arg">B</var> that is defined by mapping the list <var class="Arg">gens</var> of generators of <var class="Arg">A</var> to the list <var class="Arg">imgs</var> of images in <var class="Arg">B</var>.</p>
<p>The difference between an algebra homomorphism and an algebra-with-one homomorphism is that in the latter case, it is assumed that the identity of <var class="Arg">A</var> is mapped to the identity of <var class="Arg">B</var>, and therefore <var class="Arg">gens</var> needs to generate <var class="Arg">A</var> only as an algebra-with-one.</p>
<p>If <var class="Arg">gens</var> does not generate <var class="Arg">A</var> or if the homomorphism does not exist (i.e., if mapping the generators describes only a multi-valued mapping) then <code class="keyw">fail</code> is returned.</p>
<p>One can avoid the checks by calling <code class="func">AlgebraWithOneHomomorphismByImagesNC</code> (<a href="chap62.html#X80BF4D6A7FDC959A"><span class="RefLink">62.10-6</span></a>), and one can construct multi-valued mappings with <code class="func">AlgebraWithOneGeneralMappingByImages</code> (<a href="chap62.html#X8057E55B864567AD"><span class="RefLink">62.10-4</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m1:= NullMat( 2, 2 );; m1[1][1]:=1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m2:= NullMat( 2, 2 );; m2[2][2]:=1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= AlgebraByGenerators( Rationals, [m1,m2] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">T:= EmptySCTable( 2, 0 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( T, 1, 1, [1,1] );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( T, 2, 2, [1,2] );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= AlgebraByStructureConstants(Rationals, T);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">bA:= BasisVectors( Basis( A ) );; bB:= BasisVectors( Basis( B ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= AlgebraWithOneHomomorphismByImages( A, B, bA{[1]}, bB{[1]} );</span>
[ [ [ 1, 0 ], [ 0, 0 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ] -> [ v.1, v.1+v.2 ]
</pre></div>
<p><a id="X80BF4D6A7FDC959A" name="X80BF4D6A7FDC959A"></a></p>
<h5>62.10-6 AlgebraWithOneHomomorphismByImagesNC</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlgebraWithOneHomomorphismByImagesNC</code>( <var class="Arg">A</var>, <var class="Arg">B</var>, <var class="Arg">gens</var>, <var class="Arg">imgs</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">AlgebraWithOneHomomorphismByImagesNC</code> is the operation that is called by the function <code class="func">AlgebraWithOneHomomorphismByImages</code> (<a href="chap62.html#X866F32B5846E5857"><span class="RefLink">62.10-5</span></a>). Its methods may assume that <var class="Arg">gens</var> generates <var class="Arg">A</var> and that the mapping of <var class="Arg">gens</var> to <var class="Arg">imgs</var> defines an algebra-with-one homomorphism. Results are unpredictable if these conditions do not hold.</p>
<p>For creating a possibly multi-valued mapping from <var class="Arg">A</var> to <var class="Arg">B</var> that respects addition, multiplication, identity, and scalar multiplication, <code class="func">AlgebraWithOneGeneralMappingByImages</code> (<a href="chap62.html#X8057E55B864567AD"><span class="RefLink">62.10-4</span></a>) can be used.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m1:= NullMat( 2, 2 );; m1[1][1]:=1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m2:= NullMat( 2, 2 );; m2[2][2]:=1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= AlgebraByGenerators( Rationals, [m1,m2] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">T:= EmptySCTable( 2, 0 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( T, 1, 1, [1,1] );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( T, 2, 2, [1,2] );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= AlgebraByStructureConstants( Rationals, T);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">bA:= BasisVectors( Basis( A ) );; bB:= BasisVectors( Basis( B ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= AlgebraWithOneHomomorphismByImagesNC( A, B, bA{[1]}, bB{[1]} );</span>
[ [ [ 1, 0 ], [ 0, 0 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ] -> [ v.1, v.1+v.2 ]
</pre></div>
<p><a id="X8712E5C1861CC32C" name="X8712E5C1861CC32C"></a></p>
<h5>62.10-7 NaturalHomomorphismByIdeal</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalHomomorphismByIdeal</code>( <var class="Arg">A</var>, <var class="Arg">I</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>For an algebra <var class="Arg">A</var> and an ideal <var class="Arg">I</var> in <var class="Arg">A</var>, the return value of <code class="func">NaturalHomomorphismByIdeal</code> (<a href="chap56.html#X83D53D98809EC461"><span class="RefLink">56.8-4</span></a>) is a homomorphism of algebras, in particular the range of this mapping is also an algebra.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:= FullMatrixLieAlgebra( Rationals, 3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:= LieCentre( L );</span>
<two-sided ideal in <Lie algebra of dimension 9 over Rationals>,
(dimension 1)>
<span class="GAPprompt">gap></span> <span class="GAPinput">hom:= NaturalHomomorphismByIdeal( L, C );</span>
<linear mapping by matrix, <Lie algebra of dimension
9 over Rationals> -> <Lie algebra of dimension 8 over Rationals>>
<span class="GAPprompt">gap></span> <span class="GAPinput">ImagesSource( hom );</span>
<Lie algebra of dimension 8 over Rationals>
</pre></div>
<p><a id="X8705A9C68102FEA3" name="X8705A9C68102FEA3"></a></p>
<h5>62.10-8 OperationAlgebraHomomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OperationAlgebraHomomorphism</code>( <var class="Arg">A</var>, <var class="Arg">B</var>[, <var class="Arg">opr</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OperationAlgebraHomomorphism</code>( <var class="Arg">A</var>, <var class="Arg">V</var>[, <var class="Arg">opr</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">OperationAlgebraHomomorphism</code> returns an algebra homomorphism from the <span class="SimpleMath">F</span>-algebra <var class="Arg">A</var> into a matrix algebra over <span class="SimpleMath">F</span> that describes the <span class="SimpleMath">F</span>-linear action of <var class="Arg">A</var> on the basis <var class="Arg">B</var> of a free left module respectively on the free left module <var class="Arg">V</var> (in which case some basis of <var class="Arg">V</var> is chosen), via the operation <var class="Arg">opr</var>.</p>
<p>The homomorphism need not be surjective. The default value for <var class="Arg">opr</var> is <code class="func">OnRight</code> (<a href="chap41.html#X7960924D84B5B18F"><span class="RefLink">41.2-2</span></a>).</p>
<p>If <var class="Arg">A</var> is an algebra-with-one then the operation homomorphism is an algebra-with-one homomorphism because the identity of <var class="Arg">A</var> must act as the identity.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m1:= NullMat( 2, 2 );; m1[1][1]:= 1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m2:= NullMat( 2, 2 );; m2[2][2]:= 1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= AlgebraByGenerators( Rationals, [ m1, m2 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= FullRowSpace( Rationals, 2 );</span>
( Rationals^2 )
<span class="GAPprompt">gap></span> <span class="GAPinput">f:=OperationAlgebraHomomorphism( B, Basis( V ), OnRight );</span>
<op. hom. Algebra( Rationals,
[ [ [ 1, 0 ], [ 0, 0 ] ], [ [ 0, 0 ], [ 0, 1 ] ]
] ) -> matrices of dim. 2>
<span class="GAPprompt">gap></span> <span class="GAPinput">Image( f, m1 );</span>
[ [ 1, 0 ], [ 0, 0 ] ]
</pre></div>
<p><a id="X7B249E8E86D895F0" name="X7B249E8E86D895F0"></a></p>
<h5>62.10-9 NiceAlgebraMonomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NiceAlgebraMonomorphism</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If <var class="Arg">A</var> is an associative algebra with one, returns an isomorphism from <var class="Arg">A</var> onto a matrix algebra (see <code class="func">IsomorphismMatrixAlgebra</code> (<a href="chap62.html#X7FB760F9813B0789"><span class="RefLink">62.10-11</span></a>) for an example). If <var class="Arg">A</var> is a finitely presented Lie algebra, returns an isomorphism from <var class="Arg">A</var> onto a Lie algebra defined by a structure constants table (see <a href="chap64.html#X7B8C71E07F50B286"><span class="RefLink">64.11</span></a> for an example).</p>
<p><a id="X79D770777D873F80" name="X79D770777D873F80"></a></p>
<h5>62.10-10 IsomorphismFpAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismFpAlgebra</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>isomorphism from the algebra <var class="Arg">A</var> onto a finitely presented algebra. Currently this is only implemented for associative algebras with one.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= QuaternionAlgebra( Rationals );</span>
<algebra-with-one of dimension 4 over Rationals>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= IsomorphismFpAlgebra( A );</span>
[ e, i, j, k, e ] -> [ [(1)*x.1], [(1)*x.2], [(1)*x.3], [(1)*x.4],
[(1)*<identity ...>] ]
</pre></div>
<p><a id="X7FB760F9813B0789" name="X7FB760F9813B0789"></a></p>
<h5>62.10-11 IsomorphismMatrixAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismMatrixAlgebra</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>isomorphism from the algebra <var class="Arg">A</var> onto a matrix algebra. Currently this is only implemented for associative algebras with one.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">T:= EmptySCTable( 2, 0 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( T, 1, 1, [1,1] ); SetEntrySCTable( T, 2, 2, [1,2] );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= AlgebraByStructureConstants( Rationals, T );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= AsAlgebraWithOne( Rationals, A );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:=IsomorphismMatrixAlgebra( A );</span>
<op. hom. AlgebraWithOne( Rationals, ... ) -> matrices of dim. 2>
<span class="GAPprompt">gap></span> <span class="GAPinput">Image( f, BasisVectors( Basis( A ) )[1] );</span>
[ [ 1, 0 ], [ 0, 0 ] ]
</pre></div>
<p><a id="X7F8D3DF2863EC50D" name="X7F8D3DF2863EC50D"></a></p>
<h5>62.10-12 IsomorphismSCAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismSCAlgebra</code>( <var class="Arg">B</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismSCAlgebra</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a basis <var class="Arg">B</var> of an algebra <span class="SimpleMath">A</span>, say, <code class="func">IsomorphismSCAlgebra</code> returns an algebra isomorphism from <span class="SimpleMath">A</span> to an algebra <span class="SimpleMath">S</span> given by structure constants (see <a href="chap62.html#X7E8F45547CC07CE5"><span class="RefLink">62.4</span></a>), such that the canonical basis of <span class="SimpleMath">S</span> is the image of <var class="Arg">B</var>.</p>
<p>For an algebra <var class="Arg">A</var>, <code class="func">IsomorphismSCAlgebra</code> chooses a basis of <var class="Arg">A</var> and returns the <code class="func">IsomorphismSCAlgebra</code> value for that basis.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismSCAlgebra( GF(8) );</span>
CanonicalBasis( GF(2^3) ) -> CanonicalBasis( <algebra of dimension
3 over GF(2)> )
<span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismSCAlgebra( GF(2)^[2,2] );</span>
CanonicalBasis( ( GF(2)^
[ 2, 2 ] ) ) -> CanonicalBasis( <algebra of dimension 4 over GF(2)> )
</pre></div>
<p><a id="X7F34244B81979696" name="X7F34244B81979696"></a></p>
<h5>62.10-13 RepresentativeLinearOperation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepresentativeLinearOperation</code>( <var class="Arg">A</var>, <var class="Arg">v</var>, <var class="Arg">w</var>, <var class="Arg">opr</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>is an element of the algebra <var class="Arg">A</var> that maps the vector <var class="Arg">v</var> to the vector <var class="Arg">w</var> under the linear operation described by the function <var class="Arg">opr</var>. If no such element exists then <code class="keyw">fail</code> is returned.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m1:= NullMat( 2, 2 );; m1[1][1]:= 1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m2:= NullMat( 2, 2 );; m2[2][2]:= 1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= AlgebraByGenerators( Rationals, [ m1, m2 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativeLinearOperation( B, [1,0], [1,0], OnRight );</span>
[ [ 1, 0 ], [ 0, 0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativeLinearOperation( B, [1,0], [0,1], OnRight );</span>
fail
</pre></div>
<p><a id="X818DE6C57D1A4B33" name="X818DE6C57D1A4B33"></a></p>
<h4>62.11 <span class="Heading">Representations of Algebras</span></h4>
<p>An algebra module is a vector space together with an action of an algebra. So a module over an algebra is constructed by giving generators of a vector space, and a function for calculating the action of algebra elements on elements of the vector space. When creating an algebra module, the generators of the vector space are wrapped up and given the category <code class="code">IsLeftAlgebraModuleElement</code> or <code class="code">IsRightModuleElement</code> if the algebra acts from the left, or right respectively. (So in the case of a bi-module the elements get both categories.) Most linear algebra computations are delegated to the original vector space.</p>
<p>The transition between the original vector space and the corresponding algebra module is handled by <code class="code">ExtRepOfObj</code> and <code class="code">ObjByExtRep</code>. For an element <code class="code">v</code> of the algebra module, <code class="code">ExtRepOfObj( v )</code> returns the underlying element of the original vector space. Furthermore, if <code class="code">vec</code> is an element of the original vector space, and <code class="code">fam</code> the elements family of the corresponding algebra module, then <code class="code">ObjByExtRep( fam, vec )</code> returns the corresponding element of the algebra module. Below is an example of this.</p>
<p>The action of the algebra on elements of the algebra module is constructed by using the operator <code class="code">^</code>. If <code class="code">x</code> is an element of an algebra <code class="code">A</code>, and <code class="code">v</code> an element of a left <code class="code">A</code>-module, then <code class="code">x^v</code> calculates the result of the action of <code class="code">x</code> on <code class="code">v</code>. Similarly, if <code class="code">v</code> is an element of a right <code class="code">A</code>-module, then <code class="code">v^x</code> calculates the action of <code class="code">x</code> on <code class="code">v</code>.</p>
<p><a id="X8055B87F7ADBD66B" name="X8055B87F7ADBD66B"></a></p>
<h5>62.11-1 LeftAlgebraModuleByGenerators</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftAlgebraModuleByGenerators</code>( <var class="Arg">A</var>, <var class="Arg">op</var>, <var class="Arg">gens</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Constructs the left algebra module over <var class="Arg">A</var> generated by the list of vectors <var class="Arg">gens</var>. The action of <var class="Arg">A</var> is described by the function <var class="Arg">op</var>. This must be a function of two arguments; the first argument is the algebra element, and the second argument is a vector; it outputs the result of applying the algebra element to the vector.</p>
<p><a id="X8026B99B7955A355" name="X8026B99B7955A355"></a></p>
<h5>62.11-2 RightAlgebraModuleByGenerators</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightAlgebraModuleByGenerators</code>( <var class="Arg">A</var>, <var class="Arg">op</var>, <var class="Arg">gens</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Constructs the right algebra module over <var class="Arg">A</var> generated by the list of vectors <var class="Arg">gens</var>. The action of <var class="Arg">A</var> is described by the function <var class="Arg">op</var>. This must be a function of two arguments; the first argument is a vector, and the second argument is the algebra element; it outputs the result of applying the algebra element to the vector.</p>
<p><a id="X7F28A47E876427E0" name="X7F28A47E876427E0"></a></p>
<h5>62.11-3 BiAlgebraModuleByGenerators</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BiAlgebraModuleByGenerators</code>( <var class="Arg">A</var>, <var class="Arg">B</var>, <var class="Arg">opl</var>, <var class="Arg">opr</var>, <var class="Arg">gens</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Constructs the algebra bi-module over <var class="Arg">A</var> and <var class="Arg">B</var> generated by the list of vectors <var class="Arg">gens</var>. The left action of <var class="Arg">A</var> is described by the function <var class="Arg">opl</var>, and the right action of <var class="Arg">B</var> by the function <var class="Arg">opr</var>. <var class="Arg">opl</var> must be a function of two arguments; the first argument is the algebra element, and the second argument is a vector; it outputs the result of applying the algebra element on the left to the vector. <var class="Arg">opr</var> must be a function of two arguments; the first argument is a vector, and the second argument is the algebra element; it outputs the result of applying the algebra element on the right to the vector.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Rationals^[3,3];</span>
( Rationals^[ 3, 3 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= LeftAlgebraModuleByGenerators( A, \*, [ [ 1, 0, 0 ] ] );</span>
<left-module over ( Rationals^[ 3, 3 ] )>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:= RightAlgebraModuleByGenerators( A, \*, [ [ 1, 0, 0 ] ] );</span>
<right-module over ( Rationals^[ 3, 3 ] )>
<span class="GAPprompt">gap></span> <span class="GAPinput">M:= BiAlgebraModuleByGenerators( A, A, \*, \*, [ [ 1, 0, 0 ] ] );</span>
<bi-module over ( Rationals^[ 3, 3 ] ) (left) and ( Rationals^
[ 3, 3 ] ) (right)>
</pre></div>
<p>In the above examples, the modules <code class="code">V</code>, <code class="code">W</code>, and <code class="code">M</code> are <span class="SimpleMath">3</span>-dimensional vector spaces over the rationals. The algebra <code class="code">A</code> acts from the left on <code class="code">V</code>, from the right on <code class="code">W</code>, and from the left and from the right on <code class="code">M</code>.</p>
<p><a id="X852524F581613359" name="X852524F581613359"></a></p>
<h5>62.11-4 LeftAlgebraModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftAlgebraModule</code>( <var class="Arg">A</var>, <var class="Arg">op</var>, <var class="Arg">V</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Constructs the left algebra module over <var class="Arg">A</var> with underlying space <var class="Arg">V</var>. The action of <var class="Arg">A</var> is described by the function <var class="Arg">op</var>. This must be a function of two arguments; the first argument is the algebra element, and the second argument is a vector from <var class="Arg">V</var>; it outputs the result of applying the algebra element to the vector.</p>
<p><a id="X8222F2B67D753036" name="X8222F2B67D753036"></a></p>
<h5>62.11-5 RightAlgebraModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightAlgebraModule</code>( <var class="Arg">A</var>, <var class="Arg">op</var>, <var class="Arg">V</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Constructs the right algebra module over <var class="Arg">A</var> with underlying space <var class="Arg">V</var>. The action of <var class="Arg">A</var> is described by the function <var class="Arg">op</var>. This must be a function of two arguments; the first argument is a vector, from <var class="Arg">V</var> and the second argument is the algebra element; it outputs the result of applying the algebra element to the vector.</p>
<p><a id="X84517770868DDA02" name="X84517770868DDA02"></a></p>
<h5>62.11-6 BiAlgebraModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BiAlgebraModule</code>( <var class="Arg">A</var>, <var class="Arg">B</var>, <var class="Arg">opl</var>, <var class="Arg">opr</var>, <var class="Arg">V</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Constructs the algebra bi-module over <var class="Arg">A</var> and <var class="Arg">B</var> with underlying space <var class="Arg">V</var>. The left action of <var class="Arg">A</var> is described by the function <var class="Arg">opl</var>, and the right action of <var class="Arg">B</var> by the function <var class="Arg">opr</var>. <var class="Arg">opl</var> must be a function of two arguments; the first argument is the algebra element, and the second argument is a vector from <var class="Arg">V</var>; it outputs the result of applying the algebra element on the left to the vector. <var class="Arg">opr</var> must be a function of two arguments; the first argument is a vector from <var class="Arg">V</var>, and the second argument is the algebra element; it outputs the result of applying the algebra element on the right to the vector.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Rationals^[3,3];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= Rationals^3;</span>
( Rationals^3 )
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= Rationals^3;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M:= BiAlgebraModule( A, A, \*, \*, V );</span>
<bi-module over ( Rationals^[ 3, 3 ] ) (left) and ( Rationals^
[ 3, 3 ] ) (right)>
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( M );</span>
3
</pre></div>
<p><a id="X79AAB50D83A14A43" name="X79AAB50D83A14A43"></a></p>
<h5>62.11-7 GeneratorsOfAlgebraModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneratorsOfAlgebraModule</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>A list of elements of <var class="Arg">M</var> that generate <var class="Arg">M</var> as an algebra module.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Rationals^[3,3];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= LeftAlgebraModuleByGenerators( A, \*, [ [ 1, 0, 0 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfAlgebraModule( V );</span>
[ [ 1, 0, 0 ] ]
</pre></div>
<p><a id="X82B708BD84F3DAB1" name="X82B708BD84F3DAB1"></a></p>
<h5>62.11-8 IsAlgebraModuleElement</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAlgebraModuleElement</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAlgebraModuleElementCollection</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAlgebraModuleElementFamily</code>( <var class="Arg">fam</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Category of algebra module elements. If an object has <code class="code">IsAlgebraModuleElementCollection</code>, then it is an algebra module. If a family has <code class="code">IsAlgebraModuleElementFamily</code>, then it is a family of algebra module elements (every algebra module has its own elements family).</p>
<p><a id="X80E786467F9163F9" name="X80E786467F9163F9"></a></p>
<h5>62.11-9 IsLeftAlgebraModuleElement</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLeftAlgebraModuleElement</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLeftAlgebraModuleElementCollection</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Category of left algebra module elements. If an object has <code class="code">IsLeftAlgebraModuleElementCollection</code>, then it is a left-algebra module.</p>
<p><a id="X863756787E2B6E75" name="X863756787E2B6E75"></a></p>
<h5>62.11-10 IsRightAlgebraModuleElement</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRightAlgebraModuleElement</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRightAlgebraModuleElementCollection</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Category of right algebra module elements. If an object has <code class="code">IsRightAlgebraModuleElementCollection</code>, then it is a right-algebra module.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Rationals^[3,3];</span>
( Rationals^[ 3, 3 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">M:= BiAlgebraModuleByGenerators( A, A, \*, \*, [ [ 1, 0, 0 ] ] );</span>
<bi-module over ( Rationals^[ 3, 3 ] ) (left) and ( Rationals^
[ 3, 3 ] ) (right)>
<span class="GAPprompt">gap></span> <span class="GAPinput">vv:= BasisVectors( Basis( M ) );</span>
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsLeftAlgebraModuleElement( vv[1] );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRightAlgebraModuleElement( vv[1] );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">vv[1] = [ 1, 0, 0 ];</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">ExtRepOfObj( vv[1] ) = [ 1, 0, 0 ];</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">ObjByExtRep( ElementsFamily( FamilyObj( M ) ), [ 1, 0, 0 ] ) in M;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">xx:= BasisVectors( Basis( A ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">xx[4]^vv[1]; # left action</span>
[ 0, 1, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">vv[1]^xx[2]; # right action</span>
[ 0, 1, 0 ]
</pre></div>
<p><a id="X85654EF07F708AC3" name="X85654EF07F708AC3"></a></p>
<h5>62.11-11 LeftActingAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftActingAlgebra</code>( <var class="Arg">V</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Here <var class="Arg">V</var> is a left-algebra module; this function returns the algebra that acts from the left on <var class="Arg">V</var>.</p>
<p><a id="X826298B37E1B1520" name="X826298B37E1B1520"></a></p>
<h5>62.11-12 RightActingAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightActingAlgebra</code>( <var class="Arg">V</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Here <var class="Arg">V</var> is a right-algebra module; this function returns the algebra that acts from the right on <var class="Arg">V</var>.</p>
<p><a id="X8308408D86CFC3C9" name="X8308408D86CFC3C9"></a></p>
<h5>62.11-13 ActingAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ActingAlgebra</code>( <var class="Arg">V</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">V</var> is an algebra module; this function returns the algebra that acts on <var class="Arg">V</var> (this is the same as <code class="code">LeftActingAlgebra( <var class="Arg">V</var> )</code> if <var class="Arg">V</var> is a left module, and <code class="code">RightActingAlgebra( <var class="Arg">V</var> )</code> if <var class="Arg">V</var> is a right module; it will signal an error if <var class="Arg">V</var> is a bi-module).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Rationals^[3,3];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M:= BiAlgebraModuleByGenerators( A, A, \*, \*, [ [ 1, 0, 0 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">LeftActingAlgebra( M );</span>
( Rationals^[ 3, 3 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">RightActingAlgebra( M );</span>
( Rationals^[ 3, 3 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= RightAlgebraModuleByGenerators( A, \*, [ [ 1, 0, 0 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ActingAlgebra( V );</span>
( Rationals^[ 3, 3 ] )
</pre></div>
<p><a id="X7C325A507EC9BA18" name="X7C325A507EC9BA18"></a></p>
<h5>62.11-14 IsBasisOfAlgebraModuleElementSpace</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBasisOfAlgebraModuleElementSpace</code>( <var class="Arg">B</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>If a basis <var class="Arg">B</var> lies in the category <code class="code">IsBasisOfAlgebraModuleElementSpace</code>, then <var class="Arg">B</var> is a basis of a subspace of an algebra module. This means that <var class="Arg">B</var> has the record field <code class="code"><var class="Arg">B</var>!.delegateBasis</code> set. This last object is a basis of the corresponding subspace of the vector space underlying the algebra module (i.e., the vector space spanned by all <code class="code">ExtRepOfObj( v )</code> for <code class="code">v</code> in the algebra module).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Rationals^[3,3];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M:= BiAlgebraModuleByGenerators( A, A, \*, \*, [ [ 1, 0, 0 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Basis( M );</span>
Basis( <3-dimensional bi-module over ( Rationals^
[ 3, 3 ] ) (left) and ( Rationals^[ 3, 3 ] ) (right)>,
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">IsBasisOfAlgebraModuleElementSpace( B );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">B!.delegateBasis;</span>
SemiEchelonBasis( <vector space of dimension 3 over Rationals>,
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] )
</pre></div>
<p><a id="X789863037B0E35D2" name="X789863037B0E35D2"></a></p>
<h5>62.11-15 MatrixOfAction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MatrixOfAction</code>( <var class="Arg">B</var>, <var class="Arg">x</var>[, <var class="Arg">side</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">B</var> is a basis of an algebra module and <var class="Arg">x</var> is an element of the algebra that acts on this module. This function returns the matrix of the action of <var class="Arg">x</var> with respect to <var class="Arg">B</var>. If <var class="Arg">x</var> acts from the left, then the coefficients of the images of the basis elements of <var class="Arg">B</var> (under the action of <var class="Arg">x</var>) are the columns of the output. If <var class="Arg">x</var> acts from the right, then they are the rows of the output.</p>
<p>If the module is a bi-module, then the third parameter <var class="Arg">side</var> must be specified. This is the string <code class="code">"left"</code>, or <code class="code">"right"</code> depending whether <var class="Arg">x</var> acts from the left or the right.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">M:= LeftAlgebraModuleByGenerators( A, \*, [ [ 1, 0, 0 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:= Basis(A)[3];</span>
[ [ 0, 0, 1 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">MatrixOfAction( Basis( M ), x );</span>
[ [ 0, 0, 1 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ]
</pre></div>
<p><a id="X8742A7D27F26AFAB" name="X8742A7D27F26AFAB"></a></p>
<h5>62.11-16 SubAlgebraModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SubAlgebraModule</code>( <var class="Arg">M</var>, <var class="Arg">gens</var>[, <var class="Arg">"basis"</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>is the sub-module of the algebra module <var class="Arg">M</var>, generated by the vectors in <var class="Arg">gens</var>. If as an optional argument the string <code class="code">basis</code> is added, then it is assumed that the vectors in <var class="Arg">gens</var> form a basis of the submodule.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m1:= NullMat( 2, 2 );; m1[1][1]:= 1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m2:= NullMat( 2, 2 );; m2[2][2]:= 1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Algebra( Rationals, [ m1, m2 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M:= LeftAlgebraModuleByGenerators( A, \*, [ [ 1, 0 ], [ 0, 1 ] ] );</span>
<left-module over <algebra over Rationals, with 2 generators>>
<span class="GAPprompt">gap></span> <span class="GAPinput">bb:= BasisVectors( Basis( M ) );</span>
[ [ 1, 0 ], [ 0, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= SubAlgebraModule( M, [ bb[1] ] );</span>
<left-module over <algebra over Rationals, with 2 generators>>
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( V );</span>
1
</pre></div>
<p><a id="X86E0515987192F0E" name="X86E0515987192F0E"></a></p>
<h5>62.11-17 LeftModuleByHomomorphismToMatAlg</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftModuleByHomomorphismToMatAlg</code>( <var class="Arg">A</var>, <var class="Arg">hom</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">A</var> is an algebra and <var class="Arg">hom</var> a homomorphism from <var class="Arg">A</var> into a matrix algebra. This function returns the left <var class="Arg">A</var>-module defined by the homomorphism <var class="Arg">hom</var>.</p>
<p><a id="X7EE41297867E41A8" name="X7EE41297867E41A8"></a></p>
<h5>62.11-18 RightModuleByHomomorphismToMatAlg</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightModuleByHomomorphismToMatAlg</code>( <var class="Arg">A</var>, <var class="Arg">hom</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">A</var> is an algebra and <var class="Arg">hom</var> a homomorphism from <var class="Arg">A</var> into a matrix algebra. This function returns the right <var class="Arg">A</var>-module defined by the homomorphism <var class="Arg">hom</var>.</p>
<p>First we produce a structure constants algebra with basis elements <span class="SimpleMath">x</span>, <span class="SimpleMath">y</span>, <span class="SimpleMath">z</span> such that <span class="SimpleMath">x^2 = x</span>, <span class="SimpleMath">y^2 = y</span>, <span class="SimpleMath">xz = z</span>, <span class="SimpleMath">zy = z</span> and all other products are zero.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">T:= EmptySCTable( 3, 0 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( T, 1, 1, [ 1, 1 ]);</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( T, 2, 2, [ 1, 2 ]);</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( T, 1, 3, [ 1, 3 ]);</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( T, 3, 2, [ 1, 3 ]);</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= AlgebraByStructureConstants( Rationals, T );</span>
<algebra of dimension 3 over Rationals>
</pre></div>
<p>Now we construct an isomorphic matrix algebra.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m1:= NullMat( 2, 2 );; m1[1][1]:= 1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m2:= NullMat( 2, 2 );; m2[2][2]:= 1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m3:= NullMat( 2, 2 );; m3[1][2]:= 1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Algebra( Rationals, [ m1, m2, m3 ] );</span>
<algebra over Rationals, with 3 generators>
</pre></div>
<p>Finally we construct the homomorphism and the corresponding right module.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= AlgebraHomomorphismByImages( A, B, Basis(A), [ m1, m2, m3 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">RightModuleByHomomorphismToMatAlg( A, f );</span>
<right-module over <algebra of dimension 3 over Rationals>>
</pre></div>
<p><a id="X8729F0A678A4A09C" name="X8729F0A678A4A09C"></a></p>
<h5>62.11-19 AdjointModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AdjointModule</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the <var class="Arg">A</var>-module defined by the left action of <var class="Arg">A</var> on itself.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m1:= NullMat( 2, 2 );; m1[1][1]:= 1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m2:= NullMat( 2, 2 );; m2[2][2]:= 1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m3:= NullMat( 2, 2 );; m3[1][2]:= 1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Algebra( Rationals, [ m1, m2, m3 ] );</span>
<algebra over Rationals, with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= AdjointModule( A );</span>
<3-dimensional left-module over <algebra of dimension
3 over Rationals>>
<span class="GAPprompt">gap></span> <span class="GAPinput">v:= Basis( V )[3];</span>
[ [ 0, 1 ], [ 0, 0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">W:= SubAlgebraModule( V, [ v ] );</span>
<left-module over <algebra of dimension 3 over Rationals>>
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( W );</span>
1
</pre></div>
<p><a id="X84813BCD80BDF3C4" name="X84813BCD80BDF3C4"></a></p>
<h5>62.11-20 FaithfulModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FaithfulModule</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a faithful finite-dimensional left-module over the algebra <var class="Arg">A</var>. This is only implemented for associative algebras, and for Lie algebras of characteristic <span class="SimpleMath">0</span>. (It may also work for certain Lie algebras of characteristic <span class="SimpleMath">p > 0</span>.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">T:= EmptySCTable( 2, 0 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= AlgebraByStructureConstants( Rationals, T );</span>
<algebra of dimension 2 over Rationals>
</pre></div>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">T:= EmptySCTable( 3, 0, "antisymmetric" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( T, 1, 2, [ 1, 3 ]);</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:= LieAlgebraByStructureConstants( Rationals, T );</span>
<Lie algebra of dimension 3 over Rationals>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= FaithfulModule( L );</span>
<left-module over <Lie algebra of dimension 3 over Rationals>>
<span class="GAPprompt">gap></span> <span class="GAPinput">vv:= BasisVectors( Basis( V ) );</span>
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">x:= Basis( L )[3];</span>
v.3
<span class="GAPprompt">gap></span> <span class="GAPinput">List( vv, v -> x^v );</span>
[ [ 0, 0, 0 ], [ 1, 0, 0 ], [ 0, 0, 0 ] ]
</pre></div>
<p><code class="code">A</code> is a <span class="SimpleMath">2</span>-dimensional algebra where all products are zero.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= FaithfulModule( A );</span>
<left-module over <algebra of dimension 2 over Rationals>>
<span class="GAPprompt">gap></span> <span class="GAPinput">vv:= BasisVectors( Basis( V ) );</span>
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">xx:= BasisVectors( Basis( A ) );</span>
[ v.1, v.2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">xx[1]^vv[3];</span>
[ 1, 0, 0 ]
</pre></div>
<p><a id="X7E16630185CE2C10" name="X7E16630185CE2C10"></a></p>
<h5>62.11-21 ModuleByRestriction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ModuleByRestriction</code>( <var class="Arg">V</var>, <var class="Arg">sub1</var>[, <var class="Arg">sub2</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">V</var> is an algebra module and <var class="Arg">sub1</var> is a subalgebra of the acting algebra of <var class="Arg">V</var>. This function returns the module that is the restriction of <var class="Arg">V</var> to <var class="Arg">sub1</var>. So it has the same underlying vector space as <var class="Arg">V</var>, but the acting algebra is <var class="Arg">sub</var>. If two subalgebras <var class="Arg">sub1</var>, <var class="Arg">sub2</var> are given then <var class="Arg">V</var> is assumed to be a bi-module, and <var class="Arg">sub1</var> a subalgebra of the algebra acting on the left, and <var class="Arg">sub2</var> a subalgebra of the algebra acting on the right.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Rationals^[3,3];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= LeftAlgebraModuleByGenerators( A, \*, [ [ 1, 0, 0 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Subalgebra( A, [ Basis(A)[1] ] );</span>
<algebra over Rationals, with 1 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:= ModuleByRestriction( V, B );</span>
<left-module over <algebra over Rationals, with 1 generators>>
</pre></div>
<p><a id="X7885AAC87FDCF649" name="X7885AAC87FDCF649"></a></p>
<h5>62.11-22 NaturalHomomorphismBySubAlgebraModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalHomomorphismBySubAlgebraModule</code>( <var class="Arg">V</var>, <var class="Arg">W</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">V</var> must be a sub-algebra module of <var class="Arg">V</var>. This function returns the projection from <var class="Arg">V</var> onto <code class="code"><var class="Arg">V</var>/<var class="Arg">W</var></code>. It is a linear map, that is also a module homomorphism. As usual images can be formed with <code class="code">Image( f, v )</code> and pre-images with <code class="code">PreImagesRepresentative( f, u )</code>.</p>
<p>The quotient module can also be formed by entering <code class="code"><var class="Arg">V</var>/<var class="Arg">W</var></code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Rationals^[3,3];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= DirectSumOfAlgebras( A, A );</span>
<algebra over Rationals, with 6 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">T:= StructureConstantsTable( Basis( B ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:= AlgebraByStructureConstants( Rationals, T );</span>
<algebra of dimension 18 over Rationals>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= AdjointModule( C );</span>
<left-module over <algebra of dimension 18 over Rationals>>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:= SubAlgebraModule( V, [ Basis(V)[1] ] );</span>
<left-module over <algebra of dimension 18 over Rationals>>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= NaturalHomomorphismBySubAlgebraModule( V, W );</span>
<linear mapping by matrix, <
18-dimensional left-module over <algebra of dimension
18 over Rationals>> -> <
9-dimensional left-module over <algebra of dimension
18 over Rationals>>>
<span class="GAPprompt">gap></span> <span class="GAPinput">quo:= ImagesSource( f ); # i.e., the quotient module</span>
<9-dimensional left-module over <algebra of dimension
18 over Rationals>>
<span class="GAPprompt">gap></span> <span class="GAPinput">v:= Basis( quo )[1];</span>
[ 1, 0, 0, 0, 0, 0, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PreImagesRepresentative( f, v );</span>
v.4
<span class="GAPprompt">gap></span> <span class="GAPinput">Basis( C )[4]^v;</span>
[ 1, 0, 0, 0, 0, 0, 0, 0, 0 ]
</pre></div>
<p><a id="X85D0F3758551DADC" name="X85D0F3758551DADC"></a></p>
<h5>62.11-23 DirectSumOfAlgebraModules</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DirectSumOfAlgebraModules</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DirectSumOfAlgebraModules</code>( <var class="Arg">V</var>, <var class="Arg">W</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">list</var> must be a list of algebra modules. This function returns the direct sum of the elements in the list (as an algebra module). The modules must be defined over the same algebras.</p>
<p>In the second form is short for <code class="code">DirectSumOfAlgebraModules( [ <var class="Arg">V</var>, <var class="Arg">W</var> ] )</code></p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= FullMatrixAlgebra( Rationals, 3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= BiAlgebraModuleByGenerators( A, A, \*, \*, [ [1,0,0] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:= DirectSumOfAlgebraModules( V, V );</span>
<6-dimensional left-module over ( Rationals^[ 3, 3 ] )>
<span class="GAPprompt">gap></span> <span class="GAPinput">BasisVectors( Basis( W ) );</span>
[ ( [ 1, 0, 0 ] )(+)( [ 0, 0, 0 ] ), ( [ 0, 1, 0 ] )(+)( [ 0, 0, 0 ] )
, ( [ 0, 0, 1 ] )(+)( [ 0, 0, 0 ] ),
( [ 0, 0, 0 ] )(+)( [ 1, 0, 0 ] ), ( [ 0, 0, 0 ] )(+)( [ 0, 1, 0 ] )
, ( [ 0, 0, 0 ] )(+)( [ 0, 0, 1 ] ) ]
</pre></div>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:= SimpleLieAlgebra( "C", 3, Rationals );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= HighestWeightModule( L, [ 1, 1, 0 ] );</span>
<64-dimensional left-module over <Lie algebra of dimension
21 over Rationals>>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:= HighestWeightModule( L, [ 0, 0, 2 ] );</span>
<84-dimensional left-module over <Lie algebra of dimension
21 over Rationals>>
<span class="GAPprompt">gap></span> <span class="GAPinput">U:= DirectSumOfAlgebraModules( V, W );</span>
<148-dimensional left-module over <Lie algebra of dimension
21 over Rationals>>
</pre></div>
<p><a id="X7D7A6486803B15CE" name="X7D7A6486803B15CE"></a></p>
<h5>62.11-24 TranslatorSubalgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TranslatorSubalgebra</code>( <var class="Arg">M</var>, <var class="Arg">U</var>, <var class="Arg">W</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">M</var> is an algebra module, and <var class="Arg">U</var> and <var class="Arg">W</var> are two subspaces of <var class="Arg">M</var>. Let <var class="Arg">A</var> be the algebra acting on <var class="Arg">M</var>. This function returns the subspace of elements of <var class="Arg">A</var> that map <var class="Arg">U</var> into <var class="Arg">W</var>. If <var class="Arg">W</var> is a sub-algebra-module (i.e., closed under the action of <var class="Arg">A</var>), then this space is a subalgebra of <var class="Arg">A</var>.</p>
<p>This function works for left, or right modules over a finite-dimensional algebra. We stress that it is not checked whether <var class="Arg">U</var> and <var class="Arg">W</var> are indeed subspaces of <var class="Arg">M</var>. If this is not the case nothing is guaranteed about the behaviour of the function.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= FullMatrixAlgebra( Rationals, 3 );</span>
( Rationals^[ 3, 3 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= Rationals^[3,2];</span>
( Rationals^[ 3, 2 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">M:= LeftAlgebraModule( A, \*, V );</span>
<left-module over ( Rationals^[ 3, 3 ] )>
<span class="GAPprompt">gap></span> <span class="GAPinput">bm:= Basis(M);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">U:= SubAlgebraModule( M, [ bm[1] ] ); </span>
<left-module over ( Rationals^[ 3, 3 ] )>
<span class="GAPprompt">gap></span> <span class="GAPinput">TranslatorSubalgebra( M, U, M );</span>
<algebra of dimension 9 over Rationals>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:= SubAlgebraModule( M, [ bm[4] ] );</span>
<left-module over ( Rationals^[ 3, 3 ] )>
<span class="GAPprompt">gap></span> <span class="GAPinput">T:=TranslatorSubalgebra( M, U, W );</span>
<algebra of dimension 0 over Rationals>
</pre></div>
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