/usr/share/gap/lib/algsc.gd is in gap-libs 4r7p5-2.
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##
#W algsc.gd GAP library Thomas Breuer
##
##
#Y Copyright (C) 1997, Lehrstuhl D für Mathematik, RWTH Aachen, Germany
#Y (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
## This file contains the design of elements of algebras given by structure
## constants (s.~c.).
##
## An s.~c. algebra is a free left module $A$ of fixed dimension $n$, say,
## over a ring-with-one $R$, with multiplication defined on the vectors of
## the standard basis $B$ of $A$ by the structure constants table.
##
## A *full s.~c. algebra* is an s.c. algebra that contains $B$.
## (So a full s.~c. algebra need *not* contain the whole family of its
## elements.)
##
## The family <Fam> of s.~c. algebra elements consists of all linear
## combinations of the basis vectors of $A$, with coefficients in a
## suitable set.
## If the elements family of $R$ has a uniquely determined zero element,
## this set is the whole element family of $R$, and <Fam> has the
## category `IsFamilyOverFullCoefficientsFamily'.
## Otherwise this set is $R$ itself, stored in the component
## `coefficientsDomain' of <Fam>.
## (Note that the zero element is already part of the s.~c. table.)
##
## In any case, the value of the attribute `CoefficientsFamily' is
## the elements family of the family of $R$.
## (This is used in family predicates.)
##
## S.~c. algebra elements have an external representation, which is the
## coefficients vector w.r.t. the standard basis.
##
## The constructor for s.~c. algebra elements is `ObjByExtRep'.
##
#############################################################################
##
#F SCTableEntry
##
DeclareSynonym( "SCTableEntry", SC_TABLE_ENTRY );
#############################################################################
##
#F SCTableProduct
##
DeclareSynonym( "SCTableProduct", SC_TABLE_PRODUCT );
#############################################################################
##
#C IsFamilyOverFullCoefficientsFamily( <Fam> )
##
## If the family <Fam> has this category, all coefficients tuples over
## `CoefficientsFamily( <Fam> )' describe valid elements of <Fam>.
##
## (This tells mainly what `ObjByExtRep' can assume resp. must test.)
##
DeclareCategory( "IsFamilyOverFullCoefficientsFamily", IsFamily );
#T other file?
#############################################################################
##
#C IsSCAlgebraObj( <obj> )
#C IsSCAlgebraObjCollection( <obj> )
#C IsSCAlgebraObjFamily( <obj> )
##
## S.~c. algebra elements may have inverses, in order to allow `One' and
## `Inverse' we make them scalars.
##
DeclareCategory( "IsSCAlgebraObj", IsScalar );
DeclareCategoryCollections( "IsSCAlgebraObj" );
DeclareCategoryCollections( "IsSCAlgebraObjCollection" );
DeclareCategoryCollections( "IsSCAlgebraObjCollColl" );
DeclareCategoryFamily( "IsSCAlgebraObj" );
#############################################################################
##
#P IsFullSCAlgebra( <A> )
##
## An s.~c. algebra is a free left module $A$ over a ring-with-one $R$,
## with multiplication defined on the vectors of the standard basis $B$
## of an algebra $\hat{A}$ containing $A$ by the structure constants table
## of $\hat{A}$.
##
## $A$ is a *full s.~c. algebra* if it contains $B$.
## (So a full s.~c. algebra need *not* contain the whole family of its
## elements.)
#T Do we really need this in addition to `IsFullFPAlgebra',
#T or would it be misuse to take `IsFullFPAlgebra' here?
##
DeclareProperty( "IsFullSCAlgebra", IsFLMLOR and IsSCAlgebraObjCollection );
#############################################################################
##
#P IsCanonicalBasisFullSCAlgebra( <B> )
##
## is `true' if the underlying free left module of the basis <B> is a full
## s.~c. algebra and <B> is equal to its canonical basis,
## and `false' otherwise.
##
## The canonical basis of a full s.~c. algebra consists of elements whose
## external representations are standard basis vectors.
##
## (The canonical basis of a full s.~c. algebra is constructed together with
## the algebra.)
##
DeclareProperty( "IsCanonicalBasisFullSCAlgebra", IsBasis );
InstallTrueMethod( IsCanonicalBasis, IsCanonicalBasisFullSCAlgebra );
#############################################################################
##
#F IsSCAlgebraObjSpace( <V> )
##
## If an $F$-vector space <V> is in the filter `IsSCAlgebraObjSpace' then
## this expresses that <V> consists of elements in a s.c. algebra,
## and that <V> can be handled via the mechanism of nice bases
## (see~"Vector Spaces Handled By Nice Bases"), in the following way.
## The `NiceFreeLeftModuleInfo' value of <V> is irrelevant,
## and the `NiceVector' value of $v \in <V>$ is defined as
## $`ExtRepOfObj'( v )$.
##
DeclareHandlingByNiceBasis( "IsSCAlgebraObjSpace",
"for free left modules of s.c. algebra elements" );
#############################################################################
##
#M IsFiniteDimensional( <A> ) . . . . . S.~c. algebras are always fin. dim.
##
InstallTrueMethod( IsFiniteDimensional,
IsFreeLeftModule and IsSCAlgebraObjCollection );
#############################################################################
##
#E
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