%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %A algebra.tex GAP documentation Thomas Breuer %% %A @(#)$Id: algebra.tex,v 3.1 1994/06/10 02:38:06 vfelsch Rel $ %% %Y Copyright 1994-1995, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany %% %% This file contains the description of the algebra record and polymorphic %% functions for algebras. %% %H $Log: algebra.tex,v $ %H Revision 3.1 1994/06/10 02:38:06 vfelsch %H updated examples %H %H Revision 3.0 1994/05/19 13:59:20 sam %H Initial Revision under RCS %H %% \def\MeatAxe{\sf MeatAxe} \Chapter{Algebras} This chapter introduces the data structures and functions for algebras in {\GAP}. The word *algebra* in this manual means always *associative algebra*. At the moment {\GAP} supports only finitely presented algebras and matrix algebras. For details about implementation and special functions for the different types of algebras, see "More about Algebras" and the chapters "Finitely Presented Algebras" and "Matrix Algebras". The treatment of algebras is very similar to that of groups. For example, algebras in {\GAP} are always finitely generated, since for many questions the generators play an important role. If you are not familiar with the concepts that are used to handle groups in {\GAP} it might be useful to read the introduction and the overview sections in chapter "Groups". Algebras are created using 'Algebra' (see "Algebra") or 'UnitalAlgebra' (see "UnitalAlgebra"), subalgebras of a given algebra using 'Subalgebra' (see "Subalgebra") or 'UnitalSubalgebra' (see "UnitalSubalgebra"). See "Parent Algebras and Subalgebras", and the corresponding section "More about Groups and Subgroups" in the chapter about groups for details about the distinction between parent algebras and subalgebras. The first sections of the chapter describe the data structures (see "More about Algebras") and the concepts of unital algebras (see "Algebras and Unital Algebras") and parent algebras (see "Parent Algebras and Subalgebras"). The next sections describe the functions for the construction of algebras, and the tests for algebras (see "Algebra", "UnitalAlgebra", "IsAlgebra", "IsUnitalAlgebra", "Subalgebra", "UnitalSubalgebra", "IsSubalgebra", "AsAlgebra", "AsUnitalAlgebra", "AsSubalgebra", "AsUnitalSubalgebra"). The next sections describe the different types of functions for algebras (see "Operations for Algebras", "Zero and One for Algebras", "Set Theoretic Functions for Algebras", "Property Tests for Algebras", "Vector Space Functions for Algebras", "Algebra Functions for Algebras", "TrivialSubalgebra"). The next sections describe the operation of algebras (see "Operation for Algebras", "OperationHomomorphism for Algebras"). The next sections describe algebra homomorphisms (see "Algebra Homomorphisms", "Mapping Functions for Algebra Homomorphisms"). The next sections describe algebra elements (see "Algebra Elements", "IsAlgebraElement"). The last section describes the implementation of the data structures (see "Algebra Records"). At the moment there is no implementation for ideals, cosets, and factors of algebras in {\GAP}, and the only available algebra homomorphisms are operation homomorphisms. Also there is no implementation of bases for general algebras, this will be available as soon as it is for general vector spaces. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{More about Algebras} Let $F$ be a field. A ring $A$ is called an *$F$-algebra* if $A$ is an $F$-vector space. All algebras in {\GAP} are *associative*, that is, the multiplication is associative. An algebra always contains a *zero element* that can be obtained by subtracting an arbitrary element from itself. A discussion of *identity elements* of algebras (and of the consequences for the implementation in {\GAP}) can be found in "Algebras and Unital Algebras". *Elements of the field* $F$ are not regarded as elements of $A$. The practical reason (besides the obvious mathematical one) for this is that even if the identity matrix is contained in the matrix algebra $A$ it is not possible to write '1 + a' for adding the identity matrix to the algebra element 'a', since independent of the algebra $A$ the meaning in {\GAP} is already defined as to add '1' to all positions of the matrix 'a'. Thus one has to write 'One( A ) + a' or 'a\^0\ + a' instead. The natural *operation domains* for algebras are modules (see "Operation for Algebras", and chapter "Modules"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Algebras and Unital Algebras} Not all algebras contain a (left and right) multiplicative neutral *identity element*, but if an algebra contains such an identity element it is unique. If an algebra $A$ contains a multiplicative neutral element then in general it cannot be derived from an arbitrary element $a$ of $A$ by forming $a / a$ or $a^0$, since these operations may be not defined for the algebra $A$. More precisely, it may be possible to invert $a$ or raise it to the zero-th power, but $A$ is not necessarily closed under these operations. For example, if $a$ is a square matrix in {\GAP} then we can form $a^0$ which is the identity matrix of the same size and over the same field as $a$. On the other hand, an algebra may have a multiplicative neutral element that is *not* equal to the zero-th power of elements (see "Zero and One for Algebras"). In many cases, however, the zero-th power of algebra elements is well-defined, with the result again in the algebra. This holds for example for all finitely presented algebras (see chapter "Finitely Presented Algebras") and all those matrix algebras whose generators are the generators of a finite group. For practical purposes it is useful to distinguish general *algebras* and *unital algebras*. A unital algebra in {\GAP} is an algebra $U$ that is *known to contain* zero-th powers of elements, and all functions may assume this. A not unital algebra $A$ may contain zero-th powers of elements or not, and no function for $A$ should assume existence or nonexistence of these elements in $A$. So it may be possible to view $A$ as a unital algebra using 'AsUnitalAlgebra( )' (see "AsUnitalAlgebra"), and of course it is always possible to view a unital algebra as algebra using 'AsAlgebra( )' (see "AsAlgebra"). $A$ can have unital subalgebras, and of course $U$ can have subalgebras that are not unital. The images of unital algebras under operation homomorphisms are either unital or trivial, since the identity of the source acts trivially, so its image under the homomorphism is the identity of the image. The following example shows the main differences between algebras and unital algebras. | gap> a:= [ [ 1, 0 ], [ 0, 0 ] ];; gap> alg1:= Algebra( Rationals, [ a ] ); Algebra( Rationals, [ [ [ 1, 0 ], [ 0, 0 ] ] ] ) gap> id:= a^0; [ [ 1, 0 ], [ 0, 1 ] ] gap> id in alg1; false gap> alg2:= UnitalAlgebra( Rationals, [ a ] ); UnitalAlgebra( Rationals, [ [ [ 1, 0 ], [ 0, 0 ] ] ] ) gap> id in alg2; true gap> alg3:= AsAlgebra( alg2 ); Algebra( Rationals, [ [ [ 1, 0 ], [ 0, 0 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ] ) gap> alg3 = alg2; true gap> AsUnitalAlgebra( alg1 ); Error, is not unital | We see that if we want the identity matrix to be contained in an algebra that is not known to be unital, it might be necessary to add it to the generators. If we would not have the possibility to define unital algebras, this would lead to the strange situations that a two-generator algebra means an algebra generated by one nonidentity generator and the identity matrix, or that an algebra is free on the set $X$ but is generated as algebra by the set $X$ plus the identity. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Parent Algebras and Subalgebras} {\GAP} distinguishs between parent algebras and subalgebras of parent algebras. The concept is the same as that for groups (see "More about Groups and Subgroups"), so here it is only sketched. Each subalgebra belongs to a unique parent algebra, the so-called *parent* of the subalgebra. A parent algebra is its own parent. Parent algebras are constructed by 'Algebra' and 'UnitalAlgebra', subalgebras are constructed by 'Subalgebra' and 'UnitalSubalgebra'. The parent of the first argument of 'Subalgebra' will be the parent of the constructed subalgebra. Those algebra functions that take more than one algebra as argument require that the arguments have a common parent. Take for instance 'Centralizer'. It takes two arguments, an algebra and an algebra , where either is a parent algebra, and is a subalgebra of this parent algebra, or and are subalgebras of a common parent algebra

, and returns the centralizer of in . This is represented as a subalgebra of the common parent of and . Note that a subalgebra of a parent algebra need not be a proper subalgebra. An exception to this rule is again the set theoretic function 'Intersection' (see "Intersection"), which allows to intersect algebras with different parents. Whenever you have two subalgebras which have different parent algebras but have a common superalgebra you can use 'AsSubalgebra' or 'AsUnitalSubalgebra' (see "AsSubalgebra", "AsUnitalSubalgebra") in order to construct new subalgebras which have a common parent algebra . Note that subalgebras of unital algebras need not be unital (see "Algebras and Unital Algebras"). The following sections describe the functions related to this concept (see "Algebra", "UnitalAlgebra", "IsAlgebra", "IsUnitalAlgebra", "AsAlgebra", "AsUnitalAlgebra", "Subalgebra", "UnitalSubalgebra", "AsSubalgebra", "AsUnitalSubalgebra", and also "IsParent", "Parent"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Algebra} 'Algebra( )' returns a parent algebra $A$ which is isomorphic to the parent algebra or subalgebra . 'Algebra( , )' \\ 'Algebra( , , )' returns a parent algebra over the field and generated by the algebra elements in the list . The zero element of this algebra may be entered as ; this is necessary whenever is empty. | gap> a:= [ [ 1 ] ];; gap> alg:= Algebra( Rationals, [ a ] ); Algebra( Rationals, [ [ [ 1 ] ] ] ) gap> alg.name:= "alg";; gap> sub:= Subalgebra( alg, [] ); Subalgebra( alg, [ ] ) gap> Algebra( sub ); Algebra( Rationals, [ [ [ 0 ] ] ] ) gap> Algebra( Rationals, [], 0*a ); Algebra( Rationals, [ [ [ 0 ] ] ] ) | The algebras returned by 'Algebra' are not unital. For constructing unital algebras, use "UnitalAlgebra" 'UnitalAlgebra'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{UnitalAlgebra} 'UnitalAlgebra( )' returns a unital parent algebra $A$ which is isomorphic to the parent algebra or subalgebra . If is not unital it is checked whether the zero-th power of elements is contained in , and if not an error is signalled. 'UnitalAlgebra( , )' \\ 'UnitalAlgebra( , , )' returns a unital parent algebra over the field and generated by the algebra elements in the list . The zero element of this algebra may be entered as ; this is necessary whenever is empty. | gap> alg1:= UnitalAlgebra( Rationals, [ NullMat( 2, 2 ) ] ); UnitalAlgebra( Rationals, [ [ [ 0, 0 ], [ 0, 0 ] ] ] ) gap> alg2:= UnitalAlgebra( Rationals, [], NullMat( 2, 2 ) ); UnitalAlgebra( Rationals, [ [ [ 0, 0 ], [ 0, 0 ] ] ] ) gap> alg3:= Algebra( alg1 ); Algebra( Rationals, [ [ [ 0, 0 ], [ 0, 0 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ] ) gap> alg1 = alg3; true gap> AsUnitalAlgebra( alg3 ); UnitalAlgebra( Rationals, [ [ [ 0, 0 ], [ 0, 0 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ] ) | The algebras returned by 'UnitalAlgebra' are unital. For constructing algebras that are not unital, use "Algebra" 'Algebra'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{IsAlgebra} 'IsAlgebra( )' returns 'true' if , which can be an object of arbitrary type, is a parent algebra or a subalgebra and 'false' otherwise. The function will signal an error if is an unbound variable. | gap> IsAlgebra( FreeAlgebra( GF(2), 0 ) ); true gap> IsAlgebra( 1/2 ); false | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{IsUnitalAlgebra} 'IsUnitalAlgebra( )' returns 'true' if , which can be an object of arbitrary type, is a unital parent algebra or a unital subalgebra and 'false' otherwise. The function will signal an error if is an unbound variable. | gap> IsUnitalAlgebra( FreeAlgebra( GF(2), 0 ) ); true gap> IsUnitalAlgebra( Algebra( Rationals, [ [ [ 1 ] ] ] ) ); false | Note that the function does *not* check whether is an algebra that contains the zero-th power of elements, but just checks whether is an algebra with flag 'isUnitalAlgebra'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Subalgebra} 'Subalgebra( , )' returns the subalgebra of the algebra generated by the elements in the list . | gap> a:= [ [ 1, 0 ], [ 0, 0 ] ];; gap> b:= [ [ 0, 0 ], [ 0, 1 ] ] ;; gap> alg:= Algebra( Rationals, [ a, b ] );; gap> alg.name:= "alg";; gap> s:= Subalgebra( alg, [ a ] ); Subalgebra( alg, [ [ [ 1, 0 ], [ 0, 0 ] ] ] ) gap> s = alg; false gap> s:= UnitalSubalgebra( alg, [ a ] ); UnitalSubalgebra( alg, [ [ [ 1, 0 ], [ 0, 0 ] ] ] ) gap> s = alg; true | Note that 'Subalgebra', 'UnitalSubalgebra', 'AsSubalgebra' and 'AsUnitalSubalgebra' are the only functions in which the name 'Subalgebra' does not refer to the mathematical terms subalgebra and superalgebra but to the implementation of algebras as subalgebras and parent algebras. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{UnitalSubalgebra} 'UnitalSubalgebra( , )' returns the unital subalgebra of the algebra generated by the elements in the list . If is not (known to be) unital then first it is checked that really contains the zero-th power of elements. | gap> a:= [ [ 1, 0 ], [ 0, 0 ] ];; gap> b:= [ [ 0, 0 ], [ 0, 1 ] ] ;; gap> alg:= Algebra( Rationals, [ a, b ] );; gap> alg.name:= "alg";; gap> s:= Subalgebra( alg, [ a ] ); Subalgebra( alg, [ [ [ 1, 0 ], [ 0, 0 ] ] ] ) gap> s = alg; false gap> s:= UnitalSubalgebra( alg, [ a ] ); UnitalSubalgebra( alg, [ [ [ 1, 0 ], [ 0, 0 ] ] ] ) gap> s = alg; true | Note that 'Subalgebra', 'UnitalSubalgebra', 'AsSubalgebra' and 'AsUnitalSubalgebra' are the only functions in which the name 'Subalgebra' does not refer to the mathematical terms subalgebra and superalgebra but to the implementation of algebras as subalgebras and parent algebras. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{IsSubalgebra} 'IsSubalgebra( , )' returns 'true' if is a subalgebra of and 'false' otherwise. Note that and must have a common parent algebra. This function returns 'true' if and only if the set of elements of is a subset of the set of elements of . | gap> a:= [ [ 1, 0 ], [ 0, 0 ] ];; gap> b:= [ [ 0, 0 ], [ 0, 1 ] ] ;; gap> alg:= Algebra( Rationals, [ a, b ] );; gap> alg.name:= "alg";; gap> IsSubalgebra( alg, alg ); true gap> s:= UnitalSubalgebra( alg, [ a ] ); UnitalSubalgebra( alg, [ [ [ 1, 0 ], [ 0, 0 ] ] ] ) gap> IsSubalgebra( alg, s ); true | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{AsAlgebra} 'AsAlgebra( )' \\ 'AsAlgebra( , )' Let be a domain. 'AsAlgebra' returns an algebra $A$ over the field such that the set of elements of is the same as the set of elements of $A$ if this is possible. If is an algebra the argument may be omitted, the coefficients field of is taken as coefficients field of in this case. If is a list of algebra elements these elements must form a algebra. Otherwise an error is signalled. | gap> a:= [ [ 1, 0 ], [ 0, 0 ] ] * Z(2);; gap> AsAlgebra( GF(2), [ a, 0*a ] ); Algebra( GF(2), [ [ [ Z(2)^0, 0*Z(2) ], [ 0*Z(2), 0*Z(2) ] ] ] ) | Note that this function returns a parent algebra or a subalgebra of a parent algebra depending on . In order to convert a subalgebra into a parent algebra you must use 'Algebra' or 'UnitalAlgebra' (see "Algebra", "UnitalAlgebra"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{AsUnitalAlgebra} 'AsUnitalAlgebra( )' \\ 'AsUnitalAlgebra( , )' Let be a domain. 'AsUnitalAlgebra' returns a unital algebra $A$ over the field such that the set of elements of is the same as the set of elements of $A$ if this is possible. If is an algebra the argument may be omitted, the coefficients field of is taken as coefficients field of in this case. If is a list of algebra elements these elements must form a unital algebra. Otherwise an error is signalled. | gap> a:= [ [ 1, 0 ], [ 0, 0 ] ] * Z(2);; gap> AsUnitalAlgebra( GF(2), [ a, a^0, 0*a, a^0-a ] ); UnitalAlgebra( GF(2), [ [ [ 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2)^0 ] ], [ [ Z(2)^0, 0*Z(2) ], [ 0*Z(2), 0*Z(2) ] ] ] ) | Note that this function returns a parent algebra or a subalgebra of a parent algebra depending on . In order to convert a subalgebra into a parent algebra you must use 'Algebra' or 'UnitalAlgebra' (see "Algebra", "UnitalAlgebra"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{AsSubalgebra} 'AsSubalgebra( , )' Let be a parent algebra and be a parent algebra or a subalgebra with a possibly different parent algebra, such that the generators of are elements of . 'AsSubalgebra' returns a new subalgebra $S$ such that $S$ has parent algebra and is generated by the generators of . | gap> a:= [ [ 1, 0 ], [ 0, 0 ] ];; gap> b:= [ [ 0, 0 ], [ 0, 1 ] ] ;; gap> alg:= Algebra( Rationals, [ a, b ] );; gap> alg.name:= "alg";; gap> s:= Algebra( Rationals, [ a ] ); Algebra( Rationals, [ [ [ 1, 0 ], [ 0, 0 ] ] ] ) gap> AsSubalgebra( alg, s ); Subalgebra( alg, [ [ [ 1, 0 ], [ 0, 0 ] ] ] ) | Note that 'Subalgebra', 'UnitalSubalgebra', 'AsSubalgebra' and 'AsUnitalSubalgebra' are the only functions in which the name 'Subalgebra' does not refer to the mathematical terms subalgebra and superalgebra but to the implementation of algebras as subalgebras and parent algebras. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{AsUnitalSubalgebra} 'AsUnitalSubalgebra( , )' Let be a parent algebra and be a parent algebra or a subalgebra with a possibly different parent algebra, such that the generators of are elements of . 'AsSubalgebra' returns a new unital subalgebra $S$ such that $S$ has parent algebra and is generated by the generators of . If or do not contain the zero-th power of elements an error is signalled. | gap> a:= [ [ 1, 0 ], [ 0, 0 ] ];; gap> b:= [ [ 0, 0 ], [ 0, 1 ] ];; gap> alg:= Algebra( Rationals, [ a, b ] );; gap> alg.name:= "alg";; gap> s:= UnitalAlgebra( Rationals, [ a ] ); UnitalAlgebra( Rationals, [ [ [ 1, 0 ], [ 0, 0 ] ] ] ) gap> AsSubalgebra( alg, s ); Subalgebra( alg, [ [ [ 1, 0 ], [ 0, 0 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ] ) gap> AsUnitalSubalgebra( alg, s ); UnitalSubalgebra( alg, [ [ [ 1, 0 ], [ 0, 0 ] ] ] ) | Note that 'Subalgebra', 'UnitalSubalgebra', 'AsSubalgebra' and 'AsUnitalSubalgebra' are the only functions in which the name 'Subalgebra' does not refer to the mathematical terms subalgebra and superalgebra but to the implementation of algebras as subalgebras and parent algebras. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Operations for Algebras} ' \^\ ' The operator '\^' evaluates to the -fold direct product of , viewed as a free -module. | gap> a:= FreeAlgebra( GF(2), 2 ); UnitalAlgebra( GF(2), [ a.1, a.2 ] ) gap> a^2; Module( UnitalAlgebra( GF(2), [ a.1, a.2 ] ), [ [ a.one, a.zero ], [ a.zero, a.one ] ] ) | \vspace{5mm} ' in ' The operator 'in' evaluates to 'true' if is an element of and 'false' otherwise. must be an element of the parent algebra of . | gap> a.1^3 + a.2 in a; true gap> 1 in a; false | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Zero and One for Algebras} 'Zero( )' : \\ returns the additive neutral element of the algebra . 'One( )' : \\ returns the (right and left) multiplicative neutral element of the algebra if this exists, and 'false' otherwise. If is a unital algebra then this element is obtained on raising an arbitrary element to the zero-th power (see "Algebras and Unital Algebras"). | gap> a:= Algebra( Rationals, [ [ [ 1, 0 ], [ 0, 0 ] ] ] ); Algebra( Rationals, [ [ [ 1, 0 ], [ 0, 0 ] ] ] ) gap> Zero( a ); [ [ 0, 0 ], [ 0, 0 ] ] gap> One( a ); [ [ 1, 0 ], [ 0, 0 ] ] gap> a:= UnitalAlgebra( Rationals, [ [ [ 1, 0 ], [ 0, 0 ] ] ] ); UnitalAlgebra( Rationals, [ [ [ 1, 0 ], [ 0, 0 ] ] ] ) gap> Zero( a ); [ [ 0, 0 ], [ 0, 0 ] ] gap> One( a ); [ [ 1, 0 ], [ 0, 1 ] ] | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Set Theoretic Functions for Algebras} As already mentioned in the introduction of the chapter, algebras are domains. Thus all set theoretic functions, for example 'Intersection' and 'Size' can be applied to algebras. All set theoretic functions not mentioned here are not treated specially for algebras. 'Elements( )' : \\ computes the elements of the algebra using a Dimino algorithm. The default function for algebras computes a vector space basis at the same time. 'Intersection( , )' : \\ returns the intersection of and either as set of elements or as an algebra record. 'IsSubset( , )' : \\ If and are algebras then 'IsSubset' tests whether the generators of are elements of . Otherwise 'DomainOps.IsSubset' is used. 'Random( )' : \\ returns a random element of the algebra . This requires the computation of a vector space basis. See also "Functions for Matrix Algebras", "Functions for Finitely Presented Algebras" for the set theoretic functions for the different types of algebras. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Property Tests for Algebras} The following property tests (cf. "Properties and Property Tests") are available for algebras. 'IsAbelian( )' : \\ returns 'true' if the algebra is abelian and 'false' otherwise. An algebra is *abelian* if and only if for every $a, b\in $ the equation $a\* b = b\* a$ holds. 'IsCentral( , )' : \\ returns 'true' if the algebra centralizes the algebra and 'false' otherwise. An algebra *centralizes* an algebra if and only if for all $a\in $ and for all $u\in $ the equation $a\* u = u\* a$ holds. Note that need not to be a subalgebra of but they must have a common parent algebra. 'IsFinite( )' : \\ returns 'true' if the algebra is finite, and 'false' otherwise. 'IsTrivial( )' : \\ returns 'true' if the algebra consists only of the zero element, and 'false' otherwise. If is a unital algebra it is of course never trivial. All tests expect a parent algebra or subalgebra and return 'true' if the algebra has the property and 'false' otherwise. Some functions may not terminate if the given algebra has an infinite set of elements. A warning may be printed in such cases. | gap> IsAbelian( FreeAlgebra( GF(2), 2 ) ); false gap> a:= UnitalAlgebra( Rationals, [ [ [ 1, 0 ], [ 0, 0 ] ] ] ); UnitalAlgebra( Rationals, [ [ [ 1, 0 ], [ 0, 0 ] ] ] ) gap> a.name:= "a";; gap> s1:= Subalgebra( a, [ One(a) ] ); Subalgebra( a, [ [ [ 1, 0 ], [ 0, 1 ] ] ] ) gap> IsCentral( a, s1 ); IsFinite( s1 ); true false gap> s2:= Subalgebra( a, [] ); Subalgebra( a, [ ] ) gap> IsFinite( s2 ); IsTrivial( s2 ); true true | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Vector Space Functions for Algebras} A finite dimensional $F$-algebra $A$ is always a finite dimensional $F$-vector space. Thus in {\GAP}, an algebra is a vector space (see "IsVectorSpace"), and vector space functions such as 'Base' and 'Dimension' are applicable to algebras. | gap> a:= UnitalAlgebra( Rationals, [ [ [ 1, 0 ], [ 0, 0 ] ] ] ); UnitalAlgebra( Rationals, [ [ [ 1, 0 ], [ 0, 0 ] ] ] ) gap> Dimension( a ); 2 gap> Base( a ); [ [ [ 1, 0 ], [ 0, 1 ] ], [ [ 0, 0 ], [ 0, 1 ] ] ] | The vector space structure is used also by the set theoretic functions. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Algebra Functions for Algebras} The functions desribed in this section compute certain subalgebras of a given algebra, e.g., 'Centre' computes the centre of an algebra. They return algebra records as described in "Algebra Records" for the computed subalgebras. Some functions may not terminate if the given algebra has an infinite set of elements, while other functions may signal an error in such cases. Here the term ``subalgebra\'\'\ is used in a mathematical sense. But in {\GAP}, every algebra is either a parent algebra or a subalgebra of a unique parent algebra. If you compute the centre $C$ of an algebra $U$ with parent algebra $A$ then $C$ is a subalgebra of $U$ but its parent algebra is $A$ (see "Parent Algebras and Subalgebras"). 'Centralizer( , )' \\ 'Centralizer( , )' : \\ returns the centralizer of an element in where must be an element of the parent algebra of , resp. the centralizer of the algebra in where both algebras must have a common parent. The *centralizer* of an element in is defined as the set $C$ of elements $c$ of such that and commute. The *centralizer* of an algebra in is defined as the set $C$ of elements $c$ of $A$ such that $c$ commutes with every element of . | gap> a:= MatAlgebra( GF(2), 2 );; gap> a.name:= "a";; gap> m:= [ [ 1, 1 ], [ 0, 1 ] ] * Z(2);; gap> Centralizer( a, m ); UnitalSubalgebra( a, [ [ [ Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0 ] ], [ [ 0*Z(2), Z(2)^0 ], [ 0*Z(2), 0*Z(2) ] ] ] ) | \vspace{5mm} 'Centre( )' : \\ returns the centre of (that is, the centralizer of in ). | gap> c:= Centre( a ); UnitalSubalgebra( a, [ [ [ Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0 ] ] ] ) | \vspace{5mm} 'Closure( , )' \\ 'Closure( , )' Let be an algebra with parent algebra $A$ and let be an element of $A$. Then 'Closure' returns the closure $C$ of and as subalgebra of $A$. The closure $C$ of and is the subalgebra generated by and . Let and be two algebras with a common parent algebra $A$. Then 'Closure' returns the subalgebra of $A$ generated by and . | gap> Closure( c, m ); UnitalSubalgebra( a, [ [ [ Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0 ] ], [ [ Z(2)^0, Z(2)^0 ], [ 0*Z(2), Z(2)^0 ] ] ] ) | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{TrivialSubalgebra} 'TrivialSubalgebra( )' Let be an algebra with parent algebra $A$. Then 'TrivialSubalgebra' returns the trivial subalgebra $T$ of , as subalgebra of $A$. | gap> a:= MatAlgebra( GF(2), 2 );; gap> a.name:= "a";; gap> TrivialSubalgebra( a ); Subalgebra( a, [ ] ) | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Operation for Algebras} 'Operation( , )' Let $A$ be an $F$-algebra for a field $F$, and $M$ an $A$-module of $F$-dimension $n$. With respect to a chosen $F$-basis of $M$, the action of an element of $A$ on $M$ can be described by an $n \times n$ matrix over $F$. This induces an algebra homomorphism from $A$ onto a matrix algebra $A_M$, with action on its natural module equivalent to the action of $A$ on $M$. The matrix algebra $A_M$ can be computed as 'Operation( , )'. \vspace{5mm} 'Operation( , )' returns the operation of the algebra on an -module $M$ with respect to the vector space basis of $M$. Note that contrary to the situation for groups, the operation domains of algebras are not lists of elements but domains. For constructing the algebra homomorphism from $A$ onto $A_M$, and the module homomorphism from $M$ onto the equivalent $A_M$-module, see "OperationHomomorphism for Algebras" and "Module Homomorphisms", respectively. | gap> a:= UnitalAlgebra( Rationals, [ [ [ 1, 0 ], [ 0, 0 ] ] ] );; gap> m:= Module( a, [ [ 1, 0 ] ] );; gap> op:= Operation( a, m ); UnitalAlgebra( Rationals, [ [ [ 1 ] ] ] ) gap> mat1:= PermutationMat( (1,2,3), 3, GF(2) );; gap> mat2:= PermutationMat( (1,2), 3, GF(2) );; gap> u:= Algebra( GF(2), [ mat1, mat2 ] );; u.name:= "u";; gap> nat:= NaturalModule( u );; nat.name:= "nat";; gap> q:= nat / FixedSubmodule( nat );; gap> op1:= Operation( u, q ); UnitalAlgebra( GF(2), [ [ [ 0*Z(2), Z(2)^0 ], [ Z(2)^0, Z(2)^0 ] ], [ [ Z(2)^0, Z(2)^0 ], [ 0*Z(2), Z(2)^0 ] ] ] ) gap> b:= Basis( q, [ [ 0, 1, 1 ], [ 0, 0, 1 ] ] * Z(2) );; gap> op2:= Operation( u, b ); UnitalAlgebra( GF(2), [ [ [ Z(2)^0, Z(2)^0 ], [ Z(2)^0, 0*Z(2) ] ], [ [ Z(2)^0, Z(2)^0 ], [ 0*Z(2), Z(2)^0 ] ] ] ) gap> IsEquivalent( NaturalModule( op1 ), NaturalModule( op2 ) ); true | If the dimension of $M$ is zero then the elements of $A_M$ cannot be represented as {\GAP} matrices. The result is a null algebra, see "NullAlgebra", 'NullAlgebra'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{OperationHomomorphism for Algebras} 'OperationHomomorphism( , )' returns the algebra homomorphism (see "Algebra Homomorphisms") with source and range , provided that is a matrix algebra that was constructed as operation of on a suitable module using 'Operation( , )', see "Operation for Algebras". | gap> ophom:= OperationHomomorphism( a, op ); OperationHomomorphism( UnitalAlgebra( Rationals, [ [ [ 1, 0 ], [ 0, 0 ] ] ] ), UnitalAlgebra( Rationals, [ [ [ 1 ] ] ] ) ) gap> Image( ophom, a.1 ); [ [ 1 ] ] gap> Image( ophom, Zero( a ) ); [ [ 0 ] ] gap> PreImagesRepresentative( ophom, [ [ 2 ] ] ); [ [ 2, 0 ], [ 0, 2 ] ] | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Algebra Homomorphisms}% \index{homomorphisms!of algebras} An *algebra homomorphism* $\phi$ is a mapping that maps each element of an algebra $A$, called the source of $\phi$, to an element of an algebra $B$, called the range of $\phi$, such that for each pair $x, y \in A$ we have $(xy)^\phi = x^\phi y^\phi$ and $(x + y)^\phi = x^\phi + y^\phi$. An algebra homomorphism of unital algebras is *unital* if the zero-th power of elements in the source is mapped to the zero-th power of elements in the range. At the moment, only operation homomorphisms are supported in {\GAP} (see "OperationHomomorphism for Algebras"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Mapping Functions for Algebra Homomorphisms} This section describes how the mapping functions defined in chapter "Mappings" are implemented for algebra homomorphisms. Those functions not mentioned here are implemented by the default functions described in the respective sections. \vspace{5mm} 'Image( )' \\ 'Image( , )' \\ 'Images( , )' The image of a subalgebra under a algebra homomorphism is computed by computing the images of a set of generators of the subalgebra, and the result is the subalgebra generated by those images. \vspace{5mm} 'PreImagesRepresentative( , )' | gap> a:= UnitalAlgebra( Rationals, [ [ [ 1, 0 ], [ 0, 0 ] ] ] );; gap> a.name:= "a";; gap> m:= Module( a, [ [ 1, 0 ] ] );; gap> op:= Operation( a, m ); UnitalAlgebra( Rationals, [ [ [ 1 ] ] ] ) gap> ophom:= OperationHomomorphism( a, op ); OperationHomomorphism( a, UnitalAlgebra( Rationals, [ [ [ 1 ] ] ] ) ) gap> Image( ophom, a.1 ); [ [ 1 ] ] gap> Image( ophom, Zero( a ) ); [ [ 0 ] ] gap> PreImagesRepresentative( ophom, [ [ 2 ] ] ); [ [ 2, 0 ], [ 0, 2 ] ] | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Algebra Elements} \index{equality!of algebra elements}% \index{ordering!of algebra elements}% \index{product!of algebra elements}% \index{product!of list and algebra element}% \index{sum!of algebra elements}% \index{sum!of list and algebra element}% \index{difference!of algebra elements}% \index{difference!of list and algebra element}% \index{power!of algebra elements}% \index{test!for algebra element} This section describes the operations and functions available for algebra elements. Note that algebra elements may exist independently of an algebra, e.g., you can write down two matrices and compute their sum and product without ever defining an algebra that contains them. \vspace{5mm} *Comparisons of Algebra Elements* ' = ': \\ evaluates to 'true' if the algebra elements and are equal and to 'false' otherwise. ' \<> ': \\ evaluates to 'true' if the algebra elements and are not equal and to 'false' otherwise. ' \<\ ' \\ ' \<= ' \\ ' >= ' \\ ' > ' The operators '\<', '\<=', '>=' and '>' evaluate to 'true' if the algebra element is strictly less than, less than or equal to, greater than or equal to and strictly greater than the algebra element . There is no general ordering on all algebra elements, so and should lie in the same parent algebra. Note that for elements of finitely presented algebra, comparison means comparison with respect to the underlying free algebra (see "Elements of Finitely Presented Algebras"). \vspace{5mm} *Arithmetic Operations for Algebra Elements* ' \*\ ' \\ ' + ' \\ ' - ' The operators '\*', '+' and '-' evaluate to the product, sum and difference of the two algebra elements and . The operands must of course lie in a common parent algebra, otherwise an error is signalled. \vspace{5mm} ' / ' returns the quotient of the algebra element by the nonzero element of the base field of the algebra. \vspace{5mm} ' \^\ ' returns the -th power of an algebra element and a positive integer . If is zero or negative, perhaps the result is not defined, or not contained in the algebra generated by . \vspace{5mm} ' + ' \\ ' + ' \\ ' \*\ ' \\ ' \*\ ' In this form the operators '+' and '\*' return a new list where each entry is the sum resp. product of and the corresponding entry of . Of course addition resp. multiplication must be defined between and each entry of . %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{IsAlgebraElement} 'IsAlgebraElement( )' returns 'true' if , which may be an object of arbitrary type, is an algebra element, and 'false' otherwise. The function will signal an error if is an unbound variable. | gap> IsAlgebraElement( (1,2) ); false gap> IsAlgebraElement( NullMat( 2, 2 ) ); true gap> IsAlgebraElement( FreeAlgebra( Rationals, 1 ).1 ); true | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Algebra Records} Algebras and their subalgebras are represented by records. Once an algebra record is created you may add record components to it but you must *not* alter information already present. Algebra records must always contain the components 'isDomain' and 'isAlgebra'. Subalgebras contain an additional component 'parent'. The components 'generators', 'zero' and 'one' are not necessarily contained. The contents of important record components of an algebra $A$ is described below. The *category components* are 'isDomain': \\ is 'true'. 'isAlgebra': \\ is 'true'. 'isUnitalAlgebra': \\ is present (and then 'true') if $A$ is a unital algebra. The *identification components* are 'field': \\ is the coefficient field of $A$. 'generators': \\ is a list of algebra generators. Duplicate generators are allowed, also the algebra zero may be among the generators. Note that once created this entry must never be changed, as most of the other entries depend on 'generators'. If 'generators' is not bound it can be computed using 'Generators'. 'parent': \\ if present this contains the algebra record of the parent algebra of a subalgebra $A$, otherwise $A$ itself is a parent algebra. 'zero': \\ is the additive neutral element of $A$, can be computed using 'Zero'. The component 'operations' contains the *operations record* of $A$. This will usually be one of 'AlgebraOps', 'UnitalAlgebraOps', or a record for more specific algebras. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{FFList} 'FFList( )' returns for a finite field a list of all elements of in an ordering that is compatible with the ordering of field elements in the {\MeatAxe} share library (see chapter "The MeatAxe"). The element of corresponding to the number is '[ +1 ]', and the canonical number of the field element is 'Position( , ) -1'. | gap> FFList( GF( 8 ) ); [ 0*Z(2), Z(2)^0, Z(2^3), Z(2^3)^3, Z(2^3)^2, Z(2^3)^6, Z(2^3)^4, Z(2^3)^5 ] | (This program was originally written by Meinolf Geck.) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %E Emacs . . . . . . . . . . . . . . . . . . . . . . . local emacs variables %% %% Local Variables: %% mode: outline %% outline-regexp: "%F\\|%V\\|%E" %% fill-column: 73 %% fill-prefix: "%% " %% eval: (hide-body) %% End: