GAP (ref) - Chapter 62: Algebras

An algebra is a vector space equipped with a bilinear map (multiplication). This chapter describes the functions in GAP that deal with general algebras and associative algebras.

Algebras in GAP are vector spaces in a natural way. So all the functionality for vector spaces (see Chapter 61) is also applicable to algebras.

62.1 InfoAlgebra (Info Class)

62.1-1 InfoAlgebra

62.2 Constructing Algebras by Generators

62.2-1 Algebra

Algebra( F, gens ) is the algebra over the division ring F, generated by the vectors in the list gens.

If there are three arguments, a division ring F and a list gens and an element zero, then Algebra( F, gens, zero ) is the F-algebra generated by gens, with zero element zero.

If the last argument is the string "basis" then the vectors in gens are known to form a basis of the algebra (as an F-vector space).

gap> m:= [ [ 0, 1, 2 ], [ 0, 0, 3], [ 0, 0, 0 ] ];;
gap> A:= Algebra( Rationals, [ m ] );
<algebra over Rationals, with 1 generator>
gap> Dimension( A );
2

62.2-2 AlgebraWithOne

AlgebraWithOne( F, gens ) is the algebra-with-one over the division ring F, generated by the vectors in the list gens.

If there are three arguments, a division ring F and a list gens and an element zero, then AlgebraWithOne( F, gens, zero ) is the F-algebra-with-one generated by gens, with zero element zero.

If the last argument is the string "basis" then the vectors in gens are known to form a basis of the algebra (as an F-vector space).

gap> m:= [ [ 0, 1, 2 ], [ 0, 0, 3], [ 0, 0, 0 ] ];;
gap> A:= AlgebraWithOne( Rationals, [ m ] );
<algebra-with-one over Rationals, with 1 generator>
gap> Dimension( A );
3
gap> One(A);
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]

62.3 Constructing Algebras as Free Algebras

62.3-1 FreeAlgebra

is a free (nonassociative) algebra of rank rank over the division ring R. Here name, and name1, name2, ... are optional strings that can be used to provide names for the generators.

gap> A:= FreeAlgebra( Rationals, "a", "b" );
<algebra over Rationals, with 2 generators>
gap> g:= GeneratorsOfAlgebra( A );
[ (1)*a, (1)*b ]
gap> (g[1]*g[2])*((g[2]*g[1])*g[1]);
(1)*((a*b)*((b*a)*a))

62.3-2 FreeAlgebraWithOne

is a free (nonassociative) algebra-with-one of rank rank over the division ring R. Here name, and name1, name2, ... are optional strings that can be used to provide names for the generators.

gap> A:= FreeAlgebraWithOne( Rationals, 4, "q" );
<algebra-with-one over Rationals, with 4 generators>
gap> GeneratorsOfAlgebra( A );
[ (1)*<identity ...>, (1)*q.1, (1)*q.2, (1)*q.3, (1)*q.4 ]
gap> One( A );
(1)*<identity ...>

62.3-3 FreeAssociativeAlgebra

is a free associative algebra of rank rank over the division ring R. Here name, and name1, name2, ... are optional strings that can be used to provide names for the generators.

gap> A:= FreeAssociativeAlgebra( GF( 5 ), 4, "a" );
<algebra over GF(5), with 4 generators>

62.3-4 FreeAssociativeAlgebraWithOne

is a free associative algebra-with-one of rank rank over the division ring R. Here name, and name1, name2, ... are optional strings that can be used to provide names for the generators.

gap> A:= FreeAssociativeAlgebraWithOne( Rationals, "a", "b", "c" );
<algebra-with-one over Rationals, with 3 generators>
gap> GeneratorsOfAlgebra( A );
[ (1)*<identity ...>, (1)*a, (1)*b, (1)*c ]
gap> One( A );
(1)*<identity ...>

62.4 Constructing Algebras by Structure Constants

For an introduction into structure constants and how they are handled by GAP, we refer to Section Tutorial: Algebras of the user's tutorial.

62.4-1 AlgebraByStructureConstants

returns a free left module \(A\) over the division ring R, with multiplication defined by the structure constants table sctable. The optional argument nameinfo can be used to prescribe names for the elements of the canonical basis of \(A\); it can be either a string name (then name1, name2 etc. are chosen) or a list of strings which are then chosen. The vectors of the canonical basis of \(A\) correspond to the vectors of the basis given by sctable.

It is not checked whether the coefficients in sctable are really elements in R.

gap> T:= EmptySCTable( 2, 0 );;
gap> SetEntrySCTable( T, 1, 1, [ 1/2, 1, 2/3, 2 ] );
gap> A:= AlgebraByStructureConstants( Rationals, T );
<algebra of dimension 2 over Rationals>
gap> b:= BasisVectors( Basis( A ) );;
gap> b[1]^2;
(1/2)*v.1+(2/3)*v.2
gap> b[1]*b[2];
0*v.1

62.4-2 AlgebraWithOneByStructureConstants

The only differences between this function and AlgebraByStructureConstants (62.4-1) are that AlgebraWithOneByStructureConstants takes an additional argument onecoeffs, the coefficients vector over the ring R that describes the unique multiplicative identity element of the returned algebra w. r. t. the defining basis of this algebra, and that the returned algebra is an algebra-with-one (see IsAlgebraWithOne (62.8-4)).

gap> A:= GF(2)^[2,2];;
gap> B:= Basis( A );;
gap> onecoeffs:= Coefficients( B, One( A ) );
[ Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0 ]
gap> T:= StructureConstantsTable( B );;
gap> sc1:= AlgebraByStructureConstants( GF(2), T );
<algebra of dimension 4 over GF(2)>
gap> HasOne( sc1 );
false
gap> One( sc1 );
v.1+v.4
gap> sc2:= AlgebraWithOneByStructureConstants( GF(2), T, onecoeffs );
<algebra-with-one of dimension 4 over GF(2)>
gap> HasOne( sc2 );
true
gap> One( sc2 );
v.1+v.4

62.4-3 StructureConstantsTable

Let B be a basis of a free left module \(R\) that is also a ring. In this case StructureConstantsTable returns a structure constants table \(T\) in sparse representation, as used for structure constants algebras (see Section Tutorial: Algebras of the GAP User's Tutorial).

If B has length \(n\) then \(T\) is a list of length \(n+2\). The first \(n\) entries of \(T\) are lists of length \(n\). \(T[ n+1 ]\) is one of \(1\), \(-1\), or \(0\); in the case of \(1\) the table is known to be symmetric, in the case of \(-1\) it is known to be antisymmetric, and \(0\) occurs in all other cases. \(T[ n+2 ]\) is the zero element of the coefficient domain.

The coefficients w.r.t. B of the product of the \(i\)-th and \(j\)-th basis vector of B are stored in \(T[i][j]\) as a list of length \(2\); its first entry is the list of positions of nonzero coefficients, the second entry is the list of these coefficients themselves.

The multiplication in an algebra \(A\) with vector space basis B with basis vectors \([ v_1, \ldots, v_n ]\) is determined by the so-called structure matrices \(M_k = [ m_{ijk} ]_{ij}\), \(1 \leq k \leq n\). The \(M_k\) are defined by \(v_i v_j = \sum_k m_{ijk} v_k\). Let \(a = [ a_1, \ldots, a_n ]\) and \(b = [ b_1, \ldots, b_n ]\). Then

\[ \left( \sum_i a_i v_i \right) \left( \sum_j b_j v_j \right) = \sum_{{i,j}} a_i b_j \left( v_i v_j \right) = \sum_k \left( \sum_j \left( \sum_i a_i m_{ijk} \right) b_j \right) v_k = \sum_k \left( a M_k b^{tr} \right) v_k. \]

gap> A:= QuaternionAlgebra( Rationals );;
gap> StructureConstantsTable( Basis( A ) );
[ [ [ [ 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 ]

62.4-4 EmptySCTable

EmptySCTable returns a structure constants table for an algebra of dimension dim, describing trivial multiplication. zero 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 flag, which must be one of the strings "symmetric", "antisymmetric".

gap> EmptySCTable( 2, Zero( GF(5) ), "antisymmetric" );
[ [ [ [  ], [  ] ], [ [  ], [  ] ] ],
  [ [ [  ], [  ] ], [ [  ], [  ] ] ], -1, 0*Z(5) ]

62.4-5 SetEntrySCTable

sets the entry of the structure constants table T that describes the product of the i-th basis element with the j-th basis element to the value given by the list list.

If T is known to be antisymmetric or symmetric then also the value T[j][i] is set.

list must be of the form \([ c_{ij}^{{k_1}}, k_1, c_{ij}^{{k_2}}, k_2, \ldots ]\).

The entries at the odd positions of list must be compatible with the zero element stored in T. 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.

gap> T:= EmptySCTable( 2, 0 );;
gap> SetEntrySCTable( T, 1, 1, [ 1/2, 1, 2/3, 2 ] );
gap> T;
[ [ [ [ 1, 2 ], [ 1/2, 2/3 ] ], [ [  ], [  ] ] ],
  [ [ [  ], [  ] ], [ [  ], [  ] ] ], 0, 0 ]

62.4-6 GapInputSCTable

is a string that describes the structure constants table T in terms of EmptySCTable (62.4-4) and SetEntrySCTable (62.4-5). The assignments are made to the variable varname.

gap> T:= EmptySCTable( 2, 0 );;
gap> SetEntrySCTable( T, 1, 2, [ 1, 2 ] );
gap> SetEntrySCTable( T, 2, 1, [ 1, 2 ] );
gap> GapInputSCTable( T, "T" );
"T:= EmptySCTable( 2, 0 );\nSetEntrySCTable( T, 1, 2, [1,2] );\nSetEnt\
rySCTable( T, 2, 1, [1,2] );\n"

62.4-7 TestJacobi

tests whether the structure constants table T satisfies the Jacobi identity \(v_i * (v_j * v_k) + v_j * (v_k * v_i) + v_k * (v_i * v_j) = 0\) for all basis vectors \(v_i\) of the underlying algebra, where \(i \leq j \leq k\). (Thus antisymmetry is assumed.)

The function returns true if the Jacobi identity is satisfied, and a failing triple \([ i, j, k ]\) otherwise.

gap> T:= EmptySCTable( 2, 0, "antisymmetric" );;
gap> SetEntrySCTable( T, 1, 2, [ 1, 2 ] );;
gap> TestJacobi( T );
true

62.4-8 IdentityFromSCTable

Let T be a structure constants table of an algebra \(A\) of dimension \(n\). IdentityFromSCTable( T ) is either fail or the vector of length \(n\) that contains the coefficients of the multiplicative identity of \(A\) with respect to the basis that belongs to T.

gap> T:= EmptySCTable( 2, 0 );;
gap> SetEntrySCTable( T, 1, 1, [ 1, 1 ] );;
gap> SetEntrySCTable( T, 1, 2, [ 1, 2 ] );;
gap> SetEntrySCTable( T, 2, 1, [ 1, 2 ] );;
gap> IdentityFromSCTable( T );
[ 1, 0 ]

62.4-9 QuotientFromSCTable

Let T be a structure constants table of an algebra \(A\) of dimension \(n\). QuotientFromSCTable( T ) is either fail or the vector of length \(n\) that contains the coefficients of the quotient of num and den with respect to the basis that belongs to T.

We solve the equation system num\( = x *\) den. If no solution exists, fail is returned.

In terms of the basis \(B\) with vectors \(b_1, \ldots, b_n\) this means for \(\textit{num} = \sum_{{i = 1}}^n a_i b_i\), \(\textit{den} = \sum_{{i = 1}}^n c_i b_i\), \(x = \sum_{{i = 1}}^n x_i b_i\) that \(a_k = \sum_{{i,j}} c_i x_j c_{ijk}\) for all \(k\). Here \(c_{ijk}\) denotes the structure constants with respect to \(B\). This means that (as a vector) \(a = x M\) with \(M_{jk} = \sum_{{i = 1}}^n c_{ijk} c_i\).

gap> T:= EmptySCTable( 2, 0 );;
gap> SetEntrySCTable( T, 1, 1, [ 1, 1 ] );;
gap> SetEntrySCTable( T, 2, 1, [ 1, 2 ] );;
gap> SetEntrySCTable( T, 1, 2, [ 1, 2 ] );;
gap> QuotientFromSCTable( T, [0,1], [1,0] );
[ 0, 1 ]

62.5 Some Special Algebras

62.5-1 QuaternionAlgebra

Let F be a field or a list of field elements, let \(F\) be the field generated by F, and let a and b two elements in \(F\). QuaternionAlgebra returns a quaternion algebra over \(F\), with parameters a and b, i.e., a four-dimensional associative \(F\)-algebra with basis \((e,i,j,k)\) and multiplication defined by \(e e = e\), \(e i = i e = i\), \(e j = j e = j\), \(e k = k e = k\), \(i i = \textit{a} e\), \(i j = - j i = k\), \(i k = - k i = \textit{a} j\), \(j j = \textit{b} e\), \(k j = - j k = \textit{b} i\), \(k k = - \textit{a} \textit{b} e\). The default value for both a and b is \(-1 \in F\).

The GeneratorsOfAlgebra (62.9-1) and CanonicalBasis (61.5-3) value of an algebra constructed with QuaternionAlgebra is the list \([ e, i, j, k ]\).

Two quaternion algebras with the same parameters a, b 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 QuaternionAlgebra are cached, in the global variable QuaternionAlgebraData.)

The embedding of the field GaussianRationals (60.1-3) into a quaternion algebra \(A\) over Rationals (17.1-1) is not uniquely determined. One can specify one embedding as a vector space homomorphism that maps 1 to the first algebra generator of \(A\), and E(4) to one of the others.

gap> QuaternionAlgebra( Rationals );
<algebra-with-one of dimension 4 over Rationals>

62.5-2 ComplexificationQuat

Let \(A = e F \oplus i F \oplus j F \oplus k F\) be a quaternion algebra over the field \(F\) of cyclotomics, with basis \((e,i,j,k)\).

If \(v = v_1 + v_2 j\) is a row vector over \(A\) with \(v_1 = e w_1 + i w_2\) and \(v_2 = e w_3 + i w_4\) then ComplexificationQuat called with argument \(v\) returns the concatenation of \(w_1 + \)E(4)\( w_2\) and \(w_3 + \)E(4)\( w_4\).

If \(M = M_1 + M_2 j\) is a matrix over \(A\) with \(M_1 = e N_1 + i N_2\) and \(M_2 = e N_3 + i N_4\) then ComplexificationQuat called with argument \(M\) returns the block matrix \(A\) over \(e F \oplus i F\) such that \(A(1,1) = N_1 + \)E(4)\( N_2\), \(A(2,2) = N_1 - \)E(4)\( N_2\), \(A(1,2) = N_3 + \)E(4)\( N_4\), and \(A(2,1) = - N_3 + \)E(4)\( N_4\).

Then ComplexificationQuat(v) * ComplexificationQuat(M)= ComplexificationQuat(v * M), since

\[ v M = v_1 M_1 + v_2 j M_1 + v_1 M_2 j + v_2 j M_2 j = ( v_1 M_1 - v_2 \overline{{M_2}} ) + ( v_1 M_2 + v_2 \overline{{M_1}} ) j. \]

62.5-3 OctaveAlgebra

gap> OctaveAlgebra( Rationals );
<algebra of dimension 8 over Rationals>

62.5-4 FullMatrixAlgebra

is the full matrix algebra of \(\textit{n} \times \textit{n}\) matrices over the ring R, for a nonnegative integer n.

gap> A:=FullMatrixAlgebra( Rationals, 20 );
( Rationals^[ 20, 20 ] )
gap> Dimension( A );
400

62.5-5 NullAlgebra

gap> A:= NullAlgebra( Rationals );
<algebra of dimension 0 over Rationals>
gap> Dimension( A );
0

62.6 Subalgebras

62.6-1 Subalgebra

is the \(F\)-algebra generated by gens, with parent algebra A, where \(F\) is the left acting domain of A.

Note that being a subalgebra of A means to be an algebra, to be contained in A, and to have the same left acting domain as A.

An optional argument "basis" may be added if it is known that the generators already form a basis of the algebra. Then it is not checked whether gens really are linearly independent and whether all elements in gens lie in A.

gap> m:= [ [ 0, 1, 2 ], [ 0, 0, 3], [ 0, 0, 0 ] ];;
gap> A:= Algebra( Rationals, [ m ] );
<algebra over Rationals, with 1 generator>
gap> B:= Subalgebra( A, [ m^2 ] );
<algebra over Rationals, with 1 generator>

62.6-2 SubalgebraNC

SubalgebraNC does the same as Subalgebra (62.6-1), except that it does not check whether all elements in gens lie in A.

gap> m:= RandomMat( 3, 3 );;
gap> A:= Algebra( Rationals, [ m ] );
<algebra over Rationals, with 1 generator>
gap> SubalgebraNC( A, [ IdentityMat( 3, 3 ) ], "basis" );
<algebra of dimension 1 over Rationals>

62.6-3 SubalgebraWithOne

The optional third argument, the string "basis", may be added if it is known that the elements from gens are linearly independent. Then it is not checked whether gens really are linearly independent and whether all elements in gens lie in A.

gap> m:= [ [ 0, 1, 2 ], [ 0, 0, 3], [ 0, 0, 0 ] ];;
gap> A:= AlgebraWithOne( Rationals, [ m ] );
<algebra-with-one over Rationals, with 1 generator>
gap> B1:= SubalgebraWithOne( A, [ m ] );;
gap> B2:= Subalgebra( A, [ m ] );;
gap> Dimension( B1 );
3
gap> Dimension( B2 );
2

62.6-4 SubalgebraWithOneNC

SubalgebraWithOneNC does the same as SubalgebraWithOne (62.6-3), except that it does not check whether all elements in gens lie in A.

gap> m:= RandomMat( 3, 3 );; A:= Algebra( Rationals, [ m ] );;
gap> SubalgebraWithOneNC( A, [ m ] );
<algebra-with-one over Rationals, with 1 generator>

62.6-5 TrivialSubalgebra

gap> A:= QuaternionAlgebra( Rationals );;
gap> B:= TrivialSubalgebra( A );
<algebra of dimension 0 over Rationals>
gap> Dimension( B );
0

62.7 Ideals of Algebras

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 56. Here we give examples demonstrating the use of ideals in algebras. For an introduction into the construction of quotient algebras we refer to Chapter Tutorial: Algebras of the user's tutorial.

gap> m:= [ [ 0, 2, 3 ], [ 0, 0, 4 ], [ 0, 0, 0] ];;
gap> A:= AlgebraWithOne( Rationals, [ m ] );;
gap> I:= Ideal( A, [ m ] );  # the two-sided ideal of `A' generated by `m'
<two-sided ideal in <algebra-with-one of dimension 3 over Rationals>,
  (1 generator)>
gap> Dimension( I );
2
gap> GeneratorsOfIdeal( I );
[ [ [ 0, 2, 3 ], [ 0, 0, 4 ], [ 0, 0, 0 ] ] ]
gap> BasisVectors( Basis( I ) );
[ [ [ 0, 1, 3/2 ], [ 0, 0, 2 ], [ 0, 0, 0 ] ],
  [ [ 0, 0, 1 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ] ]
gap> A:= FullMatrixAlgebra( Rationals, 4 );;
gap> m:= NullMat( 4, 4 );; m[1][4]:=1;;
gap> I:= LeftIdeal( A, [ m ] );
<left ideal in ( Rationals^[ 4, 4 ] ), (1 generator)>
gap> Dimension( I );
4
gap> GeneratorsOfLeftIdeal( I );
[ [ [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ] ]
gap> mats:= [ [[1,0],[0,0]], [[0,1],[0,0]], [[0,0],[0,1]] ];;
gap> A:= Algebra( Rationals, mats );;
gap> # Form the two-sided ideal for which `mats[2]' is known to be
gap> # the unique basis element.
gap> I:= Ideal( A, [ mats[2] ], "basis" );
<two-sided ideal in <algebra of dimension 3 over Rationals>,
  (dimension 1)>

62.8 Categories and Properties of Algebras

62.8-1 IsFLMLOR

A FLMLOR (free left module left operator ring) in GAP is a ring that is also a free left module.

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.

gap> A:= FullMatrixAlgebra( Rationals, 2 );;
gap> IsFLMLOR ( A );
true

62.8-2 IsFLMLORWithOne

A FLMLOR-with-one in GAP is a ring-with-one that is also a free left module.

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.

Examples are magma rings-with-one or algebras-with-one (but also over the integers).

gap> A:= FullMatrixAlgebra( Rationals, 2 );;
gap> IsFLMLORWithOne ( A );
true

62.8-3 IsAlgebra

An algebra in GAP 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.

gap> A:= MatAlgebra( Rationals, 3 );;
gap> IsAlgebra( A );
true

62.8-4 IsAlgebraWithOne

An algebra-with-one in GAP 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.

gap> A:= MatAlgebra( Rationals, 3 );;
gap> IsAlgebraWithOne( A );
true

62.8-5 IsLieAlgebra

An algebra A is called Lie algebra if \(a * a = 0\) for all \(a\) in A and \(( a * ( b * c ) ) + ( b * ( c * a ) ) + ( c * ( a * b ) ) = 0\) for all \(a, b, c \in \)A (Jacobi identity).

gap> A:= FullMatrixLieAlgebra( Rationals, 3 );;
gap> IsLieAlgebra( A );
true

62.8-6 IsSimpleAlgebra

is true if the algebra A is simple, and false otherwise. This function is only implemented for the cases where A 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.

gap> A:= FullMatrixLieAlgebra( Rationals, 3 );;
gap> IsSimpleAlgebra( A );
false
gap> A:= MatAlgebra( Rationals, 3 );;
gap> IsSimpleAlgebra( A );
true

62.8-7 IsFiniteDimensional

returns true (always) for a matrix algebra matalg, since matrix algebras are always finite dimensional.

gap> A:= MatAlgebra( Rationals, 3 );;
gap> IsFiniteDimensional( A );
true

62.8-8 IsQuaternion

IsQuaternion is the category of elements in an algebra constructed by QuaternionAlgebra (62.5-1). A collection of quaternions lies in the category IsQuaternionCollection. Finally, a collection of quaternion collections (e.g., a matrix of quaternions) lies in the category IsQuaternionCollColl.

gap> A:= QuaternionAlgebra( Rationals );;
gap> b:= BasisVectors( Basis( A ) );
[ e, i, j, k ]
gap> IsQuaternion( b[1] );
true
gap> IsQuaternionCollColl( [ [ b[1], b[2] ], [ b[3], b[4] ] ] );
true

62.9 Attributes and Operations for Algebras

62.9-1 GeneratorsOfAlgebra

For a free algebra, each generator can also be accessed using the . operator (see GeneratorsOfDomain (31.9-2)).

gap> m:= [ [ 0, 1, 2 ], [ 0, 0, 3 ], [ 0, 0, 0 ] ];;
gap> A:= AlgebraWithOne( Rationals, [ m ] );
<algebra-with-one over Rationals, with 1 generator>
gap> GeneratorsOfAlgebra( A );
[ [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
  [ [ 0, 1, 2 ], [ 0, 0, 3 ], [ 0, 0, 0 ] ] ]

62.9-2 GeneratorsOfAlgebraWithOne

For a free algebra with one, each generator can also be accessed using the . operator (see GeneratorsOfDomain (31.9-2)).

gap> m:= [ [ 0, 1, 2 ], [ 0, 0, 3 ], [ 0, 0, 0 ] ];;
gap> A:= AlgebraWithOne( Rationals, [ m ] );
<algebra-with-one over Rationals, with 1 generator>
gap> GeneratorsOfAlgebraWithOne( A );
[ [ [ 0, 1, 2 ], [ 0, 0, 3 ], [ 0, 0, 0 ] ] ]

62.9-3 ProductSpace

is the vector space \(\langle u * v ; u \in U, v \in V \rangle\), where \(U\) and \(V\) are subspaces of the same algebra.

If \(\textit{U} = \textit{V}\) is known to be an algebra then the product space is also an algebra, moreover it is an ideal in U. If U and V are known to be ideals in an algebra \(A\) then the product space is known to be an algebra and an ideal in \(A\).

gap> A:= QuaternionAlgebra( Rationals );;
gap> b:= BasisVectors( Basis( A ) );;
gap> B:= Subalgebra( A, [ b[4] ] );
<algebra over Rationals, with 1 generator>
gap> ProductSpace( A, B );
<vector space of dimension 4 over Rationals>

62.9-4 PowerSubalgebraSeries

returns a list of subalgebras of A, the first term of which is A; and every next term is the product space of the previous term with itself.

gap> A:= QuaternionAlgebra( Rationals );
<algebra-with-one of dimension 4 over Rationals>
gap> PowerSubalgebraSeries( A );
[ <algebra-with-one of dimension 4 over Rationals> ]

62.9-5 AdjointBasis

The adjoint map \(ad(x)\) of an element \(x\) in an \(F\)-algebra \(A\) is the left multiplication by \(x\). This map is \(F\)-linear and thus, w.r.t. the given basis B\( = (x_1, x_2, \ldots, x_n)\) of \(A\), \(ad(x)\) can be represented by a matrix over \(F\). Let \(V\) denote the \(F\)-vector space of the matrices corresponding to \(ad(x)\), for \(x \in A\). Then AdjointBasis returns the basis of \(V\) that consists of the matrices for \(ad(x_1), \ldots, ad(x_n)\).

gap> A:= QuaternionAlgebra( Rationals );;
gap> AdjointBasis( Basis( A ) );
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 ]
     ] ] )

62.9-6 IndicesOfAdjointBasis

Let A be an algebra and let B be the basis that is output by AdjointBasis( Basis( A ) ). This function returns a list of indices. If \(i\) is an index belonging to this list, then \(ad x_i\) is a basis vector of the matrix space spanned by \(ad A\), where \(x_i\) is the \(i\)-th basis vector of the basis B.

gap> L:= FullMatrixLieAlgebra( Rationals, 3 );;
gap> B:= AdjointBasis( Basis( L ) );;
gap> IndicesOfAdjointBasis( B );
[ 1, 2, 3, 4, 5, 6, 7, 8 ]

62.9-7 AsAlgebra

gap> V:= VectorSpace( Rationals, [ IdentityMat( 2 ) ] );;
gap> AsAlgebra( Rationals, V );
<algebra of dimension 1 over Rationals>

62.9-8 AsAlgebraWithOne

If the algebra A has an identity, then it can be viewed as an algebra with one over F. This function returns this algebra with one.

gap> V:= VectorSpace( Rationals, [ IdentityMat( 2 ) ] );;
gap> A:= AsAlgebra( Rationals, V );;
gap> AsAlgebraWithOne( Rationals, A );
<algebra-with-one over Rationals, with 1 generator>

62.9-9 AsSubalgebra

If all elements of the algebra B happen to be contained in the algebra A, then B can be viewed as a subalgebra of A. This function returns this subalgebra.

gap> A:= FullMatrixAlgebra( Rationals, 2 );;
gap> V:= VectorSpace( Rationals, [ IdentityMat( 2 ) ] );;
gap> B:= AsAlgebra( Rationals, V );;
gap> BA:= AsSubalgebra( A, B );
<algebra of dimension 1 over Rationals>

62.9-10 AsSubalgebraWithOne

If B is an algebra with one, all elements of which happen to be contained in the algebra with one A, then B can be viewed as a subalgebra with one of A. This function returns this subalgebra with one.

gap> A:= FullMatrixAlgebra( Rationals, 2 );;
gap> V:= VectorSpace( Rationals, [ IdentityMat( 2 ) ] );;
gap> B:= AsAlgebra( Rationals, V );;
gap> C:= AsAlgebraWithOne( Rationals, B );;
gap> AC:= AsSubalgebraWithOne( A, C );
<algebra-with-one over Rationals, with 1 generator>

62.9-11 MutableBasisOfClosureUnderAction

Let F be a ring, Agens a list of generators for an F-algebra \(A\), and from one of "left", "right", "both"; this means that elements of \(A\) act via multiplication from the respective side(s). init must be a list of initial generating vectors, and opr the operation (a function of two arguments).

MutableBasisOfClosureUnderAction returns a mutable basis of the F-free left module generated by the vectors in init and their images under the action of Agens from the respective side(s).

zero is the zero element of the desired module. maxdim is an upper bound for the dimension of the closure; if no such upper bound is known then the value of maxdim must be infinity (18.2-1).

MutableBasisOfClosureUnderAction can be used to compute a basis of an associative algebra generated by the elements in Agens. In this case from may be "left" or "right", opr is the multiplication *, and init 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 init, whereas of course Agens need not contain the identity.)

(Note that bases of not necessarily associative algebras can be computed using MutableBasisOfNonassociativeAlgebra (62.9-12).)

Other applications of MutableBasisOfClosureUnderAction are the computations of bases for (left/ right/ two-sided) ideals \(I\) in an associative algebra \(A\) from ideal generators of \(I\); in these cases Agens is a list of algebra generators of \(A\), from denotes the appropriate side(s), init is a list of ideal generators of \(I\), and opr is again *.

(Note that bases of ideals in not necessarily associative algebras can be computed using MutableBasisOfIdealInNonassociativeAlgebra (62.9-13).)

Finally, bases of right \(A\)-modules also can be computed using MutableBasisOfClosureUnderAction. The only difference to the ideal case is that init is now a list of right module generators, and opr is the operation of the module.

gap> A:= QuaternionAlgebra( Rationals );;
gap> g:= GeneratorsOfAlgebra( A );;
gap> B:= MutableBasisOfClosureUnderAction( Rationals,
>                                g, "left", [ g[1] ], \*, Zero(A), 4 );
<mutable basis over Rationals, 4 vectors>
gap> BasisVectors( B );
[ e, i, j, k ]

62.9-12 MutableBasisOfNonassociativeAlgebra

is a mutable basis of the (not necessarily associative) F-algebra that is generated by Agens, has zero element zero, and has dimension at most maxdim. If no finite bound for the dimension is known then infinity (18.2-1) must be the value of maxdim.

The difference to MutableBasisOfClosureUnderAction (62.9-11) 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.)

gap> L:= FullMatrixLieAlgebra( Rationals, 4 );;
gap> m1:= Random( L );;
gap> m2:= Random( L );;
gap> MutableBasisOfNonassociativeAlgebra( Rationals, [ m1, m2 ],
> Zero( L ), 16 );
<mutable basis over Rationals, 16 vectors>

62.9-13 MutableBasisOfIdealInNonassociativeAlgebra

is a mutable basis of the ideal generated by Igens under the action of the (not necessarily associative) F-algebra with vector space generators Vgens. The zero element of the ideal is zero, from is one of "left", "right", "both" (with the same meaning as in MutableBasisOfClosureUnderAction (62.9-11)), and maxdim is a known upper bound on the dimension of the ideal; if no finite bound for the dimension is known then infinity (18.2-1) must be the value of maxdim.

gap> mats:= [  [[ 1, 0 ], [ 0, -1 ]], [[0,1],[0,0]] ];;
gap> A:= Algebra( Rationals, mats );;
gap> basA:= BasisVectors( Basis( A ) );;
gap> B:= MutableBasisOfIdealInNonassociativeAlgebra( Rationals, basA,
> [ mats[2] ], 0*mats[1], "both", infinity );
<mutable basis over Rationals, 1 vector>
gap> BasisVectors( B );
[ [ [ 0, 1 ], [ 0, 0 ] ] ]

62.9-14 DirectSumOfAlgebras

is the direct sum of the two algebras A1 and A2 respectively of the algebras in the list list.

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.

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.

gap> A:= QuaternionAlgebra( Rationals );;
gap> DirectSumOfAlgebras( [A, A, A] );
<algebra of dimension 12 over Rationals>

62.9-15 FullMatrixAlgebraCentralizer

Let lst be a nonempty list of square matrices of the same dimension \(n\) with entries in the field F. FullMatrixAlgebraCentralizer returns the (pointwise) centralizer of all matrices in lst, inside the full matrix algebra of \(n \times n\) matrices over F.

gap> A:= QuaternionAlgebra( Rationals );;
gap> b:= Basis( A );;
gap> mats:= List( BasisVectors( b ), x -> AdjointMatrix( b, x ) );;
gap> FullMatrixAlgebraCentralizer( Rationals, mats );
<algebra-with-one of dimension 4 over Rationals>

62.9-16 RadicalOfAlgebra

gap> m:= [ [ 0, 1, 2 ], [ 0, 0, 3 ], [ 0, 0, 0 ] ];;
gap> A:= AlgebraWithOneByGenerators( Rationals, [ m ] );
<algebra-with-one over Rationals, with 1 generator>
gap> RadicalOfAlgebra( A );
<algebra of dimension 2 over Rationals>

62.9-17 CentralIdempotentsOfAlgebra

For an associative algebra A, this function returns a list of central primitive idempotents such that their sum is the identity element of A. Therefore A is required to have an identity.

gap> A:= QuaternionAlgebra( Rationals );;
gap> B:= DirectSumOfAlgebras( [A, A, A] );
<algebra of dimension 12 over Rationals>
gap> CentralIdempotentsOfAlgebra( B );
[ v.9, v.5, v.1 ]

62.9-18 DirectSumDecomposition

This function calculates a list of ideals of the algebra L such that L is equal to their direct sum. Currently this is only implemented for semisimple associative algebras, and for Lie algebras (semisimple or not).

gap> G:= SymmetricGroup( 4 );;
gap> A:= GroupRing( Rationals, G );
<algebra-with-one over Rationals, with 2 generators>
gap> dd:= DirectSumDecomposition( A );
[ <two-sided ideal in
      <algebra-with-one of dimension 24 over Rationals>,
      (1 generator)>,
  <two-sided ideal in
      <algebra-with-one of dimension 24 over Rationals>,
      (1 generator)>,
  <two-sided ideal in
      <algebra-with-one of dimension 24 over Rationals>,
      (1 generator)>,
  <two-sided ideal in
      <algebra-with-one of dimension 24 over Rationals>,
      (1 generator)>,
  <two-sided ideal in
      <algebra-with-one of dimension 24 over Rationals>,
      (1 generator)> ]
gap> List( dd, Dimension );
[ 1, 1, 4, 9, 9 ]

gap> L:= FullMatrixLieAlgebra( Rationals, 5 );;
gap> DirectSumDecomposition( L );
[ <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)> ]

62.9-19 LeviMalcevDecomposition

A Levi-Malcev subalgebra of the algebra L is a semisimple subalgebra complementary to the radical of L. 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.

gap> m:= [ [ 1, 2, 0 ], [ 0, 1, 3 ], [ 0, 0, 1] ];;
gap> A:= Algebra( Rationals, [ m ] );;
gap> LeviMalcevDecomposition( A );
[ <algebra of dimension 1 over Rationals>,
  <algebra of dimension 2 over Rationals> ]

gap> L:= FullMatrixLieAlgebra( Rationals, 5 );;
gap> LeviMalcevDecomposition( L );
[ <Lie algebra of dimension 24 over Rationals>,
  <two-sided ideal in <Lie algebra of dimension 25 over Rationals>,
      (dimension 1)> ]

62.9-20 Grading

Let \(G\) be an Abelian group and \(A\) an algebra. Then \(A\) is said to be graded over \(G\) if for every \(g \in G\) there is a subspace \(A_g\) of \(A\) such that \(A_g \cdot A_h \subset A_{{g+h}}\) for \(g, h \in G\). In GAP 4 a grading of an algebra is a record containing the following components.

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 JenningsLieAlgebra (64.8-4), NilpotentQuotientOfFpLieAlgebra (64.11-2)).

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 \(L\) constructed by JenningsLieAlgebra (64.8-4), the length of the of the range [ Grading(L)!.min_degree .. Grading(L)!.max_degree ] may be non-polynomial in the dimension of \(L\). To handle efficiently this situation, an optional component can be used:

gap> G:= SmallGroup(3^6, 100 );
<pc group of size 729 with 6 generators>
gap> L:= JenningsLieAlgebra( G );
<Lie algebra of dimension 6 over GF(3)>
gap> g:= Grading( L );
rec( hom_components := function( d ) ... end, max_degree := 9,
  min_degree := 1, source := Integers )
gap> g.hom_components( 3 );
<vector space over GF(3), with 1 generator>
gap> g.hom_components( 14 );
<vector space of dimension 0 over GF(3)>

62.10 Homomorphisms of Algebras

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.

62.10-1 AlgebraGeneralMappingByImages

is a general mapping from the \(F\)-algebra A to the \(F\)-algebra B. This general mapping is defined by mapping the entries in the list gens (elements of A) to the entries in the list imgs (elements of B), and taking the \(F\)-linear and multiplicative closure.

gens need not generate A as an \(F\)-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 LeftModuleHomomorphismByImages (61.10-2).

gap> A:= QuaternionAlgebra( Rationals );;
gap> B:= FullMatrixAlgebra( Rationals, 2 );;
gap> bA:= BasisVectors( Basis( A ) );; bB:= BasisVectors( Basis( B ) );;
gap> f:= AlgebraGeneralMappingByImages( A, B, bA, bB );
[ e, i, j, k ] -> [ [ [ 1, 0 ], [ 0, 0 ] ], [ [ 0, 1 ], [ 0, 0 ] ],
  [ [ 0, 0 ], [ 1, 0 ] ], [ [ 0, 0 ], [ 0, 1 ] ] ]
gap> Images( f, bA[1] );
<add. coset of <algebra over Rationals, with 16 generators>>

62.10-2 AlgebraHomomorphismByImages

AlgebraHomomorphismByImages returns the algebra homomorphism with source A and range B that is defined by mapping the list gens of generators of A to the list imgs of images in B.

If gens does not generate A or if the homomorphism does not exist (i.e., if mapping the generators describes only a multi-valued mapping) then fail is returned.

One can avoid the checks by calling AlgebraHomomorphismByImagesNC (62.10-3), and one can construct multi-valued mappings with AlgebraGeneralMappingByImages (62.10-1).

gap> T:= EmptySCTable( 2, 0 );;
gap> SetEntrySCTable( T, 1, 1, [1,1] ); SetEntrySCTable( T, 2, 2, [1,2] );
gap> A:= AlgebraByStructureConstants( Rationals, T );;
gap> m1:= NullMat( 2, 2 );; m1[1][1]:= 1;;
gap> m2:= NullMat( 2, 2 );; m2[2][2]:= 1;;
gap> B:= AlgebraByGenerators( Rationals, [ m1, m2 ] );;
gap> bA:= BasisVectors( Basis( A ) );; bB:= BasisVectors( Basis( B ) );;
gap> f:= AlgebraHomomorphismByImages( A, B, bA, bB );
[ v.1, v.2 ] -> [ [ [ 1, 0 ], [ 0, 0 ] ], [ [ 0, 0 ], [ 0, 1 ] ] ]
gap> Image( f, bA[1]+bA[2] );
[ [ 1, 0 ], [ 0, 1 ] ]

62.10-3 AlgebraHomomorphismByImagesNC

AlgebraHomomorphismByImagesNC is the operation that is called by the function AlgebraHomomorphismByImages (62.10-2). Its methods may assume that gens generates A and that the mapping of gens to imgs defines an algebra homomorphism. Results are unpredictable if these conditions do not hold.

For creating a possibly multi-valued mapping from A to B that respects addition, multiplication, and scalar multiplication, AlgebraGeneralMappingByImages (62.10-1) can be used.

For the definitions of the algebras A and B in the next example we refer to the previous example.

gap> f:= AlgebraHomomorphismByImagesNC( A, B, bA, bB );
[ v.1, v.2 ] -> [ [ [ 1, 0 ], [ 0, 0 ] ], [ [ 0, 0 ], [ 0, 1 ] ] ]

62.10-4 AlgebraWithOneGeneralMappingByImages

This function is analogous to AlgebraGeneralMappingByImages (62.10-1); the only difference being that the identity of A is automatically mapped to the identity of B.

gap> A:= QuaternionAlgebra( Rationals );;
gap> B:= FullMatrixAlgebra( Rationals, 2 );;
gap> bA:= BasisVectors( Basis( A ) );; bB:= BasisVectors( Basis( B ) );;
gap> f:=AlgebraWithOneGeneralMappingByImages(A,B,bA{[2,3,4]},bB{[1,2,3]});
[ i, j, k, e ] -> [ [ [ 1, 0 ], [ 0, 0 ] ], [ [ 0, 1 ], [ 0, 0 ] ],
  [ [ 0, 0 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ]

62.10-5 AlgebraWithOneHomomorphismByImages

AlgebraWithOneHomomorphismByImages returns the algebra-with-one homomorphism with source A and range B that is defined by mapping the list gens of generators of A to the list imgs of images in B.

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 A is mapped to the identity of B, and therefore gens needs to generate A only as an algebra-with-one.

If gens does not generate A or if the homomorphism does not exist (i.e., if mapping the generators describes only a multi-valued mapping) then fail is returned.

One can avoid the checks by calling AlgebraWithOneHomomorphismByImagesNC (62.10-6), and one can construct multi-valued mappings with AlgebraWithOneGeneralMappingByImages (62.10-4).

gap> m1:= NullMat( 2, 2 );; m1[1][1]:=1;;
gap> m2:= NullMat( 2, 2 );; m2[2][2]:=1;;
gap> A:= AlgebraByGenerators( Rationals, [m1,m2] );;
gap> T:= EmptySCTable( 2, 0 );;
gap> SetEntrySCTable( T, 1, 1, [1,1] );
gap> SetEntrySCTable( T, 2, 2, [1,2] );
gap> B:= AlgebraByStructureConstants(Rationals, T);;
gap> bA:= BasisVectors( Basis( A ) );; bB:= BasisVectors( Basis( B ) );;
gap> f:= AlgebraWithOneHomomorphismByImages( A, B, bA{[1]}, bB{[1]} );
[ [ [ 1, 0 ], [ 0, 0 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ] -> [ v.1, v.1+v.2 ]

62.10-6 AlgebraWithOneHomomorphismByImagesNC

AlgebraWithOneHomomorphismByImagesNC is the operation that is called by the function AlgebraWithOneHomomorphismByImages (62.10-5). Its methods may assume that gens generates A and that the mapping of gens to imgs defines an algebra-with-one homomorphism. Results are unpredictable if these conditions do not hold.

For creating a possibly multi-valued mapping from A to B that respects addition, multiplication, identity, and scalar multiplication, AlgebraWithOneGeneralMappingByImages (62.10-4) can be used.

gap> m1:= NullMat( 2, 2 );; m1[1][1]:=1;;
gap> m2:= NullMat( 2, 2 );; m2[2][2]:=1;;
gap> A:= AlgebraByGenerators( Rationals, [m1,m2] );;
gap> T:= EmptySCTable( 2, 0 );;
gap> SetEntrySCTable( T, 1, 1, [1,1] );
gap> SetEntrySCTable( T, 2, 2, [1,2] );
gap> B:= AlgebraByStructureConstants( Rationals, T);;
gap> bA:= BasisVectors( Basis( A ) );; bB:= BasisVectors( Basis( B ) );;
gap> f:= AlgebraWithOneHomomorphismByImagesNC( A, B, bA{[1]}, bB{[1]} );
[ [ [ 1, 0 ], [ 0, 0 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ] -> [ v.1, v.1+v.2 ]

62.10-7 AlgebraHomomorphismByFunction

These functions construct an algebra homomorphism from the algebra A to the algebra B using a one-argument function f. They do not check that the function actually defines a homomorphism.

gap> A := MatrixAlgebra( Rationals, 2 );;
gap> f := AlgebraHomomorphismByFunction( Rationals, A, q->[[q,0],[0,0]] );
MappingByFunction( Rationals, ( Rationals^[ 2, 2 ] ), function( q ) ... end )
gap> 11^f;
[ [ 11, 0 ], [ 0, 0 ] ]

62.10-8 NaturalHomomorphismByIdeal

For an algebra A and an ideal I in A, the return value of NaturalHomomorphismByIdeal (56.8-4) is a homomorphism of algebras, in particular the range of this mapping is also an algebra.

gap> L:= FullMatrixLieAlgebra( Rationals, 3 );;
gap> C:= LieCentre( L );
<two-sided ideal in <Lie algebra of dimension 9 over Rationals>,
  (dimension 1)>
gap> hom:= NaturalHomomorphismByIdeal( L, C );
<linear mapping by matrix, <Lie algebra of dimension
9 over Rationals> -> <Lie algebra of dimension 8 over Rationals>>
gap> ImagesSource( hom );
<Lie algebra of dimension 8 over Rationals>

62.10-9 OperationAlgebraHomomorphism

OperationAlgebraHomomorphism returns an algebra homomorphism from the \(F\)-algebra A into a matrix algebra over \(F\) that describes the \(F\)-linear action of A on the basis B of a free left module respectively on the free left module V (in which case some basis of V is chosen), via the operation opr.

The homomorphism need not be surjective. The default value for opr is OnRight (41.2-2).

If A is an algebra-with-one then the operation homomorphism is an algebra-with-one homomorphism because the identity of A must act as the identity.

gap> m1:= NullMat( 2, 2 );; m1[1][1]:= 1;;
gap> m2:= NullMat( 2, 2 );; m2[2][2]:= 1;;
gap> B:= AlgebraByGenerators( Rationals, [ m1, m2 ] );;
gap> V:= FullRowSpace( Rationals, 2 );
( Rationals^2 )
gap> f:=OperationAlgebraHomomorphism( B, Basis( V ), OnRight );
<op. hom. Algebra( Rationals,
[ [ [ 1, 0 ], [ 0, 0 ] ], [ [ 0, 0 ], [ 0, 1 ] ]
 ] ) -> matrices of dim. 2>
gap> Image( f, m1 );
[ [ 1, 0 ], [ 0, 0 ] ]

62.10-10 NiceAlgebraMonomorphism

If A is an associative algebra with one, returns an isomorphism from A onto a matrix algebra (see IsomorphismMatrixAlgebra (62.10-12) for an example). If A is a finitely presented Lie algebra, returns an isomorphism from A onto a Lie algebra defined by a structure constants table (see 64.11 for an example).

62.10-11 IsomorphismFpAlgebra

isomorphism from the algebra A onto a finitely presented algebra. Currently this is only implemented for associative algebras with one.

gap> A:= QuaternionAlgebra( Rationals );
<algebra-with-one of dimension 4 over Rationals>
gap> f:= IsomorphismFpAlgebra( A );
[ e, i, j, k, e ] -> [ [(1)*x.1], [(1)*x.2], [(1)*x.3], [(1)*x.4],
  [(1)*<identity ...>] ]

62.10-12 IsomorphismMatrixAlgebra

isomorphism from the algebra A onto a matrix algebra. Currently this is only implemented for associative algebras with one.

gap> T:= EmptySCTable( 2, 0 );;
gap> SetEntrySCTable( T, 1, 1, [1,1] ); SetEntrySCTable( T, 2, 2, [1,2] );
gap> A:= AlgebraByStructureConstants( Rationals, T );;
gap> A:= AsAlgebraWithOne( Rationals, A );;
gap> f:=IsomorphismMatrixAlgebra( A );
<op. hom. AlgebraWithOne( Rationals, ... ) -> matrices of dim. 2>
gap> Image( f, BasisVectors( Basis( A ) )[1] );
[ [ 1, 0 ], [ 0, 0 ] ]

62.10-13 IsomorphismSCAlgebra

For a basis B of an algebra \(A\), IsomorphismSCAlgebra returns an algebra isomorphism from \(A\) to an algebra \(S\) given by structure constants (see 62.4), such that the canonical basis of \(S\) is the image of B.

For an algebra A, IsomorphismSCAlgebra chooses a basis of A and returns the IsomorphismSCAlgebra value for that basis.

gap> IsomorphismSCAlgebra( GF(8) );
CanonicalBasis( GF(2^3) ) -> CanonicalBasis( <algebra of dimension
3 over GF(2)> )
gap> IsomorphismSCAlgebra( GF(2)^[2,2] );
CanonicalBasis( ( GF(2)^
[ 2, 2 ] ) ) -> CanonicalBasis( <algebra of dimension 4 over GF(2)> )

62.10-14 RepresentativeLinearOperation

is an element of the algebra A that maps the vector v to the vector w under the linear operation described by the function opr. If no such element exists then fail is returned.

gap> m1:= NullMat( 2, 2 );; m1[1][1]:= 1;;
gap> m2:= NullMat( 2, 2 );; m2[2][2]:= 1;;
gap> B:= AlgebraByGenerators( Rationals, [ m1, m2 ] );;
gap> RepresentativeLinearOperation( B, [1,0], [1,0], OnRight );
[ [ 1, 0 ], [ 0, 0 ] ]
gap> RepresentativeLinearOperation( B, [1,0], [0,1], OnRight );
fail

62.11 Representations of Algebras

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 IsLeftAlgebraModuleElement or IsRightModuleElement 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.

The transition between the original vector space and the corresponding algebra module is handled by ExtRepOfObj and ObjByExtRep. For an element v of the algebra module, ExtRepOfObj( v ) returns the underlying element of the original vector space. Furthermore, if vec is an element of the original vector space, and fam the elements family of the corresponding algebra module, then ObjByExtRep( fam, vec ) returns the corresponding element of the algebra module. Below is an example of this.

The action of the algebra on elements of the algebra module is constructed by using the operator ^. If x is an element of an algebra A, and v an element of a left A-module, then x^v calculates the result of the action of x on v. Similarly, if v is an element of a right A-module, then v^x calculates the action of x on v.

62.11-1 LeftAlgebraModuleByGenerators

Constructs the left algebra module over A generated by the list of vectors gens. The action of A is described by the function op. 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.

62.11-2 RightAlgebraModuleByGenerators

Constructs the right algebra module over A generated by the list of vectors gens. The action of A is described by the function op. 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.

62.11-3 BiAlgebraModuleByGenerators

Constructs the algebra bi-module over A and B generated by the list of vectors gens. The left action of A is described by the function opl, and the right action of B by the function opr. opl 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. opr 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.

gap> A:= Rationals^[3,3];
( Rationals^[ 3, 3 ] )
gap> V:= LeftAlgebraModuleByGenerators( A, \*, [ [ 1, 0, 0 ] ] );
<left-module over ( Rationals^[ 3, 3 ] )>
gap> W:= RightAlgebraModuleByGenerators( A, \*, [ [ 1, 0, 0 ] ] );
<right-module over ( Rationals^[ 3, 3 ] )>
gap> M:= BiAlgebraModuleByGenerators( A, A, \*, \*, [ [ 1, 0, 0 ] ] );
<bi-module over ( Rationals^[ 3, 3 ] ) (left) and ( Rationals^
[ 3, 3 ] ) (right)>

In the above examples, the modules V, W, and M are \(3\)-dimensional vector spaces over the rationals. The algebra A acts from the left on V, from the right on W, and from the left and from the right on M.

62.11-4 LeftAlgebraModule

Constructs the left algebra module over A with underlying space V. The action of A is described by the function op. This must be a function of two arguments; the first argument is the algebra element, and the second argument is a vector from V; it outputs the result of applying the algebra element to the vector.

62.11-5 RightAlgebraModule

Constructs the right algebra module over A with underlying space V. The action of A is described by the function op. This must be a function of two arguments; the first argument is a vector, from V and the second argument is the algebra element; it outputs the result of applying the algebra element to the vector.

62.11-6 BiAlgebraModule

Constructs the algebra bi-module over A and B with underlying space V. The left action of A is described by the function opl, and the right action of B by the function opr. opl must be a function of two arguments; the first argument is the algebra element, and the second argument is a vector from V; it outputs the result of applying the algebra element on the left to the vector. opr must be a function of two arguments; the first argument is a vector from V, and the second argument is the algebra element; it outputs the result of applying the algebra element on the right to the vector.

gap> A:= Rationals^[3,3];;
gap> V:= Rationals^3;
( Rationals^3 )
gap> V:= Rationals^3;;
gap> M:= BiAlgebraModule( A, A, \*, \*, V );
<bi-module over ( Rationals^[ 3, 3 ] ) (left) and ( Rationals^
[ 3, 3 ] ) (right)>
gap> Dimension( M );
3

62.11-7 GeneratorsOfAlgebraModule

gap> A:= Rationals^[3,3];;
gap> V:= LeftAlgebraModuleByGenerators( A, \*, [ [ 1, 0, 0 ] ] );;
gap> GeneratorsOfAlgebraModule( V );
[ [ 1, 0, 0 ] ]

62.11-8 IsAlgebraModuleElement

Category of algebra module elements. If an object has IsAlgebraModuleElementCollection, then it is an algebra module. If a family has IsAlgebraModuleElementFamily, then it is a family of algebra module elements (every algebra module has its own elements family).

62.11-9 IsLeftAlgebraModuleElement

Category of left algebra module elements. If an object has IsLeftAlgebraModuleElementCollection, then it is a left-algebra module.

62.11-10 IsRightAlgebraModuleElement

Category of right algebra module elements. If an object has IsRightAlgebraModuleElementCollection, then it is a right-algebra module.

gap> A:= Rationals^[3,3];
( Rationals^[ 3, 3 ] )
gap> M:= BiAlgebraModuleByGenerators( A, A, \*, \*, [ [ 1, 0, 0 ] ] );
<bi-module over ( Rationals^[ 3, 3 ] ) (left) and ( Rationals^
[ 3, 3 ] ) (right)>
gap> vv:= BasisVectors( Basis( M ) );
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]
gap> IsLeftAlgebraModuleElement( vv[1] );
true
gap> IsRightAlgebraModuleElement( vv[1] );
true
gap> vv[1] = [ 1, 0, 0 ];
false
gap> ExtRepOfObj( vv[1] ) = [ 1, 0, 0 ];
true
gap> ObjByExtRep( ElementsFamily( FamilyObj( M ) ), [ 1, 0, 0 ] ) in M;
true
gap> xx:= BasisVectors( Basis( A ) );;
gap> xx[4]^vv[1];  # left action
[ 0, 1, 0 ]
gap> vv[1]^xx[2];  # right action
[ 0, 1, 0 ]

62.11-11 LeftActingAlgebra

Here V is a left-algebra module; this function returns the algebra that acts from the left on V.

62.11-12 RightActingAlgebra

Here V is a right-algebra module; this function returns the algebra that acts from the right on V.

62.11-13 ActingAlgebra

Here V is an algebra module; this function returns the algebra that acts on V (this is the same as LeftActingAlgebra( V ) if V is a left module, and RightActingAlgebra( V ) if V is a right module; it will signal an error if V is a bi-module).

gap> A:= Rationals^[3,3];;
gap> M:= BiAlgebraModuleByGenerators( A, A, \*, \*, [ [ 1, 0, 0 ] ] );;
gap> LeftActingAlgebra( M );
( Rationals^[ 3, 3 ] )
gap> RightActingAlgebra( M );
( Rationals^[ 3, 3 ] )
gap> V:= RightAlgebraModuleByGenerators( A, \*, [ [ 1, 0, 0 ] ] );;
gap> ActingAlgebra( V );
( Rationals^[ 3, 3 ] )

62.11-14 IsBasisOfAlgebraModuleElementSpace

If a basis B lies in the category IsBasisOfAlgebraModuleElementSpace, then B is a basis of a subspace of an algebra module. This means that B has the record field B!.delegateBasis 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 ExtRepOfObj( v ) for v in the algebra module).

gap> A:= Rationals^[3,3];;
gap> M:= BiAlgebraModuleByGenerators( A, A, \*, \*, [ [ 1, 0, 0 ] ] );;
gap> B:= Basis( M );
Basis( <3-dimensional bi-module over ( Rationals^
[ 3, 3 ] ) (left) and ( Rationals^[ 3, 3 ] ) (right)>,
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] )
gap> IsBasisOfAlgebraModuleElementSpace( B );
true
gap> B!.delegateBasis;
SemiEchelonBasis( <vector space of dimension 3 over Rationals>,
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] )

62.11-15 MatrixOfAction

Here B is a basis of an algebra module and x is an element of the algebra that acts on this module. This function returns the matrix of the action of x with respect to B. If x acts from the left, then the coefficients of the images of the basis elements of B (under the action of x) are the columns of the output. If x acts from the right, then they are the rows of the output.

If the module is a bi-module, then the third parameter side must be specified. This is the string "left", or "right" depending whether x acts from the left or the right.

gap> M:= LeftAlgebraModuleByGenerators( A, \*, [ [ 1, 0, 0 ] ] );;
gap> x:= Basis(A)[3];
[ [ 0, 0, 1 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ]
gap> MatrixOfAction( Basis( M ), x );
[ [ 0, 0, 1 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ]

62.11-16 SubAlgebraModule

is the sub-module of the algebra module M, generated by the vectors in gens. If as an optional argument the string basis is added, then it is assumed that the vectors in gens form a basis of the submodule.

gap> m1:= NullMat( 2, 2 );; m1[1][1]:= 1;;
gap> m2:= NullMat( 2, 2 );; m2[2][2]:= 1;;
gap> A:= Algebra( Rationals, [ m1, m2 ] );;
gap> M:= LeftAlgebraModuleByGenerators( A, \*, [ [ 1, 0 ], [ 0, 1 ] ] );
<left-module over <algebra over Rationals, with 2 generators>>
gap> bb:= BasisVectors( Basis( M ) );
[ [ 1, 0 ], [ 0, 1 ] ]
gap> V:= SubAlgebraModule( M, [ bb[1] ] );
<left-module over <algebra over Rationals, with 2 generators>>
gap> Dimension( V );
1

62.11-17 LeftModuleByHomomorphismToMatAlg

Here A is an algebra and hom a homomorphism from A into a matrix algebra. This function returns the left A-module defined by the homomorphism hom.

62.11-18 RightModuleByHomomorphismToMatAlg

Here A is an algebra and hom a homomorphism from A into a matrix algebra. This function returns the right A-module defined by the homomorphism hom.

First we produce a structure constants algebra with basis elements \(x\), \(y\), \(z\) such that \(x^2 = x\), \(y^2 = y\), \(xz = z\), \(zy = z\) and all other products are zero.

gap> T:= EmptySCTable( 3, 0 );;
gap> SetEntrySCTable( T, 1, 1, [ 1, 1 ]);
gap> SetEntrySCTable( T, 2, 2, [ 1, 2 ]);
gap> SetEntrySCTable( T, 1, 3, [ 1, 3 ]);
gap> SetEntrySCTable( T, 3, 2, [ 1, 3 ]);
gap> A:= AlgebraByStructureConstants( Rationals, T );
<algebra of dimension 3 over Rationals>

gap> m1:= NullMat( 2, 2 );; m1[1][1]:= 1;;
gap> m2:= NullMat( 2, 2 );; m2[2][2]:= 1;;
gap> m3:= NullMat( 2, 2 );; m3[1][2]:= 1;;
gap> B:= Algebra( Rationals, [ m1, m2, m3 ] );
<algebra over Rationals, with 3 generators>

gap> f:= AlgebraHomomorphismByImages( A, B, Basis(A), [ m1, m2, m3 ] );;
gap> RightModuleByHomomorphismToMatAlg( A, f );
<right-module over <algebra of dimension 3 over Rationals>>

62.11-19 AdjointModule

gap> m1:= NullMat( 2, 2 );; m1[1][1]:= 1;;
gap> m2:= NullMat( 2, 2 );; m2[2][2]:= 1;;
gap> m3:= NullMat( 2, 2 );; m3[1][2]:= 1;;
gap> A:= Algebra( Rationals, [ m1, m2, m3 ] );
<algebra over Rationals, with 3 generators>
gap> V:= AdjointModule( A );
<3-dimensional left-module over <algebra of dimension
3 over Rationals>>
gap> v:= Basis( V )[3];
[ [ 0, 1 ], [ 0, 0 ] ]
gap> W:= SubAlgebraModule( V, [ v ] );
<left-module over <algebra of dimension 3 over Rationals>>
gap> Dimension( W );
1

62.11-20 FaithfulModule

returns a faithful finite-dimensional left-module over the algebra A. This is only implemented for associative algebras, and for Lie algebras of characteristic \(0\). (It may also work for certain Lie algebras of characteristic \(p > 0\).)

gap> T:= EmptySCTable( 2, 0 );;
gap> A:= AlgebraByStructureConstants( Rationals, T );
<algebra of dimension 2 over Rationals>

gap> T:= EmptySCTable( 3, 0, "antisymmetric" );;
gap> SetEntrySCTable( T, 1, 2, [ 1, 3 ]);
gap> L:= LieAlgebraByStructureConstants( Rationals, T );
<Lie algebra of dimension 3 over Rationals>
gap> V:= FaithfulModule( L );
<left-module over <Lie algebra of dimension 3 over Rationals>>
gap> vv:= BasisVectors( Basis( V ) );
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]
gap> x:= Basis( L )[3];
v.3
gap> List( vv, v -> x^v );
[ [ 0, 0, 0 ], [ 1, 0, 0 ], [ 0, 0, 0 ] ]

gap> V:= FaithfulModule( A );
<left-module over <algebra of dimension 2 over Rationals>>
gap> vv:= BasisVectors( Basis( V ) );
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]
gap> xx:= BasisVectors( Basis( A ) );
[ v.1, v.2 ]
gap> xx[1]^vv[3];
[ 1, 0, 0 ]

62.11-21 ModuleByRestriction

Here V is an algebra module and sub1 is a subalgebra of the acting algebra of V. This function returns the module that is the restriction of V to sub1. So it has the same underlying vector space as V, but the acting algebra is sub. If two subalgebras sub1, sub2 are given then V is assumed to be a bi-module, and sub1 a subalgebra of the algebra acting on the left, and sub2 a subalgebra of the algebra acting on the right.

gap> A:= Rationals^[3,3];;
gap> V:= LeftAlgebraModuleByGenerators( A, \*, [ [ 1, 0, 0 ] ] );;
gap> B:= Subalgebra( A, [ Basis(A)[1] ] );
<algebra over Rationals, with 1 generator>
gap> W:= ModuleByRestriction( V, B );
<left-module over <algebra over Rationals, with 1 generator>>

62.11-22 NaturalHomomorphismBySubAlgebraModule

Here V must be a sub-algebra module of V. This function returns the projection from V onto V/W. It is a linear map, that is also a module homomorphism. As usual images can be formed with Image( f, v ) and pre-images with PreImagesRepresentative( f, u ).

gap> A:= Rationals^[3,3];;
gap> B:= DirectSumOfAlgebras( A, A );
<algebra over Rationals, with 6 generators>
gap> T:= StructureConstantsTable( Basis( B ) );;
gap> C:= AlgebraByStructureConstants( Rationals, T );
<algebra of dimension 18 over Rationals>
gap> V:= AdjointModule( C );
<left-module over <algebra of dimension 18 over Rationals>>
gap> W:= SubAlgebraModule( V, [ Basis(V)[1] ] );
<left-module over <algebra of dimension 18 over Rationals>>
gap> f:= NaturalHomomorphismBySubAlgebraModule( V, W );
<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>>>
gap> quo:= ImagesSource( f );  # i.e., the quotient module
<9-dimensional left-module over <algebra of dimension
18 over Rationals>>
gap> v:= Basis( quo )[1];
[ 1, 0, 0, 0, 0, 0, 0, 0, 0 ]
gap> PreImagesRepresentative( f, v );
v.4
gap> Basis( C )[4]^v;
[ 1, 0, 0, 0, 0, 0, 0, 0, 0 ]

62.11-23 DirectSumOfAlgebraModules

Here list 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.

gap> A:= FullMatrixAlgebra( Rationals, 3 );;
gap> V:= BiAlgebraModuleByGenerators( A, A, \*, \*, [ [1,0,0] ] );;
gap> W:= DirectSumOfAlgebraModules( V, V );
<6-dimensional left-module over ( Rationals^[ 3, 3 ] )>
gap> BasisVectors( Basis( W ) );
[ ( [ 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 ] ) ]

gap> L:= SimpleLieAlgebra( "C", 3, Rationals );;
gap> V:= HighestWeightModule( L, [ 1, 1, 0 ] );
<64-dimensional left-module over <Lie algebra of dimension
21 over Rationals>>
gap> W:= HighestWeightModule( L, [ 0, 0, 2 ] );
<84-dimensional left-module over <Lie algebra of dimension
21 over Rationals>>
gap> U:= DirectSumOfAlgebraModules( V, W );
<148-dimensional left-module over <Lie algebra of dimension
21 over Rationals>>

62.11-24 TranslatorSubalgebra

Here M is an algebra module, and U and W are two subspaces of M. Let A be the algebra acting on M. This function returns the subspace of elements of A that map U into W. If W is a sub-algebra-module (i.e., closed under the action of A), then this space is a subalgebra of A.

This function works for left, or right modules over a finite-dimensional algebra. We stress that it is not checked whether U and W are indeed subspaces of M. If this is not the case nothing is guaranteed about the behaviour of the function.

gap> A:= FullMatrixAlgebra( Rationals, 3 );
( Rationals^[ 3, 3 ] )
gap> V:= Rationals^[3,2];
( Rationals^[ 3, 2 ] )
gap> M:= LeftAlgebraModule( A, \*, V );
<left-module over ( Rationals^[ 3, 3 ] )>
gap> bm:= Basis(M);;
gap> U:= SubAlgebraModule( M, [ bm[1] ] );
<left-module over ( Rationals^[ 3, 3 ] )>
gap> TranslatorSubalgebra( M, U, M );
<algebra of dimension 9 over Rationals>
gap> W:= SubAlgebraModule( M, [ bm[4] ] );
<left-module over ( Rationals^[ 3, 3 ] )>
gap> T:=TranslatorSubalgebra( M, U, W );
<algebra of dimension 0 over Rationals>

62 Algebras