All the algebras considered in this package will be associative and commutative. Scalars belong to a commutative ring k with \(1 \neq 0\).
Why not a field? A group ring over the integers is not an algebra.
A multiplier in a commutative algebra \(A\) is a function \(\mu : A \to A\) such that
\[ \mu(ab) ~=~ (\mu a)b ~=~ a(\mu b) \quad \forall~ a,b \in A. \]
The regular multipliers of \(A\) are the functions
\[ \mu_a : A \to A ~:~ \mu_ab = ab \quad \forall~ b \in A. \]
When \(A\) has a one, it follows from the defining condition that \(\mu(b1) = (\mu 1)b\) and so \(\mu = \mu_a\) where \(a = \mu 1\). Since an ideal \(I\) of \(A\) is closed under multiplication, a multiplier \(\mu\) may be restricted to \(I\).
Question: Is there an example of an algebra \(A\) without a one which has multipliers not of the form \(\mu_a\)?
‣ RegularAlgebraMultiplier( A, I, a ) | ( operation ) |
This operation defines the multiplier \(\mu_a : I \to I\) on an ideal \(I\) of \(A\).
gap> A5c6 := GroupRing( GF(5), Group( (1,2,3,4,5,6) ) );; gap> vecA := BasisVectors( Basis( A5c6 ) );; gap> v := vecA[1] + vecA[3] + vecA[5]; (Z(5)^0)*()+(Z(5)^0)*(1,3,5)(2,4,6)+(Z(5)^0)*(1,5,3)(2,6,4) gap> I5c6 := Ideal( A5c6, [v] );; gap> v2 := vecA[2]; (Z(5)^0)*(1,2,3,4,5,6) gap> m2 := RegularAlgebraMultiplier( A5c6, I5c6, v2 ); [ (Z(5)^0)*()+(Z(5)^0)*(1,3,5)(2,4,6)+(Z(5)^0)*(1,5,3)(2,6,4), (Z(5)^0)*(1,2,3,4,5,6)+(Z(5)^0)*(1,4)(2,5)(3,6)+(Z(5)^0)*(1,6,5,4,3,2) ] -> [ (Z(5)^0)*(1,2,3,4,5,6)+(Z(5)^0)*(1,4)(2,5)(3,6)+(Z(5)^0)*(1,6,5,4,3,2), (Z(5)^0)*()+(Z(5)^0)*(1,3,5)(2,4,6)+(Z(5)^0)*(1,5,3)(2,6,4) ]
‣ IsAlgebraMultiplier( m ) | ( operation ) |
This function tests the condition \(\mu(ab) = (\mu a)b = a(\mu b)\) for all \(a,b\) in the basis for \(A\).
gap> IsAlgebraMultiplier( m2 ); true gap> one := One( A5c6 );; gap> L := List( vecA, v -> one );; gap> m1 := LeftModuleHomomorphismByImages( A5c6, A5c6, vecA, L ); [ (Z(5)^0)*(), (Z(5)^0)*(1,2,3,4,5,6), (Z(5)^0)*(1,3,5)(2,4,6), (Z(5)^0)*(1,4)(2,5)(3,6), (Z(5)^0)*(1,5,3)(2,6,4), (Z(5)^0)*(1,6,5,4,3,2) ] -> [ (Z(5)^0)*(), (Z(5)^0)*(), (Z(5)^0)*(), (Z(5)^0)*(), (Z(5)^0)*(), (Z(5)^0)*() ] gap> IsAlgebraMultiplier( m1 ); false
‣ MultiplierAlgebraOfIdealBySubalgebra ( A, I, B ) | ( operation ) |
The regular multipliers \(\mu_b : I \to I\) for all \(b \in B\), where \(I\) is an ideal in \(A\) and \(B\) is a subalgebra of \(A\), form an algebra with product \(\mu_b \circ \mu_{b'} = \mu_{bb'}\).
gap> v3 := vecA[3]; (Z(5)^0)*(1,3,5)(2,4,6) gap> B5c3 := Subalgebra( A5c6, [ v3 ] );; gap> M := MultiplierAlgebraOfIdealBySubalgebra( A5c6, I5c6, B5c3 ); <algebra of dimension 1 over GF(5)> gap> vecM := BasisVectors( Basis( M ) );; gap> vecM[1]; <linear mapping by matrix, <two-sided ideal in <algebra-with-one of dimension 6 over GF(5)>, (dimension 2 )> -> <two-sided ideal in <algebra-with-one of dimension 6 over GF(5)>, (dimension 2)>>
‣ MultiplierAlgebra( A ) | ( attribute ) |
The regular multipliers \(\mu_a : A \to A\) for all \(a \in A\) form an algebra isomorphic to \(A\) by the map \(a \mapsto \mu_a\). This operation returns MultiplierAlgebraOfIdealBySubalgebra(A,A,A);.
gap> MA5c6 := RegularMultiplierAlgebra( A5c6 ); <algebra of dimension 6 over GF(5)> gap> vecM := BasisVectors( Basis( MA5c6 ) );; gap> vecM[3]; <linear mapping by matrix, <algebra-with-one of dimension 6 over GF(5)> -> <algebra-with-one of dimension 6 over GF(5)>>
‣ MultiplierHomomorphism( M ) | ( attribute ) |
If \(M\) is a multiplier algebra with elements of algebra \(A\) multiplying an ideal \(I\) then this operation returns the homomorphism from \(A\) to \(M\) mapping \(a\) to \(\mu_a\).
gap> hom := MultiplierHomomorphism( MA5c6 );; gap> ImageElm( hom, vecA[2] ); Basis( <two-sided ideal in <algebra-with-one of dimension 6 over GF(5)>, (dimension 2)>, [ (Z(5)^0)*()+(Z(5)^0)*(1,3,5)(2,4,6)+(Z(5)^0)*(1,5,3)(2,6,4), (Z(5)^0)*(1,2,3,4,5,6)+(Z(5)^0)*(1,4)(2,5)(3,6)+(Z(5)^0)*(1,6,5,4,3,2) ] ) -> [ (Z(5)^0)*(1,2,3,4,5,6)+(Z(5)^0)*(1,4)(2,5)(3,6)+(Z(5)^0)*(1,6,5,4,3,2), (Z(5)^0)*()+(Z(5)^0)*(1,3,5)(2,4,6)+(Z(5)^0)*(1,5,3)(2,6,4) ]
If \(S\) and \(R\) are commutative k-algebras, a map
\[ R \times S ~\to~ S, \qquad (r,s) ~\mapsto~ r \cdot s \]
is a commutative action if and only if the following five axioms hold:
\(k(r \cdot s) ~=~ (kr) \cdot s ~=~ r \cdot (ks)\),
\(r \cdot (s + s') ~=~ r \cdot s + r \cdot s', \qquad\) (so \(r \cdot 0_S = 0_S ~\forall~ r \in R\)),
\((r + r') \cdot s ~=~ r \cdot s + r' \cdot s, \qquad\) (so \(0_R \cdot s = 0_S ~\forall~ s \in S\)),
\(r \cdot (ss') ~=~ (r \cdot s)s' = s(r \cdot s')\),
\((rr') \cdot s ~=~ r \cdot (r' \cdot s), \qquad\) (so \(1_R \cdot s = s ~\forall~ s \in S\) when \(R\) has a one),
for all \(k \in \)k, \(r,r' \in R\), and \(s,s' \in S\).
‣ AlgebraActionByMultipliers( A, I ) | ( operation ) |
When \(I\) is an ideal in \(A\) we have seen that the multiplier homomorphism from \(A\) to MultiplierAlgebraOf(Ideal(A,I) is an action.
In the example the algebra is the group ring of the cyclic group \(C_6\) over the field \(GF(5)\). The ideal is generated by \(v = () + (1,3,5)(2,4,6) + (1,5,3)(2,6,4)\). The generator \(r = (1,2,3,4,5,6)\) acts on \(v\) by multiplication to give the vector \(r \cdot v = (1,2,3,4,5,6) + (1,4)(2,5)(3,6) + (1,6,5,4,3,2)\).
gap> A5c6 := GroupRing( GF(5), Group( (1,2,3,4,5,6) ) );; gap> vecA := BasisVectors( Basis( A5c6 ) );; gap> v := vecA[1] + vecA[3] + vecA[5]; (Z(5)^0)*()+(Z(5)^0)*(1,3,5)(2,4,6)+(Z(5)^0)*(1,5,3)(2,6,4) gap> I5c6 := Ideal( A5c6, [v] );; gap> actm := AlgebraActionByMultipliers( A5c6, I5c6 );; gap> actm2 := Image( actm, vecA[2] );; gap> Image( actm2, v ); (Z(5)^0)*(1,2,3,4,5,6)+(Z(5)^0)*(1,4)(2,5)(3,6)+(Z(5)^0)*(1,6,5,4,3,2)
‣ AlgebraActionBySurjection( hom ) | ( operation ) |
Let \(\theta : S \to R\) be a surjective algebra homomorphism such that \(ks = 0_S ~\forall~ k \in K = \ker\theta\). Then \(R\) acts on \(S\) with \(r \cdot s = (\theta^{-1}r)s\). Note that thus action is well defined since if \(\theta p = r\) then \(\theta^{-1}r = \{ p+k ~|~ k \in \ker\theta \}\) and \((p+k)s = ps+ks = ps+0\).
Continuing with the previous example, we construct the quotient algebra \(Q5c6 = A5c6/I5c6\), and the natural homomorphism \(\theta : A5c6 \to Q5c6\). The kernel of \(\theta\) is not contained in the annihilator of \(A5c6\), so an attempt to form the action fails.
An alternative example involves a single-generator matrix algebra.
gap> theta := NaturalHomomorphismByIdeal( A5c6, I5c6 ); <linear mapping by matrix, <algebra-with-one of dimension 6 over GF(5)> -> <algebra of dimension 4 over GF(5)>> gap> List( vecA, v -> ImageElm( theta, v ) ); [ v.1, v.2, v.3, v.4, (Z(5)^2)*v.1+(Z(5)^2)*v.3, (Z(5)^2)*v.2+(Z(5)^2)*v.4 ] gap> actp := AlgebraActionBySurjection( theta ); kernel of hom is not in the annihilator of A fail gap> ## an example which does not fail: gap> m := [ [0,1,2,3], [0,0,1,2], [0,0,0,1], [0,0,0,0] ];; gap> m^2; [ [ 0, 0, 1, 4 ], [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ] gap> m^3; [ [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ] gap> A1 := Algebra( Rationals, [m] );; gap> A3 := Subalgebra( A1, [m^3] );; gap> nat3 := NaturalHomomorphismByIdeal( A1, A3 ); <linear mapping by matrix, <algebra of dimension 3 over Rationals> -> <algebra of dimension 2 over Rationals>> gap> act3 := AlgebraActionBySurjection( nat3 );; gap> a3 := Image( act3, BasisVectors( Basis( Image( nat3 ) ) )[1] );; gap> [ Image( a3, m ) = m^2, Image( a3, m^2 ) = m^3 ]; [ true, true ]
‣ SemidirectProductOfAlgebras( R, act, S ) | ( operation ) |
When \(R,S\) are commutative algebras and \(R\) acts on \(S\) then we can form the semidirect product \(R \ltimes S\), where the product is given by:
\[ (r_1,s_1)(r_2,s_2) ~=~ (r_1r_2,~ r_1 \cdot s_2 + r_2 \cdot s_1 + s_1s_2). \]
This product, as wekll as being commutative, is associative: \((r_1,s_1)(r_2,s_2)(r_3,s_3)\) expands as:
\[ (r_1r_2r_3,~ \left (r_1r_2)\cdot s3 + (r_1r_3)\cdot s_2 + (r_2r_3)\cdot s_1 + r_1 \cdot (s_2s_3) + r_2 \cdot (s_1s_3) + r_3 \cdot (s_1s_2) + s_1s_2s_3 \right). \]
If \(B_R, B_S\) are the sets of basis vectors for \(R\) and \(S\) then \(R \ltimes S\) has basis
\[ \{(r,0_S) ~|~ r \in B_R\} ~\cup~ \{(0_R,s) ~|~ s \in B_S\} \]
with defining products
\[ (r_1,0_S)(r_2,0_S) = (r_1r_2,0_S), \qquad (r,0_S)(0_R,s) = (0_R,r \cdot s), \qquad (0_R,s_1)(0_R,s_2) = (0_R,s_1s_2). \]
Continuing the example above,
gap> P := SemidirectProductOfAlgebras( A5c6, actm, I5c6 ); gap> Embedding( P, 1 ); [ (Z(5)^0)*(), (Z(5)^0)*(1,2,3,4,5,6), (Z(5)^0)*(1,3,5)(2,4,6), (Z(5)^0)*(1,4)(2,5)(3,6), (Z(5)^0)*(1,5,3)(2,6,4), (Z(5)^0)*(1,6,5,4,3,2) ] -> [ v.1, v.2, v.3, v.4, v.5, v.6 ] gap> Embedding( P, 2 ); [ (Z(5)^0)*()+(Z(5)^0)*(1,3,5)(2,4,6)+(Z(5)^0)*(1,5,3)(2,6,4), (Z(5)^0)*(1,2,3,4,5,6)+(Z(5)^0)*(1,4)(2,5)(3,6)+(Z(5)^0)*(1,6,5,4,3,2) ] -> [ v.7, v.8 ] gap> Projection( P, 1 ); [ v.1, v.2, v.3, v.4, v.5, v.6, v.7, v.8 ] -> [ (Z(5)^0)*(), (Z(5)^0)*(1,2,3,4,5,6), (Z(5)^0)*(1,3,5)(2,4,6), (Z(5)^0)*(1,4)(2,5)(3,6), (Z(5)^0)*(1,5,3)(2,6,4), (Z(5)^0)*(1,6,5,4,3,2), <zero> of ..., <zero> of ... ]
‣ SemidirectProductOfAlgebrasInfo( P ) | ( attribute ) |
The SemidirectProductOfAlgebrasInfo(P) for \(P = R \ltimes S\) is a record with fields P.action; P.algebras; P.embeddings; and P.projections.
‣ AllAlgebraHomomorphisms( A, B ) | ( operation ) |
‣ AllBijectiveAlgebraHomomorphisms( A, B ) | ( operation ) |
‣ AllIdempotentAlgebraHomomorphisms( A, B ) | ( operation ) |
These three operations list all the homomorphisms from \(A\) to \(B\) of the specified type. These lists can get very long, so the operations should only be used with small algebras.
gap> A2c6 := GroupRing( GF(2), Group( (1,2,3,4,5,6) ) );; gap> R2c3 := GroupRing( GF(2), Group( (7,8,9) ) );; gap> homAR := AllAlgebraHomomorphisms( A2c6, R2c3 );; gap> List( homAR, h -> MappingGeneratorsImages(h) ); [ [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ <zero> of ... ] ], [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ (Z(2)^0)*() ] ], [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ (Z(2)^0)*()+(Z(2)^0)*(7,8,9) ] ], [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ (Z(2)^0)*()+(Z(2)^0)*(7,8,9)+(Z(2)^0)*(7,9,8) ] ], [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ (Z(2)^0)*()+(Z(2)^0)*(7,9,8) ] ], [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ (Z(2)^0)*(7,8,9) ] ], [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ (Z(2)^0)*(7,8,9)+(Z(2)^0)*(7,9,8) ] ], [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ (Z(2)^0)*(7,9,8) ] ] ] gap> homRA := AllAlgebraHomomorphisms( R2c3, A2c6 );; gap> List( homRA, h -> MappingGeneratorsImages(h) ); [ [ [ (Z(2)^0)*(7,8,9) ], [ <zero> of ... ] ], [ [ (Z(2)^0)*(7,8,9) ], [ (Z(2)^0)*() ] ], [ [ (Z(2)^0)*(7,8,9) ], [ (Z(2)^0)*()+(Z(2)^0)*(1,3,5)(2,4,6) ] ], [ [ (Z(2)^0)*(7,8,9) ], [ (Z(2)^0)*()+(Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,5,3)(2,6,4) ] ], [ [ (Z(2)^0)*(7,8,9) ], [ (Z(2)^0)*()+(Z(2)^0)*(1,5,3)(2,6,4) ] ], [ [ (Z(2)^0)*(7,8,9) ], [ (Z(2)^0)*(1,3,5)(2,4,6) ] ], [ [ (Z(2)^0)*(7,8,9) ], [ (Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,5,3)(2,6,4) ] ], [ [ (Z(2)^0)*(7,8,9) ], [ (Z(2)^0)*(1,5,3)(2,6,4) ] ] ] gap> bijAA := AllBijectiveAlgebraHomomorphisms( A2c6, A2c6 );; gap> List( bijAA, h -> MappingGeneratorsImages(h) ); [ [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ (Z(2)^0)*()+(Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,4)(2,5)(3,6) ] ], [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ (Z(2)^0)*()+(Z(2)^0)*(1,4)(2,5)(3,6)+(Z(2)^0)*(1,5,3)(2,6,4) ] ], [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ (Z(2)^0)*(1,2,3,4,5,6) ] ], [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ (Z(2)^0)*(1,2,3,4,5,6)+(Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,5,3) (2,6,4) ] ], [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ (Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,5,3)(2,6,4)+(Z(2)^0)* (1,6,5,4,3,2) ] ], [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ (Z(2)^0)*(1,6,5,4,3,2) ] ] ] gap> ideAA := AllIdempotentAlgebraHomomorphisms( A2c6, A2c6 );; gap> Length( ideAA ); 14
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