The centre of the associative algebra A.
The centralizer of the subalgebra S of the associative algebra A, that is, the subalgebra of A commuting elementwise with S.
Side: MonStgElt Default: "Both"
Given an associative algebra A and a subalgebra B of A, compute the idealizer of B in A, that is, the largest subalgebra of A in which B is an ideal. By default the two-sided idealizer, that is, the largest subalgebra in which B is a two-sided ideal, is found; the left- or right-idealizer can be found by setting the parameter Side to "Left" or "Right" respectively.
For an associative structure constant algebra A, return the structure constant algebra L with product given by the Lie bracket (a, b) |-> a * b - b * a. As a second value the map identifying the elements of A and L is returned.
Let A and B be subalgebras of an associative algebra with underlying module M. This function returns the submodule of M which is spanned by the elements [a, b] = a * b - b * a, a ∈A, b ∈B.
For two subalgebras A and B of an associative algebra, return the ideal generated by all [a, b] = a * b - b * a, a ∈A, b ∈B.
For two subalgebras A and B of an associative algebra, return the left annihilator of B in A; that is, the subalgebra of A consisting of all elements a such that a * b = 0 for all b ∈B.
For two subalgebras A and B of an associative algebra, return the right annihilator of B in A; that is, the subalgebra of A consisting of all elements a such that b * a = 0 for all b ∈B.
We create the Lie algebra sl3(Q) as a structure constant algebra. First, we construct gl3(Q) from the full matrix algebra M3(Q) and get sl3(Q) as the derived algebra of gl3(Q).
> gl3 := LieAlgebra(Algebra(MatrixRing(Rationals(), 3))); > sl3 := gl3 * gl3; > sl3; Lie Algebra of dimension 8 with base ring Rational FieldLet's see how the first basis element acts.
> for i in [1..8] do > print sl3.i * sl3.1; > end for; (0 0 0 0 0 0 0 0) ( 0 -1 0 0 0 0 0 0) ( 0 0 -2 0 0 0 0 0) (0 0 0 1 0 0 0 0) (0 0 0 0 0 0 0 0) ( 0 0 0 0 0 -1 0 0) (0 0 0 0 0 0 2 0) (0 0 0 0 0 0 0 1)Since it acts diagonally, this element lies in a Cartan subalgebra. The next candidate seems to be the fifth basis element.
> for i in [1..8] do > print sl3.i * sl3.5; > end for; (0 0 0 0 0 0 0 0) (0 1 0 0 0 0 0 0) ( 0 0 -1 0 0 0 0 0) ( 0 0 0 -1 0 0 0 0) (0 0 0 0 0 0 0 0) ( 0 0 0 0 0 -2 0 0) (0 0 0 0 0 0 1 0) (0 0 0 0 0 0 0 2)This also acts diagonally and commutes with sl3.1, hence we have luckily found a full Cartan algebra in sl3(Q). We can now easily work out the root system. Obviously the root spaces correspond to the pairs (sl3.2, sl3.4), (sl3.3, sl3.7) and (sl3.6, sl3.8). The product of a positive root with its negative should lie in the Cartan algebra.
> sl3.2*sl3.4; ( 1 0 0 0 -1 0 0 0) > sl3.3*sl3.7; (1 0 0 0 0 0 0 0) > sl3.6*sl3.8; (0 0 0 0 1 0 0 0)Clearly some choices have to be made and we fix sl3.3 as the element eα corresponding to the first fundamental root α, sl3.7 as e - α and get sl3.1 as hα = eα * e - α. For the other fundamental root β we have to find an element eβ such that eα * eβ is non-zero.
> sl3.3*sl3.2; (0 0 0 0 0 0 0 0) > sl3.3*sl3.4; ( 0 0 0 0 0 -1 0 0) > sl3.3*sl3.6; (0 0 0 0 0 0 0 0) > sl3.3*sl3.8; (0 1 0 0 0 0 0 0)We choose sl3.8 as eβ, sl3.6 as e - β and consequently -sl3.5 as hβ. This now determines eα + β to be sl3.2 and e - α - β to be sl3.4.
Given an associative algebra A over an algebraic number or function field K and an optional coefficient field F of K, return the algebra isomorphic to A whose coefficient field is F if given otherwise the coefficient field of K.
> P<x> := PolynomialRing(Rationals()); > F<b> := NumberField(x^3-3*x-1); > Z_F := MaximalOrder(F); > A<alpha,beta,alphabeta> := QuaternionAlgebra<F | -3,b>; > A := AssociativeAlgebra(A); > A; Associative Algebra of dimension 4 with base ring F > RA, m := RestrictionOfScalars(A); > RA; Associative Algebra of dimension 12 with base ring Rational Field > m; Mapping from: AlgAss: A to AlgAss: RA given by a rule > m(A.1); (1 0 0 0 0 0 0 0 0 0 0 0) > $1 @@ m; (1 0 0 0) > RA.6 @@ m; ( 0 b^2 0 0) > m($1); (0 0 0 0 0 1 0 0 0 0 0 0)We now show an example where the scalars we are restricting to are not necessarily the coefficient field but a coefficient field of the coefficient field.
> P<x> := PolynomialRing(QuadraticField(23)); > F<b> := NumberField(x^3-3*x-1); > Z_F := MaximalOrder(F); > A<alpha,beta,alphabeta> := QuaternionAlgebra<F | -3,b>; > A := AssociativeAlgebra(A); > RA, m := RestrictionOfScalars(A); > RA; Associative Algebra of dimension 12 with base ring Quadratic Field with defining polynomial x^2 - 23 over the Rational Field > m; Mapping from: AlgAss: A to AlgAss: RA given by a rule > RA.6 @@ m; ( 0 b^2 0 0) > m($1); (0 0 0 0 0 1 0 0 0 0 0 0) > RAQ, mQ := RestrictionOfScalars(A, Rationals()); > RAQ, mQ; Associative Algebra of dimension 24 with base ring Rational Field Mapping from: AlgAss: A to AlgAss: RAQ given by a rule > mQ(A.1); (1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0) > $1 @@ mQ; (1 0 0 0) > RAQ.15 @@ mQ; ( 0 0 b^2 + 2*$.1*b + 23 0) > mQ($1); (0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0)
The centralizer of the element s of the associative algebra A, that is, the subalgebra of A commuting with s.
The Lie bracket a * b - b * a of a and b, where a and b are elements of an associative algebra A.
Returns true (and a coerced to F) iff a belongs to the base ring F of its parent algebra.
Side: MonStgElt Default: "Right"
Returns the matrix representation of Side-multiplication by the element a in the associative algebra A (which must have 1) on the A-module M.
For an associative algebra A of dimension n return an isomorphic matrix algebra. If A contains the identity-element, the matrix algebra will be of degree n, otherwise it will be of degree n + 1.
Side: MonStgElt Default: "Right"
Given a finite-dimensional R-algebra A and a Side A-module M (both free as R-modules), return the matrix algebra of A-endomorphisms of M, and the R-algebra homomorphism from A into this endomorphism ring.
Side: MonStgElt Default: "Right"
For an associative algebra A of dimension n over R return its regular representation. If B = (e1, e2, ..., en) is the stored basis for A, an element a ∈A is mapped to the matrix in Rn x n which has as its i-th row the coordinates of ei * a with respect to B. As a second map, the homomorphism of A onto the regular representation is returned.By default, the right-regular representation is computed. This can be changed to the left-regular representation (in which the i-th row of the image of a contains the coordinates of a * ei) by setting the parameter Side to "Left".
This section describes a few functions that can be used to obtain information on the structure of a finite-dimensional associative algebra.
Al: MonStgElt Default: em "Default"
This returns the largest nilpotent ideal of A. This function works for finite-dimensional associative algebras defined over a field of characteristic 0, or over a finite field.The algorithm used by default is taken from [CIW97]. The meataxe algorithm can be used by setting Al := "Meataxe".
> G:= SmallGroup( 27, 5 ); > A:= GroupAlgebra( GF(3), G ); > JacobsonRadical( A ); Ideal of dimension 26 of the group algebra A
Given an associative algebra A, return the direct sum decomposition of L as a sequence of ideals of L whose sum is L and each of which cannot be further decomposed into a direct sum of ideals. The second sequence return contains the corresponding primitive central idempotents.For a description of the algorithm we refer to [EG96].
Let Z be the centre of the associative algebra A, and let J(Z) denote its Jacobson radical. This function returns a sequence of primitive orthogonal idempotents in Z such that their images in Z/J(Z) span J(Z). Each such idempotent generates a two-sided ideal in A. The second return value is the sequence of these ideals.In particular, if A is a semisimple algebra, then this function returns a sequence of primitive orthogonal idempotents spanning Z. Furthermore, the ideals in the second sequence returned are simple algebras, and their direct sum equals A.
For a description of the algorithm we refer to [EG96].
> G:= SmallGroup( 10, 2 );
> A:= GroupAlgebra( Rationals(), G );
> ee, II:= CentralIdempotents( A );
> ee[1];
1/10*Id(G) + 1/10*G.2 + 1/10*G.2^2 + 1/10*G.2^3 + 1/10*G.2^4 + 1/10*G.1 +
1/10*G.1 * G.2 + 1/10*G.1 * G.2^2 + 1/10*G.1 * G.2^3 + 1/10*G.1 * G.2^4
> II;
[
Ideal of dimension 1 of the group algebra A
Basis:
Id(G) + G.2 + G.2^2 + G.2^3 + G.2^4 + G.1 + G.1 * G.2 + G.1 * G.2^2 +
G.1 * G.2^3 + G.1 * G.2^4,
Ideal of dimension 1 of the group algebra A
Basis:
Id(G) + G.2 + G.2^2 + G.2^3 + G.2^4 - G.1 - G.1 * G.2 - G.1 * G.2^2 -
G.1 * G.2^3 - G.1 * G.2^4,
Ideal of dimension 4 of the group algebra A
Basis:
Id(G) - G.2^4 + G.1 - G.1 * G.2^4
G.2 - G.2^4 + G.1 * G.2 - G.1 * G.2^4
G.2^2 - G.2^4 + G.1 * G.2^2 - G.1 * G.2^4
G.2^3 - G.2^4 + G.1 * G.2^3 - G.1 * G.2^4,
Ideal of dimension 4 of the group algebra A
Basis:
Id(G) - G.2^4 - G.1 + G.1 * G.2^4
G.2 - G.2^4 - G.1 * G.2 + G.1 * G.2^4
G.2^2 - G.2^4 - G.1 * G.2^2 + G.1 * G.2^4
G.2^3 - G.2^4 - G.1 * G.2^3 + G.1 * G.2^4
]
We see that here the group algebra is the direct sum of two 1-dimensional
and two 4-dimensional ideals. The first idempotent is the sum over all group
elements divided by the group order.