Subsections

• Operations on Associative Algebras
• Operations on Elements
• Representations of Associative Algebras

The centre of the associative algebra A.

Centraliser(A, S) : AlgAss, AlgAss -> AlgAss

The centralizer of the subalgebra S of the associative algebra A, that is, the subalgebra of A commuting elementwise with S.

Idealiser(A, B: parameters) : AlgAss, AlgAss -> AlgAss

    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 in A, b in B.

For two subalgebras A and B of an associative algebra, return the ideal generated by all [a, b] = a * b - b * a, a in A, b in 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 in 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 in B.

We create the Lie algebra sl_3(Q) as a structure constant algebra. First, we construct gl_3(Q) from the full matrix algebra M_3(Q) and get sl_3(Q) as the derived algebra of gl_3(Q).

> gl3 := LieAlgebra(Algebra(MatrixRing(Rationals(), 3)));
> sl3 := gl3 * gl3;
> sl3;
Lie Algebra of dimension 8 with base ring Rational Field

> 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)

> 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)

> 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)

> 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)

Centraliser(A, s) : AlgAss, AlgAssElt -> AlgAss

The centralizer of the element s of the associative algebra A, that is, the subalgebra of A commuting with s.

(a, b) : AlgAssElt, AlgAssElt -> AlgAssElt

The Lie bracket a * b - b * a of a and b, where a and b are elements of an associative algebra A.

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"

For an associative algebra A of dimension n over R return its regular representation. If B = (e_1, e_2, ..., e_n) is the stored basis for A, an element a in A is mapped to the matrix in R^(n x n) which has as its i-th row the coordinates of e_i * 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 * e_i) by setting the parameter Side to "Left".