Subsections

• The Construction of Direct Sums and Tensor Products
• Construction of Elements of Direct Sums and Tensor Products

Given two matrix algebras R and T, where R and T have the same coefficient ring S, return the direct sum D of R and T (with the action given by the direct sum of the action of R and the action of T).

Given two matrix algebras R and T, where R and T have the same coefficient ring S, construct the tensor product of R and T.

> Q := RationalField();
> A := MatrixAlgebra< Q, 3 | [ 1/3,0,0, 3/2,3,0, -1/2,4,3],
>        [ 3,0,0, 1/2,-5,0, 8,-1/2,4] >;
> AplusA := DirectSum(A, A);
> print AplusA: Maximal;
Matrix Algebra of degree 6 and dimension 12 with 4 generators over Rational 
Field
Generators:
[ 1/3    0    0    0    0    0]
[ 3/2    3    0    0    0    0]
[-1/2    4    3    0    0    0]
[   0    0    0    0    0    0]
[   0    0    0    0    0    0]
[   0    0    0    0    0    0]

[   3    0    0    0    0    0]
[ 1/2   -5    0    0    0    0]
[   8 -1/2    4    0    0    0]
[   0    0    0    0    0    0]
[   0    0    0    0    0    0]
[   0    0    0    0    0    0]

[   0    0    0    0    0    0]
[   0    0    0    0    0    0]
[   0    0    0    0    0    0]
[   0    0    0  1/3    0    0]
[   0    0    0  3/2    3    0]
[   0    0    0 -1/2    4    3]

[   0    0    0    0    0    0]
[   0    0    0    0    0    0]
[   0    0    0    0    0    0]
[   0    0    0    3    0    0]
[   0    0    0  1/2   -5    0]
[   0    0    0    8 -1/2    4]
> AtimesA := TensorProduct(A, A);
> print AtimesA:Maximal;                                                       
Matrix Algebra of degree 9 with 4 generators over Rational Field
Generators:
[ 1/9    0    0    0    0    0    0    0    0]
[ 1/2    1    0    0    0    0    0    0    0]
[-1/6  4/3    1    0    0    0    0    0    0]
[ 1/2    0    0    1    0    0    0    0    0]
[ 9/4  9/2    0  9/2    9    0    0    0    0]
[-3/4    6  9/2 -3/2   12    9    0    0    0]
[-1/6    0    0  4/3    0    0    1    0    0]
[-3/4 -3/2    0    6   12    0  9/2    9    0]
[ 1/4   -2 -3/2   -2   16   12 -3/2   12    9]

[    1     0     0     0     0     0     0     0     0]
[  1/6  -5/3     0     0     0     0     0     0     0]
[  8/3  -1/6   4/3     0     0     0     0     0     0]
[  9/2     0     0     9     0     0     0     0     0]
[  3/4 -15/2     0   3/2   -15     0     0     0     0]
[   12  -3/4     6    24  -3/2    12     0     0     0]
[ -3/2     0     0    12     0     0     9     0     0]
[ -1/4   5/2     0     2   -20     0   3/2   -15     0]
[   -4   1/4    -2    32    -2    16    24  -3/2    12]

[    1     0     0     0     0     0     0     0     0]
[  9/2     9     0     0     0     0     0     0     0]
[ -3/2    12     9     0     0     0     0     0     0]
[  1/6     0     0  -5/3     0     0     0     0     0]
[  3/4   3/2     0 -15/2   -15     0     0     0     0]
[ -1/4     2   3/2   5/2   -20   -15     0     0     0]
[  8/3     0     0  -1/6     0     0   4/3     0     0]
[   12    24     0  -3/4  -3/2     0     6    12     0]
[   -4    32    24   1/4    -2  -3/2    -2    16    12]

[   9    0    0    0    0    0    0    0    0]
[ 3/2  -15    0    0    0    0    0    0    0]
[  24 -3/2   12    0    0    0    0    0    0]
[ 3/2    0    0  -15    0    0    0    0    0]
[ 1/4 -5/2    0 -5/2   25    0    0    0    0]
[   4 -1/4    2  -40  5/2  -20    0    0    0]
[  24    0    0 -3/2    0    0   12    0    0]
[   4  -40    0 -1/4  5/2    0    2  -20    0]
[  64   -4   32   -4  1/4   -2   32   -2   16]

Given an element a of the matrix algebra Q and an element b of the matrix algebra R, form the direct sum of matrices a and b. The square is returned as an element of the matrix algebra T, which must be the direct sum of the parent of a and the parent of b.

Given an element a of the matrix algebra M_n(S), form the exterior square of a as an element of M_m(S), where m = n(n - 1)/2.

Given an element a of the matrix algebra M_n(S), form the symmetric square of a as an element of M_m(S), where m = n(n + 1)/2.

Given an element a belonging to a subalgebra of M_(n_1)(S) and an element b belonging to a subalgebra of M_(n_2)(S), construct the tensor product of a and b as an element of the matrix algebra M_n(S), where n = n_1 * n_2.