The Construction of Extensions and their Elements

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The Construction of Direct Sums and Tensor Products

DirectSum(R, T) : AlgMat, AlgMat -> AlgMat
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).
TensorProduct(A, B) : AlgMat, AlgMat -> AlgMat
Given two unital matrix algebras A and B, where A and B have the same coefficient ring S, construct the tensor product of A and B.

Example AlgMat_Products (H90E5)

We construct the direct product and tensor product of the matrix algebra A (defined above) with itself.
> 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);
> AplusA: Maximal;
Matrix Algebra of degree 6 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);
> AtimesA: Maximal;
Matrix Algebra of degree 9 with 4 generators over
    Rational Field
Generators:
[ 1/3    0    0    0    0    0    0    0    0]
[   0  1/3    0    0    0    0    0    0    0]
[   0    0  1/3    0    0    0    0    0    0]
[ 3/2    0    0    3    0    0    0    0    0]
[   0  3/2    0    0    3    0    0    0    0]
[   0    0  3/2    0    0    3    0    0    0]
[-1/2    0    0    4    0    0    3    0    0]
[   0 -1/2    0    0    4    0    0    3    0]
[   0    0 -1/2    0    0    4    0    0    3]
[   3    0    0    0    0    0    0    0    0]
[   0    3    0    0    0    0    0    0    0]
[   0    0    3    0    0    0    0    0    0]
[ 1/2    0    0   -5    0    0    0    0    0]
[   0  1/2    0    0   -5    0    0    0    0]
[   0    0  1/2    0    0   -5    0    0    0]
[   8    0    0 -1/2    0    0    4    0    0]
[   0    8    0    0 -1/2    0    0    4    0]
[   0    0    8    0    0 -1/2    0    0    4]
[ 1/3    0    0    0    0    0    0    0    0]
[ 3/2    3    0    0    0    0    0    0    0]
[-1/2    4    3    0    0    0    0    0    0]
[   0    0    0  1/3    0    0    0    0    0]
[   0    0    0  3/2    3    0    0    0    0]
[   0    0    0 -1/2    4    3    0    0    0]
[   0    0    0    0    0    0  1/3    0    0]
[   0    0    0    0    0    0  3/2    3    0]
[   0    0    0    0    0    0 -1/2    4    3]
[   3    0    0    0    0    0    0    0    0]
[ 1/2   -5    0    0    0    0    0    0    0]
[   8 -1/2    4    0    0    0    0    0    0]
[   0    0    0    3    0    0    0    0    0]
[   0    0    0  1/2   -5    0    0    0    0]
[   0    0    0    8 -1/2    4    0    0    0]
[   0    0    0    0    0    0    3    0    0]
[   0    0    0    0    0    0  1/2   -5    0]
[   0    0    0    0    0    0    8 -1/2    4]

Construction of Direct Sums and Tensor Products of Elements

DirectSum(a, b) : AlgMatElt, AlgMatElt -> AlgMatElt
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.
ExteriorSquare(a) : AlgMatElt -> AlgMatElt
Given an element a of the matrix algebra Mn(S), form the exterior square of a as an element of Mm(S), where m = n(n - 1)/2.
ExteriorPower(a,r) : AlgMat, RngIntElt -> AlgMatElt
Given an element a of the matrix algebra Mn(S), form the rth exterior power of a as an element of Mm(S), where m = n choose r.
SymmetricSquare(a) : AlgMatElt -> AlgMatElt
Given an element a of the matrix algebra Mn(S), form the symmetric square of a as an element of Mm(S), where m = n(n + 1)/2.
SymmetricPower(a,r) : AlgMatElt, RngIntElt -> AlgMatElt
Given an element a of the matrix algebra Mn(S), form the rth symmetric power of a as an element of Mm(S), for the appropriate m.
TensorProduct(a, b) : AlgMatElt, AlgMatElt -> AlgMatElt
Given an element a belonging to a subalgebra of Mn1(S) and an element b belonging to a subalgebra of Mn2(S), construct the tensor product of a and b as an element of the matrix algebra Mn(S), where n = n1 * n2.
V2.28, 13 July 2023