The Construction of Extensions and their Elements
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: Direct Sum and Tensor Product (ex-548c90)
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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 \(M_n(S)\), form the exterior square of \(a\) as an element of \(M_m(S)\), where \(m = n(n-1)/2\).
- ExteriorPower(a, r): AlgMat, RngIntElt -> AlgMatElt
Given an element \(a\) of the matrix algebra \(M_n(S)\), form the \(r\)th exterior power of \(a\) as an element of \(M_m(S)\), where \(m = n\choose r\).
- SymmetricSquare(a): AlgMatElt -> AlgMatElt
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 \(m = n(n+1)/2\).
- SymmetricPower(a, r): AlgMatElt, RngIntElt -> AlgMatElt
Given an element \(a\) of the matrix algebra \(M_n(S)\), form the \(r\)th symmetric power of \(a\) as an element of \(M_m(S)\), for the appropriate \(m\).
- TensorProduct(a, b): AlgMatElt, AlgMatElt -> AlgMatElt
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 \(n = n_1 * n_2\).