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LieAlgebras[TensorProduct] - form the tensor product representation for a list of representations of a Lie algebra; form various tensor product representations from a single representation of a Lie algebra
Calling Sequences
TensorProduct(R, W)
TensorProduct(rho, T, W)
Parameters
R - a list R = [rho1, rho2, ...] of representations rho1, rho2, ... of a Lie algebra g on vector spaces V1, V2, ...
W - a Maple name or string, the name of the frame for the representation space for the tensor product representation
rho - a representation of a Lie algebra g on a vector space V
T - a list of linearly independent type (r,s) tensors on V defining a subspace of tensors invariant under the induced representation of rho
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Description
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Let W = V1 * V2 *... be the tensor product of the vector spaces V1, V2, ... The dimension of W is the product of the dimensions of the vector spaces V1, V2, ... Then the tensor product of the representations rho1, rho2,... is the representation of Lie algebra rho of g on W defined by rho(x)(y1*y2* ...) = rho(x)(y1)*y2* ... + y1 *rho(x)(y2)*... + ... where y1 in V1, y2 in V2, ... and x in g.
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The second calling sequence returns a p dimensional representation of rho, where p is the number of elements in the list T, defined by the restriction to T of the representation of rho on the space T^r_s(V) of type (r,s) tensors on V. For example, T may be a basis for all symmetric or skew-symmetric tensors of a given rank.
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Examples
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Example 1.
Define the standard representation and the adjoint representation for sl2. Then form the tensor product representation. First, setup the representation spaces.
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![DGsetup([x1, x2], V1)](/support/helpjp/helpview.aspx?si=6634/file05848/math146.png)
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V1 >
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![DGsetup([y1, y2, y2], V2)](/support/helpjp/helpview.aspx?si=6634/file05848/math150.png)
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Define the standard representation.
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V2 >
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![M1 := [Matrix([[0, 1], [0, 0]]), Matrix([[1, 0], [0, -1]]), Matrix([[0, 0], [1, 0]])]](/support/helpjp/helpview.aspx?si=6634/file05848/math158.png)
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![M1 := [Matrix(2, 2, {(1, 1) = 0, (1, 2) = 1, (2, 1) = 0, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 1, (1, 2) = 0, (2, 1) = 0, (2, 2) = -1}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 1, (2, 2) = 0})]](/support/helpjp/helpview.aspx?si=6634/file05848/math161.png)
| (2.1) |
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V2 >
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![_DG([["LieAlgebra", sl2, [3, table( [ ] )]], [[[1, 2, 1], -2], [[1, 3, 2], 1], [[2, 3, 3], -2]]])](/support/helpjp/helpview.aspx?si=6634/file05848/math168.png)
| (2.2) |
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V2 >
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sl2 >
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![rho1 := [[e1, Matrix(2, 2, {(1, 1) = 0, (1, 2) = 1, (2, 1) = 0, (2, 2) = 0})], [e2, Matrix(2, 2, {(1, 1) = 1, (1, 2) = 0, (2, 1) = 0, (2, 2) = -1})], [e3, Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 1, (2, 2) = 0})]]](/support/helpjp/helpview.aspx?si=6634/file05848/math179.png)
| (2.3) |
Define the adjoint representation using the Adjoint command.
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sl2 >
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![rho2 := [[e1, Matrix(3, 3, {(1, 1) = 0, (1, 2) = -2, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0})], [e2, Matrix(3, 3, {(1, 1) = 2, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -2})], [e3, Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = -1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 2, (3, 3) = 0})]]](/support/helpjp/helpview.aspx?si=6634/file05848/math193.png)
| (2.4) |
We will need a 6 dimensional vector space for the representation space for the tensor product of rho1 and rho2.
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sl2 >
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![DGsetup([z1, z2, z3, z4, z5, z6], W1)](/support/helpjp/helpview.aspx?si=6634/file05848/math205.png)
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W1 >
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![phi1 := TensorProduct([rho1, rho2], W1)](/support/helpjp/helpview.aspx?si=6634/file05848/math209.png)
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| (2.5) |
Use the Query command to verify that rho1 is a representation.
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sl2 >
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Example 2.
Compute the representation of rho1 (the standard representation of sl2) on the 3rd symmetric product Sym^3(V1) of V1. Use the GenerateSymmetricTensors command to generate a basis T1 for this tensor space.
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sl2 >
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V1 >
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![T1 := Tensor:-GenerateSymmetricTensors([D_x1, D_x2], 3)](/support/helpjp/helpview.aspx?si=6634/file05848/math255.png)
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| (2.7) |
We will need a 4 dimensional vector space for the representation space.
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V1 >
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![DGsetup([z1, z2, z3, z4], W2)](/support/helpjp/helpview.aspx?si=6634/file05848/math266.png)
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W2 >
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Example 3.
Compute the representation of rho1 (the standard representation of sl2) on the 2nd exterior product of the 3rd symmetric product Lambda^2(Sym^3(V1)).
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sl2 >
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W2 >
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![T3 := Tools:-GenerateForms([dz1, dz2, dz3, dz4], 2)](/support/helpjp/helpview.aspx?si=6634/file05848/math293.png)
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![[_DG([["form", W2, 2], [[[1, 2], 1]]]), _DG([["form", W2, 2], [[[1, 3], 1]]]), _DG([["form", W2, 2], [[[1, 4], 1]]]), _DG([["form", W2, 2], [[[2, 3], 1]]]), _DG([["form", W2, 2], [[[2, 4], 1]]]), _DG([["form", W2, 2], [[[3, 4], 1]]])]](/support/helpjp/helpview.aspx?si=6634/file05848/math296.png)
| (2.9) |
We will need a 6 dimensional vector space for the representation space.
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W2 >
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![DGsetup([p1, p2, p3, p4, p5, p6], W3)](/support/helpjp/helpview.aspx?si=6634/file05848/math304.png)
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W3 >
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| (2.10) |
Use the Invariants command to calculate the invariants of this representation.
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sl2 >
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![[_DG([["vector", W3, []], [[[3], -3], [[4], 1]]])]](/support/helpjp/helpview.aspx?si=6634/file05848/math325.png)
| (2.11) |
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