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

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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

Description

•

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.

•

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.

Examples

Example 1.

Define the standard representation and the adjoint representation for sl2. Then form the tensor product representation. First, setup the representation spaces.

V1 >

Define the standard representation.

V2 >

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})]

(2.1)

V2 >

(2.2)

V2 >

sl2 >

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})]]

(2.3)

Define the adjoint representation using the Adjoint command.

sl2 >

(2.4)

We will need a 6 dimensional vector space for the representation space for the tensor product of rho1 and rho2.

sl2 >

W1 >

(2.5)

Use the Query command to verify that rho1 is a representation.

sl2 >

(2.6)

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.