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 >
|
|
| (2.1) |
V2 >
|
|
| (2.2) |
V2 >
|
|
sl2 >
|
|
| (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.
sl2 >
|
|
V1 >
|
|
| (2.7) |
We will need a 4 dimensional vector space for the representation space.
V1 >
|
|
W2 >
|
|
| (2.8) |
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)).
sl2 >
|
|
W2 >
|
|
| (2.9) |
We will need a 6 dimensional vector space for the representation space.
W2 >
|
|
W3 >
|
|
| (2.10) |
Use the Invariants command to calculate the invariants of this representation.
sl2 >
|
|
| (2.11) |
|
|