LieAlgebras[TensorProductOfRepresentations] - 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
TensorProductOfRepresentations(ρ, T, W)
R - a list ρ1,ρ2, ... of representations 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
ρ - 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 ρ
Let ρ1: 𝔤 → glV1, ρ2: 𝔤 → glV2, ... be a list of representations of a Lie algebra 𝔤. Let W = V1 ⊗V2⊗⋅⋅⋅ be the tensor product of the vector spaces V1, V2, ... . The tensor product of the representations ρ1, ρ2, ... is the representation ρ: 𝔤 → gl(W) defined by
ρxy 1⊗ y2 ⊗ ... = ρ1xy1 ⊗ y2⊗⋅⋅⋅ + y1 ⊗ ρ2x y2⊗⋅⋅⋅+⋅⋅⋅ where x ∈ 𝔤 and y1 ∈ V1, y2 ∈ V2 , ... .
Let ρ: 𝔤 → glVbe a representation. Then ρ determines a representation τ of 𝔤 on TsrV, the space of type r, s tensors on V. The representation τ , in turn, the restricts to any τ-invariant subspace, spanned by a list T of p type r,s tensors. The second calling sequence returns this p−dimensional representation of ρ.
Define the standard representation and the adjoint representation for sl2. Then form the tensor product representation. First, set up the representation spaces.
Define the standard representation.
M1 ≔ Matrix⁡0,1,0,0,Matrix⁡1,0,0,−1,Matrix⁡0,0,1,0
L ≔ LieAlgebraData⁡M1,sl2
ρ1 ≔ Representation⁡sl2,V1,M1
Define the adjoint representation using the Adjoint command.
ρ2 ≔ Representation⁡sl2,V2,Adjoint⁡
We will need a 6-dimensional vector space to represent the tensor product of rho1 and rho2.
φ1 ≔ TensorProductOfRepresentations⁡ρ1,ρ2,W1
Use the Query command to verify that rho1 is a representation.
Compute the representation of rho1 (the standard representation of sl2) on the 3rd symmetric product Sym3V1of V1. First, use the GenerateSymmetricTensors command to generate a basis T1 for Sym3V1.
T1 ≔ Tensor:-GenerateSymmetricTensors⁡D_x1,D_x2,3
We will need a - dimensional representation space.
φ2 ≔ TensorProductOfRepresentations⁡ρ1,T1,W2
Compute the representation of rho1 (the standard representation of sl2) on the 2nd exterior product of the 3rd symmetric product ∧2Sym3V1.
T3 ≔ Tools:-GenerateForms⁡dz1,dz2,dz3,dz4,2
We will need a 6-dimensional representation space.
φ3 ≔ TensorProductOfRepresentations⁡φ2,T3,W3
Use the Invariants command to calculate the invariants of this representation.
DifferentialGeometry, Tensor, Tools, LieAlgebras, Invariants, GenerateForms, GenerateSymmetricTensors, Query, Representation
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