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
Calling Sequences
TensorProductOfRepresentations(R, W)
TensorProductOfRepresentations(ρ, T, W)
Parameters
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
Description
Examples
See Also
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.
withDifferentialGeometry:withLieAlgebras:
Example 1.
Define the standard representation and the adjoint representation for sl2. Then form the tensor product representation. First, set up the representation spaces.
DGsetupx1,x2,V1:
DGsetupy1,y2,y2,V2:
Define the standard representation.
M1≔Matrix0,1,0,0,Matrix1,0,0,−1,Matrix0,0,1,0
L≔LieAlgebraDataM1,sl2
L:=e1,e2=−2e1,e1,e3=e2,e2,e3=−2e3
DGsetupL:
ρ1≔Representationsl2,V1,M1
Define the adjoint representation using the Adjoint command.
ρ2≔Representationsl2,V2,Adjoint
We will need a 6-dimensional vector space to represent the tensor product of rho1 and rho2.
DGsetupz1,z2,z3,z4,z5,z6,W1:
φ1≔TensorProductOfRepresentationsρ1,ρ2,W1
Use the Query command to verify that rho1 is a representation.
Queryφ1,Representation
true
Example 2.
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.
ChangeFrameV1:
T1≔Tensor:-GenerateSymmetricTensorsD_x1,D_x2,3
T1:=D_x1D_x1D_x1,13D_x1D_x1D_x2+13D_x1D_x2D_x1+13D_x2D_x1D_x1,13D_x1D_x2D_x2+13D_x2D_x1D_x2+13D_x2D_x2D_x1,D_x2D_x2D_x2
We will need a - dimensional representation space.
DGsetupz1,z2,z3,z4,W2:
φ2≔TensorProductOfRepresentationsρ1,T1,W2
Example 3.
Compute the representation of rho1 (the standard representation of sl2) on the 2nd exterior product of the 3rd symmetric product ∧2Sym3V1.
ChangeFrameW2:
T3≔Tools:-GenerateFormsdz1,dz2,dz3,dz4,2
T3:=dz1⋀dz2,dz1⋀dz3,dz1⋀dz4,dz2⋀dz3,dz2⋀dz4,dz3⋀dz4
We will need a 6-dimensional representation space.
DGsetupp1,p2,p3,p4,p5,p6,W3:
φ3≔TensorProductOfRepresentationsφ2,T3,W3
Use the Invariants command to calculate the invariants of this representation.
Invariantsφ3
−3D_p3+D_p4
DifferentialGeometry, Tensor, Tools, LieAlgebras, Invariants, GenerateForms, GenerateSymmetricTensors, Query, Representation
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