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 - Maple Programming Help

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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

Description

• 

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 pdimensional representation of ρ.

Examples

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:

V1 > 

DGsetupy1,y2,y2,V2:

 

Define the standard representation.

V2 > 

M1Matrix0,1,0,0,Matrix1,0,0,1,Matrix0,0,1,0

M1:=0100,1001,0010

(2.1)
V2 > 

LLieAlgebraDataM1,sl2

L:=e1,e2=2e1,e1,e3=e2,e2,e3=2e3

(2.2)
V2 > 

DGsetupL:

sl2 > 

ρ1Representationsl2,V1,M1

ρ1:=e1,0100,e2,1001,e3,0010

(2.3)

 

Define the adjoint representation using the Adjoint command.

sl2 > 

ρ2Representationsl2,V2,Adjoint

ρ2:=e1,020001000,e2,200000002,e3,000100020

(2.4)

 

We will need a 6-dimensional vector space to represent the tensor product of rho1 and rho2.

sl2 > 

DGsetupz1,z2,z3,z4,z5,z6,W1:

W1 > 

φ1TensorProductOfRepresentationsρ1,ρ2,W1

φ1:=e1,020100001010000001000020000001000000,e2,300000010000001000000100000010000003,e3,000000100000020000100000010100001020

(2.5)

 

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

sl2 > 

Queryφ1,Representation

true

(2.6)

 

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.

sl2 > 

ChangeFrameV1:

V1 > 

T1Tensor:-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

(2.7)

 

We will need a - dimensional representation space.

V1 > 

DGsetupz1,z2,z3,z4,W2:

W2 > 

φ2TensorProductOfRepresentationsρ1,T1,W2

φ2:=e1,0100002000030000,e2,3000010000100003,e3,0000300002000010

(2.8)

 

Example 3.

Compute the representation of rho1 (the standard representation of sl2) on the 2nd exterior product of the 3rd symmetric product 2Sym3V1.

sl2 > 

ChangeFrameW2:

W2 > 

T3Tools:-GenerateFormsdz1,dz2,dz3,dz4,2

T3:=dz1dz2,dz1dz3,dz1dz4,dz2dz3,dz2dz4,dz3dz4

(2.9)

 

We will need a 6-dimensional representation space.

W2 > 

DGsetupp1,p2,p3,p4,p5,p6,W3:

W3 > 

φ3TensorProductOfRepresentationsφ2,T3,W3

φ3:=e1,000000200000030000010000001300000020,e2,400000020000000000000000000020000004,e3,020000001300000030000010000002000000

(2.10)

 

Use the Invariants command to calculate the invariants of this representation.

sl2 > 

Invariantsφ3

3D_p3+D_p4

(2.11)

See Also

DifferentialGeometry, Tensor, Tools, LieAlgebras, Invariants, GenerateForms, GenerateSymmetricTensors, Query, Representation