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LieAlgebras[DirectSumOfRepresentations] - form the direct sum representation for a list of representations of a Lie algebra

Calling Sequences

     DirectSumOfRepresentations(R, W)

Parameters

     R         - a list of representations ρ1, ρ2, ... 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 direct sum representation

 

Description

Examples

Description

• 

 Let 𝔤 be a Lie algebra and let ρi : 𝔤  Vi , i = 1, 2, ...,  p be a sequence of representations of 𝔤. Then the direct sum representation of the representationsρi is the representation σ : 𝔤  W, where W = V1V2  Vp  and

σxY  = ρ1xY1 + ρ2xY2 ++ ρpxYp   for  Y = Y1 + Y2 +  + Yp with Yi Vi.

• 

The command DirectSumOfRepresentations(R, W) returns the representation σ.

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

Define the standard representation and the adjoint representation for sl2. Then form the direct sum representation. First, setup the representation spaces.

DGsetupx1,x2,V1:

V1 > 

DGsetupy1,y2,y2,V2:

V2 > 

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

W1 > 

DGsetupz1,z2,z3,z4,z5,z6,W2:

 

Define the standard representation.

W2 > 

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

M1:=0100,1001,0010

(2.1)
W2 > 

LLieAlgebraDataM1,sl2

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

(2.2)
W2 > 

DGsetupL

Lie algebra: sl2

(2.3)
sl2 > 

ρ1Representationsl2,V1,M1

ρ1:=e1,0100,e2,1001,e3,0010

(2.4)

 

Define the adjoint representation.

sl2 > 

ρ2Representationsl2,V2,Adjoint

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

(2.5)

 

Define the direct sum representation of ρ1and ρ2

sl2 > 

φ1DirectSumOfRepresentationsρ1,ρ2,W1

φ1:=e1,0100000000000200000100000,e2,1000001000002000000000002,e3,0000010000000000010000020

(2.6)
sl2 > 

Queryφ1,Representation

true

(2.7)

 

Define the direct sum of 3 copies of ρ1.

sl2 > 

φ2DirectSumOfRepresentationsρ1,ρ1,ρ1,W2

φ2:=e1,010000000000000100000000000001000000,e2,100000010000001000000100000010000001,e3,000000100000000000001000000000000010

(2.8)
sl2 > 

Queryφ2,Representation

true

(2.9)

See Also

DifferentialGeometry

LieAlgebras

Representation

 


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