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LieAlgebras[QuotientRepresentation] - find the induced representation defined on the quotient space of the representation space by an invariant subspace

Calling Sequences

     QuotientRepresentation(ρ, S, C, W)

Parameters

     ρ       - a representation of a Lie algebra 𝔤 on a vector space V

     S       - a list of vectors in V whose span defines a ρ invariant subspace of V

     C       - a list of vectors in V defining a complementary subspace to S

     W       - a Maple name or string, giving the frame name for the representation space for the quotient representation

 

Description

Examples

Description

• 

If  ρ:𝔤 glV is a representation and S is a subspace of V, then S is ρ invariant if ρxY  S for all x 𝔤 and Y  S. For any Y V,let Y = Y + S  denote the coset of Y in the quotient space V/S. The induced representation  ρ:𝔤 glV/S is defined

by ρxY = ρxY.

• 

The command QuotientRepresentation(rho, S, C, W) returns the representation ρ. The coset representatives of the vectors in C in the quotient space V/S give the basis used to calculate the matrices for the linear transformation ρ.

Examples

withDifferentialGeometry:withLieAlgebras:withLibrary:

 

Example 1.

LRetrieveWinternitz,1,4,7,Alg1

L:=e1,e4=2e1,e2,e3=e1,e2,e4=e2,e3,e4=e2+e3

(2.1)

 

Initialize the Lie algebra Alg1.

DGsetupL:

 

Initialize the representation space V.

Alg1 > 

DGsetupx1,x2,x3,x4,V:

 

Define the matrices M which specify a representation of Alg1 on V.

V > 

MMatrix0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,Matrix0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,Matrix0,1,0,0,0,0,0,1,0,0,0,1,0,0,0,0,Matrix2,0,0,0,0,1,1,0,0,0,1,0,0,0,0,0:

 

Define the representation.

V > 

ρRepresentationAlg1,V,M

ρ:=e1,0002000000000000,e2,0010000100000000,e3,0100000100010000,e4,2000011000100000

(2.2)

 

Define a subspace S  V and use the Query command to check that it is invariant.

Alg1 > 

SD_x1

S:=D_x1

(2.3)
V > 

Queryρ,S,InvariantSubspace

true

(2.4)

 

Pick a complement C=span[Dx2, Dx3, Dx4].This complement need not be invariant.

V > 

Queryρ,D_x2,D_x3,D_x4,InvariantSubspace

false

(2.5)

 

Define a vector space for the induced representation of ρ on V/S.

V > 

DGsetupy1,y2,y3,W

frame name: W

(2.6)

 

Compute the quotient representation. Note that in this example the matrices are just the lower 3 ×3 blocks of the matrices in the original representation.

W > 

φQuotientRepresentationρ,S,D_x2,D_x3,D_x4,W

φ:=e1,000000000,e2,001000000,e3,001001000,e4,110010000

(2.7)
Alg1 > 

Queryφ,Representation

true

(2.8)

See Also

DifferentialGeometry

Library

LieAlgebras

Query

Retrieve