find the induced representation on an invariant subspace of the representation space - Maple Programming Help

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LieAlgebras[SubRepresentation] - find the induced representation on an invariant subspace of the representation space

Calling Sequences

     SubRepresentation(ρ,S, 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

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

 

Description

Examples

Description

• 

If ρ: 𝔤  glV is a representation of a Lie algebra 𝔤 on a vector space V, then S is a ρ-invariant subspace of V if ρxY  S for all x  𝔤 and Y S. 

• 

The command SubRepresentation(ρ,S,W) returns the representation φ of 𝔤 on the vector space S defined by φxy =ρxY, where x  𝔤 and Y  S.

Examples

withDifferentialGeometry:withLieAlgebras:withLibrary:

 

Example 1.

We shall define a 4-dimensional representation ρ of a 4-dimensional Lie algebra taken from the DifferentialGeometry Library, find an invariant subspace S of ρ, and calculate the subrepresentation of ρ on S.

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.

V > 

DGsetupL:

 

Initialize the representation space V.

Alg1 > 

DGsetupx1,x2,x3,x4,V:

 

Define the matrices 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 with the Representation command.

V > 

ρRepresentationAlg1,V,M

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

(2.2)

 

Define a subspace S of V.

Alg1 > 

SD_x1,D_x2,D_x3

S:=D_x1,D_x2,D_x3

(2.3)

 

We can use the Query command to check that S is a ρ-invariant subspace.

V > 

Queryρ,S,InvariantSubspace

true

(2.4)

 

Define a frame for the induced representation of ρ on S.

V > 

DGsetupy1,y2,y3,W:

W > 

φSubRepresentationρ,S,W

φ:=e1,000000000,e2,001000000,e3,010000000,e4,200011001

(2.5)
Alg1 > 

Queryφ,Representation

true

(2.6)

See Also

DifferentialGeometry

Library

LieAlgebras

Query

Representation

Retrieve