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LieAlgebras[RestrictedRepresentation] - find the restriction of a representation of a subalgebra

Calling Sequences

     RestrictedRepresentation(ρ, alg,W)

Parameters

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

     alg       - a Maple name or string, giving the frame name of an initialized algebra, corresponding to a subalgebra of 𝔤

     H         - (optional) a list of vectors in 𝔤 defining a basis for a subalgebra of 𝔤

 

Description

Examples

Description

• 

If ρ:𝔤  glV is a representation and 𝔥 is a subalgebra of 𝔤 , then the restriction of ρ to 𝔥 is the representation φ:𝔥  glV defined by φxY =ρxY, where x  𝔥 and Y  V. 

• 

The command RestrictedRepresentation(rho, alg, H) returns the restriction of the representation ρ to the subalgebra defined by the vectors in the list H. The subalgebra defined by the vectors H must be initialized as a Lie algebra in its own right with the name alg.

• 

If the basis e1, e2 ,e3, ... , en  for 𝔤 is adapted to the subalgebra defined by H in the sense that H = [e1, e2 ,... ,ep ], then the list H need not be specified in the calling sequence for RestrictedRepresentation.

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, define a subalgebra, and calculate the restricted representation of ρ to the subalgebra..

LRetrieveWinternitz,1,4,7,Alg

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 adjoint representation.

V > 

ρAdjointAlg,representationspace=V

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

(2.2)

 

Define a 2-dimensional abelian subalgebra of Alg1 using the command LieAlgebraData.

Alg1 > 

H1e1,e2

H1:=e1,e2

(2.3)
Alg1 > 

L1LieAlgebraDataH1,Alg1

L1:=

(2.4)
Alg1 > 

DGsetupL1,P,p

Lie algebra: Alg1

(2.5)
Alg1 > 

ρ1RestrictedRepresentationρ,Alg1

ρ1:=P1,0002000000000000,P2,0010000100000000

(2.6)
Alg1 > 

Queryρ1,Representation

true

(2.7)

 

Example 2.

Define a 2 dimensional solvable subalgebra of Alg1, one that is not adapted to the basis e1, e2, e3, e4.

Alg2 > 

H2e4+e2,e2

H2:=e4+e2,e2

(2.8)
Alg1 > 

L2LieAlgebraDataH2,Alg2

L2:=e1,e2=e2

(2.9)
Alg1 > 

DGsetupL2,Q,q

Lie algebra: Alg2

(2.10)
Alg1 > 

ρ2RestrictedRepresentationρ,Alg2,H2

ρ2:=Q1,2010011100100000,Q2,0010000100000000

(2.11)
Alg1 > 

Queryρ2,Representation

true

(2.12)

 

See Also

DifferentialGeometry

Library

LieAlgebras

LieAlgebraData

Query

Representation

Retrieve