LieAlgebras[RestrictedRepresentation] - find the restriction of a representation of a subalgebra
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
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.
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..
L ≔ Retrieve⁡Winternitz,1,4,7,Alg
Initialize the Lie algebra Alg1.
Initialize the representation space V.
Define the adjoint representation.
ρ ≔ Adjoint⁡Alg,representationspace=V
Define a 2-dimensional abelian subalgebra of Alg1 using the command LieAlgebraData.
H1 ≔ e1,e2
L1 ≔ LieAlgebraData⁡H1,Alg1
Lie algebra: Alg1
ρ1 ≔ RestrictedRepresentation⁡ρ,Alg1
Define a 2 dimensional solvable subalgebra of Alg1, one that is not adapted to the basis e1, e2, e3, e4.
H2 ≔ e4+e2,e2
L2 ≔ LieAlgebraData⁡H2,Alg2
Lie algebra: Alg2
ρ2 ≔ RestrictedRepresentation⁡ρ,Alg2,H2
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