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
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.
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.
L≔RetrieveWinternitz,1,4,7,Alg1
L:=e1,e4=2e1,e2,e3=e1,e2,e4=e2,e3,e4=e2+e3
Initialize the Lie algebra Alg1.
DGsetupL:
Initialize the representation space V.
DGsetupx1,x2,x3,x4,V:
Define the matrices which specify a representation of Alg1 on V.
M≔Matrix0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,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,Matrix−2,0,0,0,0,−1,−1,0,0,0,−1,0,0,0,0,0:
Define the representation with the Representation command.
ρ≔RepresentationAlg1,V,M
Define a subspace S of V.
S≔D_x1,D_x2,D_x3
S:=D_x1,D_x2,D_x3
We can use the Query command to check that S is a ρ-invariant subspace.
Queryρ,S,InvariantSubspace
true
Define a frame for the induced representation of ρ on S.
DGsetupy1,y2,y3,W:
φ≔SubRepresentationρ,S,W
Queryφ,Representation
See Also
DifferentialGeometry
Library
LieAlgebras
Query
Representation
Retrieve
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