LieAlgebras[QuotientRepresentation] - find the induced representation defined on the quotient space of the representation space by an invariant subspace
QuotientRepresentation(ρ, S, C, W)
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
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 ρ‾xY = 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‾.
L ≔ Retrieve⁡Winternitz,1,4,7,Alg1
Initialize the Lie algebra Alg1.
Initialize the representation space V.
Define the matrices M which specify a representation of Alg1 on V.
M ≔ Matrix⁡0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,Matrix⁡0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,Matrix⁡0,−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.
ρ ≔ Representation⁡Alg1,V,M
Define a subspace S ⊂ V and use the Query command to check that it is invariant.
S ≔ D_x1
Pick a complement C=span[Dx2, Dx3, Dx4].This complement need not be invariant.
Define a vector space for the induced representation of ρ on V/S.
frame name: W
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.
φ ≔ QuotientRepresentation⁡ρ,S,D_x2,D_x3,D_x4,W
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