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LieAlgebras[QuotientRepresentation] - find the induced representation on the quotient space of the representation space by an invariant subspace
Calling Sequences
QuotientRepresentation(rho, S, C, W)
Parameters
rho - a representation of a Lie algebra g on a vector space V
S - a list of vectors in V whose span defines a rho 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
Description
If rho: g -> gl(V) is a representation and S is a subspace of V, then S is rho invariant if rho(x)(y) in S for all x in g and y in S. For any y in V, let [y] = y + S denote the coset of y in the quotient space V/S.
The command QuotientRepresentation(rho, S, C, W) returns the representation phi of g on the vector space V/S defined by phi(x)([y]) = [rho(x)(y)] for all x in g and [y] in V/S. The coset representatives of the vectors in C in the quotient space V/S give the basis used in V/S to calculate the matrices for the linear transformation phi(x).
Examples
Example 1.
Initialize the Lie algebra Alg1.
Initialize the representation space V.
Define the Matrices which specify a representation of Alg1 on V.
Define the representation.
Define a subspace S of V and use the Query command to check that it is invariant.
Pick a complement C to S in V. This complement need not be invariant.
Define a vector space for the induced representation of rho on V/S.
Compute the quotient representation. Note that in this example the matrices are just the lower 3x3 blocks of the matrices in the original representation.
See Also
DifferentialGeometry, Library, LieAlgebras, Query, Retrieve
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