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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
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Description
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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.
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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).
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Examples
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Example 1.
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![L := Retrieve("Winternitz", 1, [4, 7], Alg1)](/support/helpjp/helpview.aspx?si=6629/file05827/math118.png)
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![_DG([["LieAlgebra", Alg1, [4]], [[[2, 3, 1], 1], [[1, 4, 1], 2], [[2, 4, 2], 1], [[3, 4, 2], 1], [[3, 4, 3], 1]]])](/support/helpjp/helpview.aspx?si=6629/file05827/math121.png)
| (2.1) |
Initialize the Lie algebra Alg1.
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Initialize the representation space V.
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Alg1 >
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![DGsetup([x1, x2, x3, x4], V)](/support/helpjp/helpview.aspx?si=6629/file05827/math141.png)
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Define the Matrices which specify a representation of Alg1 on V.
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V >
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![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]])]](/support/helpjp/helpview.aspx?si=6629/file05827/math153.png)
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Define the representation.
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V >
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| (2.2) |
Define a subspace S of V and use the Query command to check that it is invariant.
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Alg1 >
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![S := [D_x1]](/support/helpjp/helpview.aspx?si=6629/file05827/math179.png)
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![[_DG([["vector", V, []], [[[1], 1]]])]](/support/helpjp/helpview.aspx?si=6629/file05827/math182.png)
| (2.3) |
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V >
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| (2.4) |
Pick a complement C to S in V. This complement need not be invariant.
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V >
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![Query(rho, [D_x2, D_x3, D_x4], "InvariantSubspace")](/support/helpjp/helpview.aspx?si=6629/file05827/math203.png)
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| (2.5) |
Define a vector space for the induced representation of rho on V/S.
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V >
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![DGsetup([y1, y2, y3], W)](/support/helpjp/helpview.aspx?si=6629/file05827/math218.png)
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| (2.6) |
Compute the quotient representation. Note that in this example the matrices are just the lower 3x3 blocks of the matrices in the original representation.
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W >
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![phi := QuotientRepresentation(rho, S, [D_x2, D_x3, D_x4], W)](/support/helpjp/helpview.aspx?si=6629/file05827/math229.png)
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| (2.7) |
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Alg1 >
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| (2.8) |
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