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LieAlgebras[SubRepresentation] - find the induced representation on an invariant subspace of the representation space
Calling Sequences
SubRepresentation(rho, S, 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
W - a Maple name or string, giving the frame name for the representation space for the subrepresentation
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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. The command SubRepresentation(rho, W) returns the representation phi of g on the vector space S defined by phi(x)(y) = rho(x)(y) for all x in g and y in S.
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Examples
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Example 1.
We shall define a 4-dimensional representation rho of a 4 dimensional Lie algebra taken from the DifferentialGeometry Library, find an invariant subspace S of rho, and calculate the subrepresentation of rho on S.
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![L := Retrieve("Winternitz", 1, [4, 7], Alg1)](/support/helpjp/helpview.aspx?si=6633/file05846/math106.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=6633/file05846/math109.png)
| (2.1) |
Initialize the Lie algebra Alg1.
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V >
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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=6633/file05846/math129.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=6633/file05846/math141.png)
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Define the representation with the Representation command.
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V >
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| (2.2) |
Define a subspace of V.
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Alg1 >
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![S := [D_x1, D_x2, D_x3]](/support/helpjp/helpview.aspx?si=6633/file05846/math165.png)
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![[_DG([["vector", V, []], [[[1], 1]]]), _DG([["vector", V, []], [[[2], 1]]]), _DG([["vector", V, []], [[[3], 1]]])]](/support/helpjp/helpview.aspx?si=6633/file05846/math168.png)
| (2.3) |
We can use the Query command to check that S is a rho invariant subspace.
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V >
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| (2.4) |
Define a frame for the induced representation of rho on S.
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V >
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![DGsetup([y1, y2, y3], W)](/support/helpjp/helpview.aspx?si=6633/file05846/math194.png)
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W >
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| (2.5) |
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Alg1 >
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| (2.6) |
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