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LieAlgebras[DirectSumOfRepresentations] - form the direct sum representation for a pair of representations of a Lie algebra
Calling Sequences
DirectSum(R, W)
Parameters
R - a list R = [rho1, rho2, ...] of representations rho1, rho2, ... of a Lie algebra g on vector spaces V1, V2, ...
W - a Maple name or string, the name of the frame for the representation space for the direct sum representation
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Description
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Let W = V1 + V2 + ... (direct sum). The command DirectSum(R, W) returns the representation phi on W defined by phi(x)(y) = rho1(x)(y1) + rho2(x)(y2) + ..., where y = y1 + y2 + ..., y1 in V1, y2 in V2, ... and x in g.
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Examples
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Example 1.
Define the standard representation and the adjoint representation for sl2. Then form the direct sum representation. First, setup the representation spaces.
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![DGsetup([x1, x2], V1)](/support/helpjp/helpview.aspx?si=6627/file05801/math82.png)
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V1 >
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![DGsetup([y1, y2, y2], V2)](/support/helpjp/helpview.aspx?si=6627/file05801/math86.png)
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V2 >
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![DGsetup([z1, z2, z3, z4, z5], W1)](/support/helpjp/helpview.aspx?si=6627/file05801/math90.png)
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W1 >
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![DGsetup([z1, z2, z3, z4, z5, z6], W2)](/support/helpjp/helpview.aspx?si=6627/file05801/math94.png)
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Define the standard representation.
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W2 >
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![M1 := [Matrix([[0, 1], [0, 0]]), Matrix([[1, 0], [0, -1]]), Matrix([[0, 0], [1, 0]])]](/support/helpjp/helpview.aspx?si=6627/file05801/math102.png)
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![M1 := [Matrix(2, 2, {(1, 1) = 0, (1, 2) = 1, (2, 1) = 0, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 1, (1, 2) = 0, (2, 1) = 0, (2, 2) = -1}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 1, (2, 2) = 0})]](/support/helpjp/helpview.aspx?si=6627/file05801/math105.png)
| (2.1) |
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W2 >
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![_DG([["LieAlgebra", sl2, [3, table( [ ] )]], [[[1, 2, 1], -2], [[1, 3, 2], 1], [[2, 3, 3], -2]]])](/support/helpjp/helpview.aspx?si=6627/file05801/math112.png)
| (2.2) |
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W2 >
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| (2.3) |
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sl2 >
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![rho1 := [[e1, Matrix(2, 2, {(1, 1) = 0, (1, 2) = 1, (2, 1) = 0, (2, 2) = 0})], [e2, Matrix(2, 2, {(1, 1) = 1, (1, 2) = 0, (2, 1) = 0, (2, 2) = -1})], [e3, Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 1, (2, 2) = 0})]]](/support/helpjp/helpview.aspx?si=6627/file05801/math126.png)
| (2.4) |
Define the adjoint representation.
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sl2 >
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![rho2 := [[e1, Matrix(3, 3, {(1, 1) = 0, (1, 2) = -2, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0})], [e2, Matrix(3, 3, {(1, 1) = 2, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -2})], [e3, Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = -1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 2, (3, 3) = 0})]]](/support/helpjp/helpview.aspx?si=6627/file05801/math137.png)
| (2.5) |
Define the direct sum representation of rho1 and rho2.
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sl2 >
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![phi1 := DirectSumOfRepresentations([rho1, rho2], W1)](/support/helpjp/helpview.aspx?si=6627/file05801/math149.png)
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| (2.6) |
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sl2 >
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| (2.7) |
Define the direct sum of 3 copies of rho1.
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sl2 >
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![phi2 := DirectSumOfRepresentations([rho1, rho1, rho1], W2)](/support/helpjp/helpview.aspx?si=6627/file05801/math169.png)
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| (2.8) |
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sl2 >
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| (2.9) |
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