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LieAlgebras[Killing] - find the Killing form (matrix) of a Lie algebra, evaluate the Killing form on a pair of vectors, evaluate the Killing form on a subspace
LieAlgebras[KillingForm] - find the Killing form (symmetric tensor) of a Lie algebra
Calling Sequences
Killing(x, y)
Killing(Alg)
Killing(h)
KilllingForm(Alg)
Parameters
x,y - a pair of vectors in a Lie algebra g
Alg - (optional) the name of a Lie algebra
h - a list of vectors defining a basis for a subspace of a Lie algebra g
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Description
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The Killing form on a Lie algebra g is the symmetric quadratic form defined by Killing(x, y) = trace(ad(x).ad(y)) for any x, y in g. Here ad(x) and ad(y) are the ad matrices for the vector x and y.
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In terms of the structure constants C^k_{ij} with respect to the basis e_i for g, Killing(e_i, e_j) = C^k_{il} C^l_{jk} (summation on l).
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Killing() calculates the Matrix [Killing(e_i, e_j)] = [C^k_{il} C^l_{jk}], where the C^k_{ij} are the structure constants for the current Lie algebra.
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Killing(Alg) calculates the Killing Matrix for the Lie algebra Alg.
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Killing(h) calculates the Killing Matrix restricted to the subalgebra h.
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The command Killing is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form Killing(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Killing(...).
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Examples
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Example 1.
First initialize a Lie algebra and display the Lie bracket multiplication table.
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![L1 := _DG([["LieAlgebra", Alg1, [3]], [[[2, 3, 1], 1], [[1, 3, 2], -1], [[1, 2, 3], 1]]])](/support/helpjp/helpview.aspx?si=6609/file05810/math114.png)
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Alg1 >
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![[[e1, e2] = e3, [e1, e3] = -e2, [e2, e3] = e1]](/support/helpjp/helpview.aspx?si=6609/file05810/math123.png)
| (2.1) |
Compute the Killing form on the vectors x = e1 + e2 and y = e1 - e2 + e3.
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Alg1 >
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![_DG([["vector", Alg1, []], [[[1], 1], [[3], 1]]])](/support/helpjp/helpview.aspx?si=6609/file05810/math138.png)
| (2.2) |
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Alg1 >
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![_DG([["vector", Alg1, []], [[[1], 1], [[2], -1], [[3], 1]]])](/support/helpjp/helpview.aspx?si=6609/file05810/math145.png)
| (2.3) |
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Alg1 >
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| (2.4) |
Compute the Killing form for the current Lie algebra.
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Alg1 >
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| (2.5) |
Compute the Killing form restricted to the subspace S = [e2, e3].
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Alg1 >
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![S := [e2, e3]](/support/helpjp/helpview.aspx?si=6609/file05810/math173.png)
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Alg1 >
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Example 2.
Here is the Killing form for the Lie algebra from Example 1, given as a symmetric, covariant tensor on the Lie algebra.
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![_DG([["tensor", Alg1, [["cov_bas", "cov_bas"], []]], [[[1, 1], -2], [[2, 2], -2], [[3, 3], -2]]])](/support/helpjp/helpview.aspx?si=6609/file05810/math202.png)
| (2.7) |
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