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Example 1.
We consider the Lie algebra
This is the 24-dimensional real Lie algebra of 6×6 complex matrices
which are trace-free and skew-Hermitian with respect to the quadratic form
. We use the command SimpleLieAlgebraData to initialize this Lie algebra.
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| (2.1) |
We use the command SimpleLieAlgebraProperties to obtain the Cartan subalgebra, the root space decomposition, and the simple roots.
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The result
is a table. Here is the Cartan subalgebra for
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| (2.2) |
Here is the root space decomposition for
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| (2.3) |
Here are the positive roots.
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| (2.4) |
Let us find
where
is the first root (2.4)
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| (2.5) |
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| (2.6) |
We check that
is in the Cartan subalgebra.
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| (2.7) |
Here are the root spaces for
and
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| (2.8) |
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| (2.9) |
We check that
defines a Lie subalgebra.
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| (2.10) |
If we scale the vectors X and Y then the structure equations take the standard form for
.
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| (2.11) |
Example 2.
We illustrute how to use RootToCartanSubalgebraElementH to calculate the Cartan matrix for
We first calculate the
for the simple roots
.
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| (2.12) |
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![[_DG([["vector", su33, []], [[[1], -(1/2)*I], [[2], (1/2)*I], [[3], 1/2], [[4], -1/2]]]), _DG([["vector", su33, []], [[[2], -(1/2)*I], [[4], 1/2], [[5], -1/2]]]), _DG([["vector", su33, []], [[[5], 1]]]), _DG([["vector", su33, []], [[[2], (1/2)*I], [[4], 1/2], [[5], -1/2]]]), _DG([["vector", su33, []], [[[1], (1/2)*I], [[2], -(1/2)*I], [[3], 1/2], [[4], -1/2]]])]](/support/helpjp/helpview.aspx?si=6617/file05834/math276.png)
| (2.13) |
Then we calculate the Killing form , restricted to subspace [
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| (2.14) |
The Cartan matrix is given by normalizing the entries of
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| (2.15) |
The Lie algebra 
is a rank 5 simple Lie algebra of type "A". The matrix in (2.15) is therefore correct.
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| (2.16) |