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LieAlgebras[SimpleRoots] - find the simple roots for a set of positive roots
Calling Sequences
SimpleRoots(PR
Parameters
PR - a list of vectors, giving the positive roots of a simple Lie algebra
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Description
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If  is set of positive roots,then a root  is called a simple root if it is not a sum of any other 2 positive roots.
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The number of simple roots equals the rank of the Lie algebra.
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Examples
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Example 1.
We calculate the simple roots for the Lie algebra  This is the 36-dimensional Lie algebra of  matrices  which are skew-symmetric with respect to the skew form ![Q = [[[0,I[4]],[-I[4],0]]].](/support/helpjp/helpview.aspx?si=6592/file05843/math91.png)
We use the command SimpleLieAlgebraData to obtain the structure equations for this Lie algebra.
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![LD := SimpleLieAlgebraData("sp(8, R))", sp8R, labelformat = "gl", labels = ['E', 'omega'])](/support/helpjp/helpview.aspx?si=6592/file05843/math100.png)
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| (2.1) |
The following diagonal elements define a Cartan subalgebra. (This can be calculated using the command CartanSubalgebra).
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sp8R >
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![CSA_sp8R := [E11, E22, E33, E44]](/support/helpjp/helpview.aspx?si=6592/file05843/math118.png)
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![[_DG([["vector", sp8R, []], [[[1], 1]]]), _DG([["vector", sp8R, []], [[[6], 1]]]), _DG([["vector", sp8R, []], [[[11], 1]]]), _DG([["vector", sp8R, []], [[[16], 1]]])]](/support/helpjp/helpview.aspx?si=6592/file05843/math121.png)
| (2.2) |
Here is the corresponding root space decomposition.
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sp8R >
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| (2.3) |
We calculate the positive roots for  .
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sp8R >
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| (2.4) |
The rank of  is 4 so we should find 4 positive roots.
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sp8R >
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![SR_sp8R := [Vector(4, {(1) = 0, (2) = 0, (3) = -1, (4) = 1}), Vector(4, {(1) = 0, (2) = -1, (3) = 1, (4) = 0}), Vector(4, {(1) = -1, (2) = 1, (3) = 0, (4) = 0}), Vector(4, {(1) = 2, (2) = 0, (3) = 0, (4) = 0})]](/support/helpjp/helpview.aspx?si=6592/file05843/math164.png)
| (2.5) |
We check that the positive roots are positive integer linear combinations of the simple roots with the GetComponents command.
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sp8R >
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![[[0, 1, 1, 0], [1, 0, 0, 0], [0, 0, 1, 1], [0, 1, 0, 0], [2, 2, 2, 1], [0, 0, 1, 0], [0, 0, 2, 1], [1, 1, 1, 1], [0, 0, 0, 1], [1, 2, 2, 1], [0, 1, 1, 1], [0, 2, 2, 1], [1, 1, 2, 1], [1, 1, 1, 0], [0, 1, 2, 1], [1, 1, 0, 0]]](/support/helpjp/helpview.aspx?si=6592/file05843/math178.png)
| (2.6) |
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