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
We use the command SimpleLieAlgebraData to obtain the structure equations for this Lie algebra.
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| (2.1) |
The following diagonal elements define a Cartan subalgebra. (This can be calculated using the command CartanSubalgebra).
sp8R >
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| (2.2) |
Here is the corresponding root space decomposition.
sp8R >
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| (2.3) |
We calculate the positive roots for .
sp8R >
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The rank of is 4 so we should find 4 positive roots.
sp8R >
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| (2.5) |
We check that the positive roots are positive integer linear combinations of the simple roots with the GetComponents command.
sp8R >
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| (2.6) |
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