LieAlgebras[RootSpace] - find a root space for a semi-simple Lie algebra from a Cartan subalgebra or a root space decomposition
Calling Sequences
RootSpace()
RootSpace()
Parameters
RV - a column vector
CSA - a list of vectors in a semi-simple Lie algebra, defining a Cartan subalgebra
RSD - a table, defining a root space decomposition of a semi-simple Lie algebra
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Description
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The second calling sequence simply returns the table entry in the table of root spaces corresponding to the root .
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Examples
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Example 1.
Use the command SimpleLieAlgebraData to obtain the Lie algebra data for the simple Lie algebra This is the 15-dimensional Lie algebra of trace-free, skew-Hermitian matrices.
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Initialize the Lie algebra
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The command StandardRepresentation will produce the actual matrices defining . (This command only applies to Lie algebras constructed by the procedure.)
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The Lie algebra elements corresponding to the complex diagonal matrices define a Cartan subalgebra.
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We check this is indeed a Cartan subalgebra using the Query command
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Here is the root space corresponding to the root <I, I, -I>.
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We check that the X is an eigenvector for the elements of the Cartan subalgebra.
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The column vector <I, I, I> is not a root
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Example 2.
Here is the full root space decomposition for the Lie algebra from Example 1.
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The second calling sequence for simply converts the given root vector to a list and extracts the corresponding root space from the root space decomposition table.
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