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LieAlgebras[RootSpace] - find a root space for a semi-simple Lie algebra from a Cartan subalgebra or a root space decomposition
Calling Sequences
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
Description
Let g be a Lie algebra and h a Cartan subalgebra. Let be a basis for . A root for g with respect to this basis is a non-zero -tuple of complex numbers such that (*) for some .
The set of which satisfy (*) is called the root space of g defined by and denoted by A basic theorem in the structure theorem of semi-simple Lie algebras asserts that the root spaces are 1-dimensional.
The first call sequence calculates the root space for a given root. If is not a root, then the zero vector (in ) is returned.
The second calling sequence simply returns the table entry in the table of root spaces corresponding to the root .
Examples
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.
Initialize the Lie algebra
The command StandardRepresentation will produce the actual matrices defining . (This command only applies to Lie algebras constructed by the procedure.)
The Lie algebra elements corresponding to the complex diagonal matrices define a Cartan subalgebra.
We check this is indeed a Cartan subalgebra using the Query command
Here is the root space corresponding to the root <I, I, -I>.
We check that the X is an eigenvector for the elements of the Cartan subalgebra.
The column vector <I, I, I> is not a root
Example 2.
Here is the full root space decomposition for the Lie algebra from Example 1.
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.
See Also
DifferentialGeometry, CartanSubalgebra, GetComponents, Query, RootSpaceDecomposition, SimpleLieAlgebraData, SimpleLieAlgebraProperties, StandardRepresentation
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