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LieAlgebras[RestrictedRootSpaceDecomposition] - find the real root space decomposition of a non-compact semi-simple Lie algebra with respect to an Abelian subalgebra
Calling Sequences
RestrictedRootSpaceDecomposition()
Parameters
A - a list of vectors, defining an Abelian subalgebra of a non-compact, semi-simple Lie algebra
RSD - a table, definining a root space decomposition
CSA - a list of vectors, defining the Cartan subalgebra used to calculate the root space decomposition
Description
Let g be a semi-simple Lie algebra and h a Cartan subalgebra. Let be a basis forThe linear transformations are simultaneously diagonalizable over C -- if x 2 g is a common eigenvector for all these transformations, then . The -tuples are called the roots of with respect to the Cartan subalgebra and the root space decomposition of g with respect to h.
Now suppose that is non-compact. To obtain the restricted root space decomposition, let g = k 4p be a Cartan decomposition of g. Let a={ be a maximal Abelian subalgebra of p. Then the matrices for all commute and have real common eigenspaces. The resulting eigenspace decomposition , where is the centralizer of a in g, called the restricted root space decomposition. The restricted roots are q-tuples of real numbers. The common eigenspacesneed not be 1-dimensional. Eigenspaces associated to different roots are orthogonal with respect to the Killing form.
If is a Cartan involution which preserves , then a can be chosen as a subalgebra of h.
For the second calling sequence, the restricted roots are determined by restricting the roots in the root space decomposition (as functionals on h) to the subalgebra a.
Examples
Example 1.
We find the restricted root space decomposition for the Lie algebra . This is the 15-dimensional Lie algebra of matrices which are skew-symmetric with respect to the quadratic form . We use the command SimpleLieAlgebraData to initialize .
To find a suitable candidate for the subspace a, we first calculate a Cartan subalgebra.
Now we shall use the Signature command to find a subalgebra on which the Killing form is positive-definite.
We can use the subspace a to find the restricted root space decomposition for .
First calling sequence.
Second calling sequence.
For the second calling sequence we first need the root space decomposition with respect to the Cartan subalgebra .
It is instructive to compare the root space decomposition (equation (2.6) and the restricted root space decomposition (equation (2.8)). First, note that the roots for are vectors in 3-dimensions (since the Cartan subalgebra is 3-dimensional) while the roots for RRSD are vectors in 2-dimensions (since the subspace a is 2-dimensional). Second, we see that the first 2 components of the roots for are all real and the 3rd component is pure imaginary. This reflects the fact that the basis we have used for the Cartan subalgebra is adapted to the Cartan decomposition. Third, we see that the restricted roots are just the projections [The restricted root space for is just the direct sum of the root spaces for the roots of the form Finally, and this is the whole point, the restricted root spaces have a real basis.
Example 2
We find a restricted root space decomposition for so*(8).
We calculate a Cartan subalgebra and a subspace on which the Killing form is positive-definite.
Note here that the restricted root spaces for have dimensions 1or 4.
See Also
DifferentialGeometry, CartanSubalgebra, Killing, LieAlgebras, RootSpaceDecomposition, Signature, SimpleLieAlgebraData, SimpleLieAlgebraProperties
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