Example 1
We obtain the structural properties for the Lie algebra . This is the split real form for Lie algebras of root type A. First, we use the command SimpleLieAlgebraData to initialize this Lie algebra.
We use the command SimpleLieAlgebraProperties to obtain the properties of the Lie algebra .
Here are the indices for the table Properties.
It is convenient to use the map and op commands to display the indices as a list of strings.
Here are some of the individual properties for the Lie algebra .
The command LieAlgebraRoots lists the roots associated to this root space decomposition. Note that the roots are all real.
Note that the first non-zero component of each positive root is positive and that the first non-zero component of each negative root is negative.
It is easy to check that positive roots are positive linear combinations of the simple roots.
We check that the Killing form is positive-definite on the first list of vectors CD[1] and negative-definitive on the second list of vectors.
Example 2
We obtain the structural properties for the Lie algebra . This is the compact form for Lie algebras of root type A. First, we use the command SimpleLieAlgebraData to initialize this Lie algebra.
We use the command SimpleLieAlgebraProperties to obtain the properties of the Lie algebra .
It is convenient to use the map and op commands to display the indices as a list of strings.
Here are some of the individual properties for the Lie algebra .
The roots are all pure imaginary numbers so that this is indeed the compact form.
The first non-zero coefficient of in each positive root is positive.
Example 3
We obtain the structural properties for the Lie algebra . First, we use the command SimpleLieAlgebraData to initialize this Lie algebra.
We use the command SimpleLieAlgebraProperties to obtain the properties of the Lie algebra .
It is convenient to use the map and op commands to display the indices as a list of strings.
Here are some of the individual properties for the Lie algebra .
Note that first two components of the roots are real and the third component is pure imaginary.
Since the root vectors are neither real nor pure imaginary, we have a restricted root space decomposition.
The restricted roots are the projections of the roots which yield real vectors. Since the restricted root [1,1] is the projection of the 2 roots [1, 1, 2I] and [1, 1, -2I], the restricted root space for [1,1] is 2-dimensional. Note also that while the root spaces are defined over C, the restricted root space are real subspaces of .
Let's us check the properties of this KAN decomposition. The first list of vectors defines a subalgebra with negative-definite Killing form.
The second list of vectors defines an abelian subalgebra.
The third list of vectors defines a nilpotent Lie algebra.