Example 1.
We calculate a Chevalley basis for the rank 2 Lie algebra . We begin with the basis provided by the command SimpleLieAlgebraData.
We will use the choices of the Cartan subalgebra, root space decomposition, and positive roots for contained in SimpleLieAlgebraProperties. (For Lie algebras not created by the SimpleLieAlgebraData command, use CartanSubalgebra, RootSpaceDecomposition, PositiveRoots.)
The Chevalley basis for determined by this Cartan subalgebra and choice of positive roots is:
We calculate the structure equations for in the Chevalley basis and initialize the Lie algebra in this new basis.
To display the multiplication table for this Lie algebra we use interface to increase the maximum inline array display size.
Let us focus in on various parts of the multiplication table. From the first two rows
it is clear that act diagonally and so form a Cartan subalgebra. From the 3rd and 4th columns we can read off the Cartan matrix for as the coefficients of:
The vectors correspond to the roots with being the simple roots. Therefore, from the multiplication table
we can read off the root pattern as , . Finally we note that the vectors satisfy the same structure equations as .
