A computational approach to the kostant-sekiguchi correspondence

摘要

Let g be a real form of a simple complex Lie algebra. Based on ideas of Dokovíc and Vinberg, we describe an algorithm to compute representatives of the nilpotent orbits of g using the Kostant-Sekiguchi correspondence. Our algorithms are implemented for the computer algebra system GAP and, as an application, we have built a database of nilpotent orbits of all real forms of simple complex Lie algebras of rank at most 8. In addition, we consider two real forms g and g'of a complex simple Lie algebra g~c with Cartan decompositions g = t + p and g' = t' + P'. We describe an explicit construction of an isomorphism g → g', respecting the given Cartan decompositions, which fails if and only if g and g' are not isomorphic. This isomorphism can be used to map the representatives of the nilpotent orbits of g to other realizations of the same algebra.