A computational approach to the Kostant–Sekiguchi correspondence

Abstract

Let g be a real form of a simple complex Lie algebra. Based on ideas of ‐okovi´ c 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 0 of a complex simple Lie algebra g c with Cartan decompositions gD k p and g 0 D k 0 p 0 . We describe an explicit construction of an isomorphism g! g 0 , respecting the given Cartan decompositions, which fails if and only if g and g 0 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.