A computational approach to the Kostant-Sekiguchi correspondence

TL;DR

This paper presents an algorithmic method, implemented in GAP, for computing nilpotent orbits in real forms of simple Lie algebras using the Kostant-Sekiguchi correspondence, and constructs explicit isomorphisms between different real forms.

Contribution

It introduces a novel algorithm for nilpotent orbit computation, implements it in GAP, and creates a comprehensive database for real forms of simple Lie algebras of rank up to 8.

Findings

Successfully computed nilpotent orbits for all real forms of rank ≤ 8

Developed explicit isomorphisms respecting Cartan decompositions

Built a database of nilpotent orbits for these Lie algebras

Abstract

Let g be a real form of a simple complex Lie algebra. Based on ideas of Djokovic 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= k+p and g'=k'+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 realisations of the same algebra.