Let g be a real form of a
simple complex Lie algebra. Based on ideas of Ðoković 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 gc 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 realizations of the same
algebra.
Keywords
real Lie algebra, real nilpotent orbit, computational
methods, Kostant–Sekiguchi correspondence
