Vol. 265, No. 2, 2013

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A computational approach to the Kostant–Sekiguchi correspondence

Heiko Dietrich and Willem A. de Graaf

Vol. 265 (2013), No. 2, 349–379
Abstract

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 gof 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 gare 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
Mathematical Subject Classification 2010
Primary: 17B45, 20G20
Milestones
Received: 27 July 2012
Revised: 17 September 2012
Accepted: 24 September 2012
Published: 28 August 2013
Authors
Heiko Dietrich
School of Mathematical Sciences
Monash University
Clayton, VIC 3800
Australia
Willem A. de Graaf
Department of Mathematics
University of Trento
I-38050 Povo
Italy