Regular subalgebras and nilpotent orbits of real graded Lie algebras

Heiko Dietrich, Paolo Faccin, Willem Adriaan de Graaf

Research output: Contribution to journalArticleResearchpeer-review

Abstract

For a semisimple Lie algebra over the complex numbers, Dynkin (1952) developed an algorithm to classify the regular semisimple subalgebras, up to conjugacy by the inner automorphism group. For a graded semisimple Lie algebra over the complex numbers, Vinberg (1979) showed that a classification of a certain type of regular subalgebras (called carrier algebras) yields a classification of the nilpotent orbits in a homogeneous component of that Lie algebra. Here we consider these problems for (graded) semisimple Lie algebras over the real numbers. First, we describe an algorithm to classify the regular semisimple subalgebras of a real semisimple Lie algebra. This also yields an algorithm for listing, up to conjugacy, the carrier algebras in a real graded semisimple real algebra. We then discuss what needs to be done to obtain a classification of the nilpotent orbits from that; such classifications have applications in differential geometry and theoretical physics. Our algorithms are implemented in the language of the computer algebra system GAP, using our package CoReLG; we report on example computations.
Original languageEnglish
Pages (from-to)1044-1079
Number of pages36
JournalJournal of Algebra
Volume423
DOIs
Publication statusPublished - 2015

Cite this

Dietrich, Heiko ; Faccin, Paolo ; de Graaf, Willem Adriaan. / Regular subalgebras and nilpotent orbits of real graded Lie algebras. In: Journal of Algebra. 2015 ; Vol. 423. pp. 1044-1079.
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Regular subalgebras and nilpotent orbits of real graded Lie algebras. / Dietrich, Heiko; Faccin, Paolo; de Graaf, Willem Adriaan.

In: Journal of Algebra, Vol. 423, 2015, p. 1044-1079.

Research output: Contribution to journalArticleResearchpeer-review

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