Computing Split Maximal Toral Subalgebras of Lie algebras over Fields of Small Characteristic

Details Computing Split Maximal Toral Subalgebras of Lie algebras over Fields of Small Characteristic Computing Split Maximal Toral Subalgebras of Lie algebras over Fields of Small Characteristic

2011-08-15

arxiv.org/abs/1108.2932v3

Mathematics

Rings and Algebras

18 pages, 4 figures

Scientific paper

Important subalgebras of a Lie algebra of an algebraic group are its toral subalgebras, or equivalently (over fields of characteristic 0) its Cartan subalgebras. Of great importance among these are ones that are split: their action on the Lie algebra splits completely over the field of definition. While algorithms to compute split maximal toral subalgebras exist and have been implemented [Ryb07, CM09], these algorithms fail when the Lie algebra is defined over a field of characteristic 2 or 3. We present heuristic algorithms that, given a reductive Lie algebra L over a finite field of characteristic 2 or 3, find a split maximal toral subalgebra of L. Together with earlier work [CR09] these algorithms are very useful for the recognition of reductive Lie algebras over such fields.

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