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The Jacobson--Morozov theorem and complete reducibility of Lie subalgebras

Lookup NU author(s): Dr David Stewart



This is the authors' accepted manuscript of an article that has been published in its final definitive form by Wiley-Blackwell, 2018.

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In this paper we determine the precise extent to which the classical sl_2-theory of complex semisimple finite-dimensional Lie algebras due to Jacobson--Morozov and Kostant can be extended to positive characteristic. This builds on work of Pommerening and improves significantly upon previous attempts due to Springer--Steinberg and Carter/Spaltenstein. Our main advance arises by investigating quite fully the extent to which subalgebras of the Lie algebras of semisimple algebraic groups over algebraically closed fields k are G-completely reducible, a notion essentially due to Serre. For example if G is exceptional and char k=p\geq 5, we classify the triples (\h,\g,p) such that there exists a non-G-completely reducible subalgebra of \g isomorphic to \h. We do this also under the restriction that \h be a p-subalgebra of \g. We find that the notion of subalgebras being G-completely reducible effectively characterises when it is possible to find bijections between the conjugacy classes of sl_2-subalgebras and nilpotent orbits and it is this which allows us to prove our main theorems. For absolute completeness, we also show that there is essentially only one occasion in which a nilpotent element cannot be extended to an sl_2-triple when p\geq 3: this happens for the exceptional orbit in G_2 when p=3.

Publication metadata

Author(s): Stewart DI, Thomas AR

Publication type: Article

Publication status: Published

Journal: Proceedings of the London Mathematical Society

Year: 2018

Volume: 116

Issue: 1

Pages: 68-100

Print publication date: 01/01/2018

Online publication date: 29/09/2017

Acceptance date: 26/07/2017

Date deposited: 02/08/2017

ISSN (print): 0024-6115

ISSN (electronic): 1460-244X

Publisher: Wiley-Blackwell


DOI: 10.1112/plms.12067


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