The JacobsonMorozov theorem and complete reducibility of Lie subalgebras
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 Dr David Stewart




Author(s)   Stewart DI, Thomas AR 
Publication type   Article 
Journal   Proceedings of the London Mathematical Society 
Year   2017 
Volume   
Issue   
Pages   Epub ahead of print 
ISSN (print)   00246115 
ISSN (electronic)   1460244X 



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In this paper we determine the precise extent to which the classical sl_2theory of complex semisimple finitedimensional Lie algebras due to JacobsonMorozov and Kostant can be extended to positive characteristic. This builds on work of Pommerening and improves significantly upon previous attempts due to SpringerSteinberg 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 Gcompletely 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 nonGcompletely reducible subalgebra of \g isomorphic to \h. We do this also under the restriction that \h be a psubalgebra of \g. We find that the notion of subalgebras being Gcompletely reducible effectively characterises when it is possible to find bijections between the conjugacy classes of sl_2subalgebras 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_2triple when p\geq 3: this happens for the exceptional orbit in G_2 when p=3. 



Publisher   WileyBlackwell 
URL   https://doi.org/10.1112/plms.12067 
DOI   10.1112/plms.12067 

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