On the smoothness of normalisers, the subalgebra structure of modular Lie algebras, and the cohomology of small representations

  • Sebastian Herpel

  • David I. Stewart

On the smoothness of normalisers, the subalgebra structure of modular Lie algebras, and the cohomology of small representations cover
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Abstract

We provide results on the smoothness of normalisers in connected reductive algebraic groups GG over fields kk of positive characteristic pp. Specifically we we give bounds on pp which guarantee that normalisers of subalgebras of g\mathfrak{g} in GG are smooth, i.e.so that the Lie algebras of these normalisers coincide with the infinitesimal normalisers. One of our main tools is to exploit cohomology vanishing of small dimensional modules. Along the way, we obtain complete reducibility results for small dimensional modules in the spirit of similar results due to Jantzen, Guralnick, Serre and Bendel--Nakano--Pillen.

Cite this article

Sebastian Herpel, David I. Stewart, On the smoothness of normalisers, the subalgebra structure of modular Lie algebras, and the cohomology of small representations. Doc. Math. 21 (2016), pp. 1–37

DOI 10.4171/DM/525