On the smoothness of normalisers and the subalgebra structure of modular Lie algebras

Abstract

We provide results on the smoothness of normalisers in connected reductive algebraic groups $G$ over fields $k$ of positive characteristic $p$. Specifically we we give bounds on $p$ which guarantee that normalisers of subalgebras of $\mathfrak{g}$ in $G$ 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.

31 pages. Final version to appear in Doc. Math

References in corpus (1)

• Simple Irreducible Subgroups of Exceptional Algebraic Groups

Cited by in corpus (1)

• Generic property and conjugacy classes of homogeneous Borel subalgebras of restricted Lie algebras