Classification of the maximal subalgebras of exceptional Lie algebras over fields of good characteristic

2017/11/19 by Premet, Alexander, Stewart, David I.
#17B45 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1711.06988

Abstract

Let G be an exceptional simple algebraic group over an algebraically closed field k and suppose that the characteristic p of k is a good prime for G. In this paper we classify the maximal Lie subalgebras \mathfrakm of the Lie algebra \mathfrakg=\rm Lie(G). Specifically, we show that one of the following holds: \mathfrakm=\rm Lie(M) for some maximal connected subgroup M of G, or \mathfrakm is a maximal Witt subalgebra of \mathfrakg, or \mathfrakm is a maximal \itexotic semidirect product. The conjugacy classes of maximal connected subgroups of G are known thanks to the work of Seitz, Testerman and Liebeck--Seitz. All maximal Witt subalgebras of \mathfrakg are G-conjugate and they occur when G is not of type \rm E₆ and p-1 coincides with the Coxeter number of G. We show that there are two conjugacy classes of maximal exotic semidirect products in \mathfrakg, one in characteristic 5 and one in characteristic 7, and both occur when G is a group of type \rm E₇.

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