Existence of Levi factors in certain algebraic groups

Jim Humphreys

Vol. 23 (1967), No. 3, 543–546

DOI: 10.2140/pjm.1967.23.543

Abstract

If G is a connected algebraic linear group with unipotent radical U, Borel and Tits define a Levi factor of G to be any connected reductive subgroup L of G such that G = L.U (semidirect product in the sense of algebraic groups). This differs from the usual notion of Levi decomposition in Lie theory but leads to equivalent results at characteristic 0. The existence of Levi factors at characteristic p is problematic, in view of an example of a group having no Levi factor constructed by Chevalley (unpublished). In this note sufficient conditions are given for a Levi factor to exist, based on the structure of the Lie algebra of G.

Mathematical Subject Classification

Primary: 14.50

Secondary: 22.00

Milestones

Received: 10 January 1967

Published: 1 December 1967