Vol. 23, No. 3, 1967

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Existence of Levi factors in certain algebraic groups

Jim Humphreys

Vol. 23 (1967), No. 3, 543–546

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at characteristic p is problematic, in view of an example of a group having no Levi factor constructed by Chevalley (unpublished).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
Received: 10 January 1967
Published: 1 December 1967
Jim Humphreys
Department of Mathematics and Statistics
University of Massachusetts
Lederle Graduate Research Tower
Amherst MA 01003-9305
United States