Vanishing of trace forms in low characteristics

Garibaldi, Skip

doi:10.2140/ant.2009.3.543

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Vanishing of trace forms in low characteristics

Skip Garibaldi

Appendix: Alexander Premet

Vol. 3 (2009), No. 5, 543–566

DOI: 10.2140/ant.2009.3.543

Abstract

Every finite-dimensional representation of an algebraic group $G$ gives a trace symmetric bilinear form on the Lie algebra of $G$ . We give criteria in terms of root system data for the existence of a representation such that this form is nonzero or nondegenerate. As a corollary, we show that a Lie algebra of type $E_{8}$ over a field of characteristic 5 does not have a “quotient trace form”, answering a question posed in the 1960s.

Keywords

trace form, E8, Richardson's condition, Dynkin index

Mathematical Subject Classification 2000

Primary: 20G05

Secondary: 17B50, 17B25

Milestones

Received: 16 July 2008

Revised: 5 March 2009

Accepted: 6 April 2009

Published: 9 November 2009

Authors

Skip Garibaldi
Department of Mathematics and Computer Science Emory University Atlanta, GA 30322 United States
http://www.mathcs.emory.edu/~skip/
Alexander Premet
School of Mathematics The University of Manchester Oxford Rd. Manchester, M13 9PL United Kingdom

Vol. 3, No. 5, 2009