Vol. 3, No. 5, 2009

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Vanishing of trace forms in low characteristics

Skip Garibaldi

Appendix: Alexander Premet

Vol. 3 (2009), No. 5, 543–566

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 E8 over a field of characteristic 5 does not have a “quotient trace form”, answering a question posed in the 1960s.

trace form, E8, Richardson's condition, Dynkin index
Mathematical Subject Classification 2000
Primary: 20G05
Secondary: 17B50, 17B25
Received: 16 July 2008
Revised: 5 March 2009
Accepted: 6 April 2009
Published: 9 November 2009
Skip Garibaldi
Department of Mathematics and Computer Science
Emory University
Atlanta, GA 30322
United States
Alexander Premet
School of Mathematics
The University of Manchester
Oxford Rd.
Manchester, M13 9PL
United Kingdom