#### Vol. 3, No. 5, 2009

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Vanishing of trace forms in low characteristics

### Appendix: Alexander Premet

Vol. 3 (2009), No. 5, 543–566
##### 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