Algebra & Number Theory

Vanishing of trace forms in low characteristics

Skip Garibaldi

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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.

Article information

Algebra Number Theory, Volume 3, Number 5 (2009), 543-566.

Received: 16 July 2008
Revised: 5 March 2009
Accepted: 6 April 2009
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20G05: Representation theory
Secondary: 17B50: Modular Lie (super)algebras 17B25: Exceptional (super)algebras

trace form E8 Richardson's condition Dynkin index


Garibaldi, Skip. Vanishing of trace forms in low characteristics. Algebra Number Theory 3 (2009), no. 5, 543--566. doi:10.2140/ant.2009.3.543.

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