Vanishing of trace forms in low characteristics

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.