Vanishing of trace forms in low characteristics

TL;DR

This paper investigates conditions under which trace forms derived from algebraic group representations are nonzero or nondegenerate, with specific results for Lie algebras of type E8 in characteristic 5, answering longstanding questions.

Contribution

It provides criteria based on root system data for the existence of nondegenerate trace forms and shows that E8 Lie algebra over characteristic 5 lacks a quotient trace form.

Findings

Criteria for nonzero and nondegenerate trace forms

E8 Lie algebra in characteristic 5 has no quotient trace form

Answers a question from the 1960s about trace forms

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 E8 over a field of characteristic 5 does not have a so-called "quotient trace form", answering a question posed in the 1960s.