Restricted graph Lie algebras in characteristic two

TL;DR

This paper studies restricted Lie algebras related to decorated graphs over fields of characteristic two, revealing unique cohomological phenomena and field dependence not seen in other characteristics.

Contribution

It introduces graph-based restricted Lie algebras in characteristic two, computes their cohomology, and characterizes when certain Lie-theoretic analogues hold based on the base field.

Findings

Cohomology of these Lie algebras depends on the base field in characteristic two.

The prime field $ extbf{F}_2$ uniquely characterizes the validity of a twisted Droms theorem analogue.

Generalizations of graph Lie algebras are discussed.

Abstract

We investigate restricted Lie algebras arising as analogues of (twisted) right-angled Artin groups and right-angled Coxeter groups over fields of characteristic two. These algebras are defined via quadratic relations determined by decorated graphs. We compute their cohomology rings with trivial coefficients and uncover phenomena specific to characteristic two: unlike in zero/odd characteristics, where quadratically defined ordinary and restricted Lie algebras have equivalent cohomology theories, the characteristic two case exhibits dependence on the base field. In particular, we prove that the ground field being the prime field $\mathbb F_2$ characterizes when a Lie-theoretic analogue of the twisted Droms theorem holds. Generalizations of graph Lie algebras are also discussed.