Root graded groups revisited

TL;DR

This paper classifies algebraic structures underlying root graded groups associated with irreducible root systems of rank at least 3, and provides a construction method for these groups.

Contribution

It offers a complete description of the varieties of algebraic structures for root graded groups and a construction method for all such groups.

Findings

Complete classification of algebraic structures for root graded groups of rank ≥ 3.

Construction method for root graded groups from classified algebraic structures.

Excludes root systems of types H3 and H4 from classification.

Abstract

A group $G$ is called root graded if it has a family of subgroups $G_\alpha$ indexed by roots from a root system $\Phi$ satisfying natural conditions similar to Chevalley groups over commutative unital rings. For any such group there is a corresponding algebraic structure (commutative unital ring, associative unital ring, etc.) encoding the commutator relations between $G_\alpha$. We give a complete description of varieties of such structures for irreducible root systems of rank $\geq 3$ excluding $\mathsf H_3$ and $\mathsf H_4$. Moreover, we provide a construction of root graded groups for all algebraic structures from these varieties.