Groups with $\mathsf A_\ell$-commutator relations

TL;DR

This paper generalizes a known result about groups with root subgroups related to $ ext{A}_ ext{ell}$-type root systems, constructing a ring with a Peirce decomposition without relying on Weyl elements.

Contribution

It extends previous work by recovering a ring structure from groups with root subgroups without using Weyl elements, employing Peirce decomposition instead.

Findings

Constructed a non-unital associative ring with Peirce decomposition

Generalized the relation between groups with root subgroups and rings

Provided a new approach avoiding Weyl elements

Abstract

If $A$ is a unital associative ring and $\ell \geq 2$, then the general linear group $\mathrm{GL}(\ell, A)$ has root subgroups $U_\alpha$ and Weyl elements $n_\alpha$ for $\alpha$ from the root system of type $\mathsf A_{\ell - 1}$. Conversely, if an arbitrary group has such root subgroups and Weyl elements for $\ell \geq 4$ satisfying natural conditions, then there is a way to recover the ring $A$. We prove a generalization of this result not using the Weyl elements, so instead of the matrix ring $\mathrm M(\ell, A)$ we construct a non-unital associative ring with a well-behaved Peirce decomposition.