Reductive monoids and cluster algebras

TL;DR

This paper demonstrates that the coordinate ring of the Vinberg monoid of a simply connected semisimple complex group is an upper cluster algebra, and constructs cluster structures on various reductive monoids and groups.

Contribution

It establishes the cluster algebra structure on the coordinate ring of the Vinberg monoid and extends this structure to a broad class of reductive monoids and groups.

Findings

Coordinate ring of Vinberg monoid is an upper cluster algebra.

Constructs cluster structures on flat reductive monoids.

Obtains cluster structures on connected reductive groups after localization.

Abstract

We show that the coordinate ring of the Vinberg monoid of a simply connected semisimple complex group is an upper cluster algebra. As an application, we construct cluster structures on a large class of flat reductive monoids. After localization, we obtain cluster structures on any connected reductive group whose commutator group is simply connected.