---
abstract: 'We survey some recent constructions of cluster algebra structures on coordinate rings of unipotent subgroups and unipotent cells of Kac-Moody groups. We also review a quantized version of these results.'
author:
- 'Christof Geiss, Bernard Leclerc, Jan Schröer'
title: Cluster algebras in algebraic Lie theory
---

Introduction
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Cluster algebras were invented by Fomin and Zelevinsky [@FZ1] as an abstraction of certain combinatorial structures which they had previously discovered while studying total positivity in semisimple algebraic groups. A cluster algebra is a commutative ring with a distinguished set of generators and a particular type of relations. Although there can be infinitely many generators and relations, they are all obtained from a finite number of them by means of an inductive procedure called mutation. The precise definition of a cluster algebra will be recalled in §\[sect2\] below.

Several examples arising in Lie theory were already mentioned in [@FZ1], like $\C[\SL_4/N]$, $\C[\Sp_4/N]$, $\C[{\operatorname{Gr}}_{2,n+3}]$. Moreover, Fomin and Zelevinsky [@FZ1 p. 498] conjectured that

> *“the above examples can be extensively generalized: for any simply-connected connected semisimple group $G$, the coordinate rings $\C[G]$ and $\C[G/N]$, as well as coordinate rings of many other interesting varieties related to $G$, have a natural structure 