The cluster variety face of quantum groups

TL;DR

This paper shows that quantum groups, specifically $A(n)_q$, can be understood as cluster varieties using free-field formalism, revealing new insights into their structure and classical limits.

Contribution

It demonstrates that quantum groups are naturally cluster varieties and derives mutation formulas from root order independence.

Findings

Quantum groups $A(n)_q$ are cluster varieties.

Mutation formulas follow from root order independence.

Classical limits yield Poisson leaves with specific coordinate conditions.

Abstract

Using the well-known free-field formalism for quantum groups, we demonstrate in case of $A(n)_q$, that quantum group is naturally also a cluster variety. Widely used formulae for mutations are direct consequence of independence of group element on the order of simple roots. Usual formulae for $2 n$ Poisson leaf emerge in classical limit, if all but few ($2n$) coordinates vanish.