The cluster variety face of quantum groups

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 $2n$ Poisson leaf emerge in classical limit, if all but few ($2n$) coordinates vanish.

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