Duals of Semisimple Poisson–Lie Groups and Cluster Theory of Moduli Spaces of G-local Systems

Abstract

Abstract We study the dual $\textrm{G}^{\ast }$ of a standard semisimple Poisson–Lie group $\textrm{G}$ from a perspective of cluster theory. We show that the coordinate ring $\mathcal{O}(\textrm{G}^{\ast })$ can be naturally embedded into a quotient algebra of a cluster Poisson algebra with a Weyl group action. The coordinate ring $\mathcal{O}(\textrm{G}^{\ast })$ admits a natural basis, which has positive integer structure coefficients and satisfies an invariance property under a braid group action. We continue the study of the moduli space $\mathscr{P}_{\textrm{G},{{\mathbb{S}}}}$ of $\textrm{G}$-local systems introduced in [ 16] and prove that the coordinate ring of $\mathscr{P}_{\textrm{G}, {{\mathbb{S}}}}$ coincides with its underlying cluster Poisson algebra.