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

We study the dual G of a standard semisimple Poisson–Lie group G from a perspective of cluster theory. We show that the coordinate ring O(G) can be naturally embedded into a quotient algebra of a cluster Poisson algebra with a Weyl group action. The coordinate ring O(G) 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 P_G,S of G-local systems introduced in [ 16] and prove that the coordinate ring of P_G, S coincides with its underlying cluster Poisson algebra.