Duals of semisimple Poisson-Lie groups and cluster theory of moduli spaces of G-local systems

Abstract

We study the dual ${\mathrm{G}}^{\ast }$ of a standard semisimple Poisson-Lie group $\mathrm{G}$ from a perspective of cluster theory. We show that the coordinate ring $\mathcal{O}({\mathrm{G}}^{\ast })$ can be naturally embedded into a cluster Poisson algebra with a Weyl group action. We prove that $\mathcal{O}({\mathrm{G}}^{\ast })$ admits a natural basis which has positive integer structure coefficients and satisfies an invariance property with respect to a braid group action. We continue the study of the moduli space ${\mathcal{P}}_{\mathrm{G},\mathbb{S}}$ of $\mathrm{G}$-local systems introduced in \cite{GS3}, and prove that the coordinate ring of ${\mathcal{P}}_{\mathrm{G},\mathbb{S}}$ coincides with its underlying cluster Poisson algebra.

21 pages, 5 figures, to appear in IMRN

References in corpus (4)

• The quantum dilogarithm and representations quantum cluster varieties
• Quantum geometry of moduli spaces of local systems and representation theory
• Quantum cluster algebras and their specializations
• Cluster Structures on Double Bott-Samelson Cells

Cited by in corpus (1)

• Newton-Okounkov polytopes of Schubert varieties arising from cluster structures