Cluster algebra structures on Poisson nilpotent algebras

Abstract

Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert cells for symmetrizable Kac--Moody groups. We prove that every Poisson nilpotent algebra satisfying a mild condition on certain scalars is canonically isomorphic to a cluster algebra which coincides with the corresponding upper cluster algebra, without additional localizations by frozen variables. The constructed cluster structure is compatible with the Poisson structure in the sense of Gekhtman, Shapiro and Vainshtein. All Poisson nilpotent algebras are proved to be equivariant Poisson Unique Factorization Domains. Their seeds are constructed from sequences of Poisson-prime elements for chains of Poisson UFDs; mutation matrices are effectively determined from linear systems in terms of the underlying Poisson structure. Uniqueness, existence, mutation, and other properties are established for these sequences of Poisson-prime elements.

minor edits in version 2. arXiv admin note: text overlap with arXiv:1309.7869

References in corpus (7)

• Kac-Moody groups and cluster algebras
• Locally acyclic cluster algebras
• Quantum cluster algebras and quantum nilpotent algebras
• Quantum cluster algebra structures on quantum nilpotent algebras
• On the Spectra of Quantum Groups
• From quantum Ore extensions to quantum tori via noncommutative UFDs
• Bott-Samelson varieties and Poisson Ore extensions

Cited by in corpus (6)

• Factoriality and class groups of cluster algebras
• Cluster Structures on Double Bott-Samelson Cells
• Reddening Sequences for Banff Quivers and the Class $\mathcal{P}$
• Plethora of cluster structures on $G{L}_n$
• Newton-Okounkov polytopes of Schubert varieties arising from cluster structures
• Semi-toric degenerations of Richardson varieties arising from cluster structures on flag varieties