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 KacMoody 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 Poissonprime 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 Poissonprime elements.
 Publication:

arXiv eprints
 Pub Date:
 January 2018
 DOI:
 10.48550/arXiv.1801.01963
 arXiv:
 arXiv:1801.01963
 Bibcode:
 2018arXiv180101963G
 Keywords:

 Mathematics  Commutative Algebra;
 Mathematics  Quantum Algebra;
 Mathematics  Symplectic Geometry;
 Primary 13F60;
 Secondary 17B63;
 17B37
 EPrint:
 minor edits in version 2. arXiv admin note: text overlap with arXiv:1309.7869