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Cluster algebra structures on Poisson nilpotent algebras
About this Title
K. R. Goodearl and M. T. Yakimov
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 290, Number 1445
ISBNs: 978-1-4704-6735-7 (print); 978-1-4704-7631-1 (online)
DOI: https://doi.org/10.1090/memo/1445
Published electronically: October 12, 2023
Keywords: Cluster algebras,
Poisson algebras,
Poisson-prime elements,
Poisson-Ore extensions,
Poisson-CGL extensions
Table of Contents
Chapters
- 1. Introduction
- 2. Poisson algebras
- 3. Cluster algebras and Poisson cluster algebras
- 4. Poisson-primes in Poisson-Ore extensions
- 5. Iterated Poisson-Ore extensions
- 6. Symmetry and maximal tori for Poisson-CGL extensions
- 7. One-step mutations in Poisson-CGL extensions
- 8. Homogeneous Poisson-prime elements for subalgebras of symmetric Poisson-CGL extensions
- 9. Chains of mutations in symmetric Poisson-CGL extensions
- 10. Division properties of mutations between Poisson-CGL extension presentations
- 11. Symmetric Poisson nilpotent algebras and cluster 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, affine charts of Bott-Samelson varieties, coordinate rings of double Bruhat cells (in the last case after a localization). We prove that every symmetric 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.- Jason Bell, Stéphane Launois, Omar León Sánchez, and Rahim Moosa, Poisson algebras via model theory and differential-algebraic geometry, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 7, 2019–2049. MR 3656478, DOI 10.4171/JEMS/712
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