Cluster algebra structures on Poisson nilpotent algebras / K.R. Goodearl, M.T. Yakimov.

Format:

Book

Author/Creator:

Goodearl, K. R., author.

Yakimov, Milen, 1973- author.

Series:

Memoirs of the American Mathematical Society ; no.1445.

Memoirs of the American Mathematical Society, 0065-9266 ; number 1445

Language:

English

Subjects (All):

Cluster algebras.

Poisson algebras.

Physical Description:

v, 100 pages ; 26 cm.

Place of Publication:

Providence, RI : American Mathematical Society, 2023.

Summary:

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.

Contents:

Chapter 1. Introduction

Chapter 2. Poisson algebras

Chapter 3. Cluster algebras and Poisson cluster algebras

Chapter 4. Poisson-primes in Poisson-Ore extensions

Chapter 5. Iterated Poisson-Ore extensions

Chapter 6. Symmetry and maximal tori for Poisson-CGL extensions

Chapter 7. One-step mutations in Poisson-CGL extensions

Chapter 8. Homogenous Poisson-prime elements for subalgebras of symmetric Poisson-CGL extensions

Chapter 9. Chains of mutations in symmetric Poisson-CGL extensions

Chapter 10. Division properties of mutations between Poisson-CGL extension presenations

Chapter 11. Symmetric Poisson nilpotent algebras and cluster algebras

Bibliography

Index.

Notes:

"October 2023, volume 290, number 1445 (fifth of 5 numbers)"

Includes bibliographical references (pages 95-97) and index.

ISBN:

9781470467357

1470467356

OCLC:

1416884824