Generalized cluster structures related to the Drinfeld double of GLn

Misha Gekhtman; Michael Shapiro; Alek Vainshtein

doi:10.1112/jlms.12542

Generalized cluster structures related to the Drinfeld double of GL_n

Misha Gekhtman, Michael Shapiro, Alek Vainshtein

Research output: Contribution to journal › Article › peer-review

Abstract

We prove that the regular generalized cluster structure on the Drinfeld double of (Formula presented.) constructed in Gekhtman, Shapiro, and Vainshtein (Int. Math. Res. Notes, 2022, to appear, arXiv:1912.00453) is complete and compatible with the standard Poisson–Lie structure on the double. Moreover, we show that for (Formula presented.) this structure is distinct from a previously known regular generalized cluster structure on the Drinfeld double, even though they have the same compatible Poisson structure and the same collection of frozen variables. Further, we prove that the regular generalized cluster structure on band periodic matrices constructed in Gekhtman, Shapiro, and Vainshtein (Int. Math. Res. Notes, 2022, to appear, arXiv:1912.00453) possesses similar compatibility and completeness properties.

Original language	English
Pages (from-to)	1601-1633
Number of pages	33
Journal	Journal of the London Mathematical Society
Volume	105
Issue number	3
DOIs	https://doi.org/10.1112/jlms.12542
State	Published - Apr 2022

Bibliographical note

Publisher Copyright:
© 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1112/jlms.12542

Cite this

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abstract = "We prove that the regular generalized cluster structure on the Drinfeld double of (Formula presented.) constructed in Gekhtman, Shapiro, and Vainshtein (Int. Math. Res. Notes, 2022, to appear, arXiv:1912.00453) is complete and compatible with the standard Poisson–Lie structure on the double. Moreover, we show that for (Formula presented.) this structure is distinct from a previously known regular generalized cluster structure on the Drinfeld double, even though they have the same compatible Poisson structure and the same collection of frozen variables. Further, we prove that the regular generalized cluster structure on band periodic matrices constructed in Gekhtman, Shapiro, and Vainshtein (Int. Math. Res. Notes, 2022, to appear, arXiv:1912.00453) possesses similar compatibility and completeness properties.",

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