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.
Bibliographical noteFunding Information:
Our research was supported in part by the NSF research grant DMS #1702054 (M.G.), NSF research grant DMS #1702115 and International Laboratory of Cluster Geometry NRU HSE, RF Government grant, ag. # 075‐15‐2021‐608 from 08.06.2021 (M.S.), and ISF grant #1144/16 (A.V.). While working on this project, we benefited from support of the following institutions and programs: Research Institute for Mathematical Sciences, Kyoto (M.G., M.S., A.V., Spring 2019), Research in Pairs Program at the Mathematisches Forschungsinstitut Oberwolfach (M.S. and A.V., Summer 2019), Istituto Nazionale di Alta Matematica Francesco Severi and the Sapienza University of Rome (A.V., Fall 2019), University of Haifa (M.G., Fall 2019), Mathematical Science Research Institute, Berkeley (M.S., Fall 2019), Michigan State University (A.V., Spring 2020), University of Notre Dame (A.V., Spring 2020). We are grateful to all these institutions for their hospitality and outstanding working conditions they provided. Special thanks are due to Peigen Cao and Fang Li who in response to our request provided a generalization [ 3 ] of their previous results, to Linhui Shen for pointing out to us the preprint [ 21 ], to Alexander Shapiro and Gus Schrader for many fruitful discussions, and to the reviewers for constructive suggestions.
© 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.
ASJC Scopus subject areas
- Mathematics (all)