Generalized cluster structures related to the Drinfeld double of GLn$GL_n$

Abstract

We prove that the regular generalized cluster structure on the Drinfeld double of 𝐺𝐿 𝑛 constructed in 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 𝑛 = 4 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, Vainshtein (Int. Math. Res. Notes, 2022, to appear, arXiv:1912.00453) possesses similar compatibility and completeness properties.

Multiple generalized cluster structures on

“…They were used to define a generalized cluster structure on the Drinfeld double D(GL n ) = GL n × GL n , which is an alternative to the use of the φ-functions in [20]. It was shown in [21] that the resulting constructions with ϕ-and φ-functions (for the trivial BD triples) provide equivalent generalized cluster structures in n = 2 and n = 3; however, they are not equivalent in n = 4 (in other words, the initial extended seeds are not mutation equivalent). In the supplementary note [33], we show that GC † g (Γ) and GC † h (Γ op ) are equivalent for n = 3 and any BD triple Γ.…”

“…At present, we know that Any simple complex Poisson–Lie group endowed with the standard Poisson bracket possesses a compatible cluster structure; see [16]; For any Belavin–Drinfeld data, the existence of a compatible cluster structure for for all was shown in [16]; for , the conjecture was proved by Eisner in [8]; For a large class of the so-called aperiodic oriented BD triples, the conjecture was proved for in [20]; For other Poisson–Lie groups, the conjecture was established for the Drinfeld double of in [18] for the standard Poisson structure, as well as for , which is an image of the dual group in . An alternative construction on the Drinfeld double of was also given in [19, 21]. The above results naturally extend to .…”

“…For other Poisson–Lie groups, the conjecture was established for the Drinfeld double of in [18] for the standard Poisson structure, as well as for , which is an image of the dual group in . An alternative construction on the Drinfeld double of was also given in [19, 21].…”