A plethora of cluster structures on GLn / M. Gekhtman, M. Shapiro, A. Vainshtein.
Material type: TextSeries: Memoirs of the American Mathematical Society ; v. 1486.Publisher: Providence, RI : American Mathematical Society, 2024Copyright date: ©2024Description: v, 104 pages : illustrations ; 26 cmContent type:- text
- unmediated
- volume
- 1470469707
- 9781470469702
- G313p
Item type | Current library | Collection | Call number | Status | Date due | Barcode |
---|---|---|---|---|---|---|
Books | Castorina | Coleções de Monografias (Monographs Collections) | Available | 39063000809602 |
Includes bibliographical references (pages 103-104).
We continue the study of multiple cluster structures in the rings of regular functions on GLn, SLn and Matn that are compatible with Poisson-Lie and Poisson-homogeneous structures. According to our initial conjecture, each class in the Belavin-Drinfeld classification of Poisson--Lie structures on a semisimple complex group G corresponds to a cluster structure in O(G). Here we prove this conjecture for a large subset of Belavin-Drinfeld (BD) data of An type, which includes all the previously known examples. Namely, we subdivide all possible An type BD data into oriented and non-oriented kinds. In the oriented case, we single out BD data satisfying a certain combinatorial condition that we call aperiodicity and prove that for any BD data of this kind there exists a regular cluster structure compatible with the corresponding Poisson-Lie bracket. In fact, we extend the aperiodicity condition to pairs of oriented BD data and prove a more general result that establishes an existence of a regular cluster structure on SLn compatible with a Poisson bracket homogeneous with respect to the right and left action of two copies of SLn equipped with two different Poisson-Lie brackets. If the aperiodicity condition is not satisfied, a compatible cluster structure has to be replaced with a generalized cluster structure. We will address this situation in future publications.
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