Cluster Structures on Simple Complex Lie Groups and Belavin–Drinfeld Classification
| Autor: | Misha Gekhtman, Alek Vainshtein, Michael Shapiro |
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| Rok vydání: | 2012 |
| Předmět: | |
| Zdroj: | Moscow Mathematical Journal. 12:293-312 |
| ISSN: | 1609-4514 1609-3321 |
| DOI: | 10.17323/1609-4514-2012-12-2-293-312 |
| Popis: | We study natural cluster structures in the rings of regular func- tions on simple complex Lie groups and Poisson-Lie structures compatible with these cluster structures. According to our main conjecture, each class in the Belavin-Drinfeld classification of Poisson-Lie structures on G corresponds to a cluster structure in O(G). We prove a reduction theorem explaining how different parts of the conjecture are related to each other. The conjecture is established for SLn, n < 5, and for any G in the case of the standard Poisson- Lie structure. Since the invention of cluster algebras in 2001, a large part of research in the field has been devoted to uncovering cluster structures in rings of regular functions on various algebraic varieties arising in algebraic geometry, representation theory, and mathematical physics. Once the existence of such a structure was established, abstract features of cluster algebras were used to study geometric properties of underlying objects. Research in this direction led to many exciting results (SSVZ, FoGo1, FoGo2). It also created an impression that, given an algebraic variety, there is a unique (if at all) natural cluster structure associated with it. The main goal of the current paper is to establish the following phenomenon: in certain situations, the same ring may have multiple natural cluster structures. More exactly, we engage into a systematic study of multiple cluster structures in the rings of regular functions on simple Lie groups (in what follows we will shorten that to cluster structures on simple Lie groups). Consistent with the philosophy advocated in (GSV1, GSV2, GSV3, GSV4, GSV5, GSV6), we will focus on compatible Poisson structures on the Lie groups, that is, on compatible Poisson-Lie structures. The notion of a Poisson bracket compatible with a cluster structure was intro- duced in (GSV1). It was used there to interpret cluster transformations and matrix mutations from a viewpoint of Poisson geometry. In addition, it was shown that if a Poisson algebraic variety (M,f�,ŧ ) possesses a coordinate chart that consists of regular functions whose logarithms have pairwise constant Poisson brackets, then one can use this chart to define a cluster structure CM compatible with f�,ŧ . Al- gebraic structures corresponding to CM (the cluster algebra and the upper cluster algebra) are closely related to the ring O(M) of regular functions on M. More pre- cisely, under certain rather mild conditions, O(M) can be obtained by tensoring one of these algebras by C. |
| Databáze: | OpenAIRE |
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