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Cremmer–Gervais Cluster Structure on Sln SPECIAL FEATURE Cremmer–Gervais cluster structure on SLn SPECIAL FEATURE Michael Gekhtmana, Michael Shapirob, and Alek Vainshteinc,1 aDepartment of Mathematics, University of Notre Dame, Notre Dame, IN 46556, bDepartment of Mathematics, Michigan State University, East Lansing, MI 48823, and cDepartment of Mathematics and Department of Computer Science, University of Haifa, Haifa, Mount Carmel 31905, Israel Edited by Bernard Leclerc, University of Caen, Caen, France, and accepted by the Editorial Board December 19, 2013 (received for review August 13, 2013) We study natural cluster structures in the rings of regular functions Cremmer–Gervais case. Our result allows equipping SLn, GLn, on simple complex Lie groups and Poisson–Lie structures compat- and the affine space Matn of n × n matrices with an alternative ible with these cluster structures. According to our main conjec- cluster structure, CCG. ture, each class in the Belavin–Drinfeld classification of Poisson–Lie In the first section below, we collect the necessary information structures on G corresponds to a cluster structure in OðGÞ. We have on cluster algebras, compatible Poisson brackets, and the toric shown before that this conjecture holds for any G in the case of the action. In the next section, we formulate the main conjecture standard Poisson–Lie structure and for all Belavin–Drinfeld classes from ref. 1, present the definition of the Cremmer–Gervais in SLn, n < 5. In this paper we establish it for the Cremmer–Gervais Poisson bracket, and formulate our main result. We introduce – SL Poisson Lie structure on n, which is the least similar to the stan- the cluster structure CCG and outline the proof of the main SL dard one. Besides, we prove that on 3 the cluster algebra and the theorem by breaking it into a series of intermediate results about upper cluster algebra corresponding to the Cremmer–Gervais cluster CCG. In the following section, we discuss the relation between structure do not coincide, unlike the case of the standard cluster cluster algebras and upper cluster algebras on SLn. In the stan- structure. Finally, we show that the positive locus with respect to dard case these two algebras coincide. We show that for the the Cremmer–Gervais cluster structure is contained in the set of Cremmer–Gervais cluster structure on SL3 this is not the case. totally positive matrices. The next section treats positivity for the exotic cluster structure C . Finally, in the last section we formulate several directions – – CG Poisson Lie group | Belavin Drinfeld triple for future research. n ref. 1 we initiated a systematic study of multiple cluster Cluster Structures and Compatible Poisson Brackets MATHEMATICS Istructures in the rings of regular functions on simple Lie groups We start with the basics on cluster algebras of geometric type. following an approach developed and implemented in refs. 2–4 The definition that we present below is not the most general one, for constructing cluster structures on algebraic Poisson varieties. see, e.g., refs. 5, 7 for a detailed exposition. In what follows, we It is based on the notion of a Poisson bracket compatible with a will use a notation ½i; j for an interval fi; i + 1; ...; jg in N, and we cluster structure. The key point is that if an algebraic Poisson will denote ½1; n by ½n. variety ðM; f·;·gÞ possesses a coordinate chart that consists of The coefficient group P is a free multiplicative abelian group of regular functions whose logarithms have pairwise constant Poisson finite rank m with generators g1; ...; gm.Anambient field is the brackets, then one can try to use this chart to define a cluster field F of rational functions in n independent variables with ·;· structure CM compatible with f g. Algebraic structures cor- coefficients in the field of fractions of the integer group ring responding to (the cluster algebra and the upper cluster al- ± ± ± − CM ZP = Z½ g 1; ...; g 1 (here we write x 1 instead of x; x 1). gebra) are closely related to the ring OðMÞ of regular functions 1 m ~ A seed (of geometric type)inF is a pair Σ = ðx; BÞ, where on M. In fact, under certain rather mild conditions, OðMÞ can x = ðx ; ...; x Þ is a transcendence basis of F over the field of be obtained by tensoring one of these algebras with C. 1 n fractions of Z and B~ is an n × n + m integer matrix whose This construction was applied in ref. 4, Ch. 4.3 to double P ð Þ Bruhat cells in semisimple Lie groups equipped with (the re- principal part B is skew-symmetrizable (recall that the principal part of a rectangular matrix is its maximal leading square sub- striction of) the standard Poisson–Lie structure. It was shown ~ that the resulting cluster structure coincides with the one built in matrix). Matrices B and B are called the exchange matrix and the ref. 5. Recall that it was proved in ref. 5 that the corresponding extended exchange matrix, respectively. upper cluster algebra is isomorphic to the ring of regular func- tions on the double Bruhat cell. Because the open double Bruhat Significance cell is dense in the corresponding Lie group, one can equip the ring of regular functions on the Lie group with the same cluster Coexistence of diverse mathematical structures supported on structure. The standard Poisson–Lie structure is a particular case the same variety leads to deeper understanding of its features. of Poisson–Lie structures corresponding to quasitriangular Lie If the manifold is a Lie group, endowing it with a Poisson bialgebras. Such structures are associated with solutions to the structure that respects group multiplication (Poisson–Lie struc- classical Yang–Baxter equation. Their complete classification ture) is instrumental in a study of classical and quantum- was obtained by Belavin and Drinfeld in ref. 6. In ref. 1 we mechanical systems with symmetries. In turn, a Poisson struc- conjectured that any such solution gives rise to a compatible ture on a variety can be compatible with a cluster structure— cluster structure on the Lie group and provided several examples a useful combinatorial tool that organizes generators of the ring supporting this conjecture by showing that it holds true for the of regular functions into a collection of overlapping clusters class of the standard Poisson–Lie structure in any simple com- connected via rational transformations. We conjectured that this plex Lie group, and for the whole Belavin–Drinfeld classification is the case for an important class of Poisson–Lie groups. The in SLn for n = 2; 3; 4. We call the cluster structures associated paper verifies this conjecture for a group of invertible matrices with the nontrivial Belavin–Drinfeld data exotic. equipped with a nonstandard Poisson–Lie structure. In this paper, we outline the proof of the conjecture of ref. 1 in the case of the Cremmer–Gervais Poisson structure on SLn.We Author contributions: M.G., M.S., and A.V. performed research and wrote the paper. chose to consider this case because, in a sense, the Poisson The authors declare no conflict of interest. structure in question differs the most from the standard: the This article is a PNAS Direct Submission. B.L. is a guest editor invited by the Editorial discrete data featured in the Belavin–Drinfeld classification are Board. trivial in the standard case and have the “maximal size” in the 1To whom correspondence should be addressed. Email: [email protected]. www.pnas.org/cgi/doi/10.1073/pnas.1315283111 PNAS Early Edition | 1of8 Downloaded by guest on October 3, 2021 The n-tuple x is called a cluster, and its elements x1; ...; xn are counterparts in F C obtained by extension of scalars; they are called cluster variables. Denote xn+i = gi for i ∈ ½m. We say that denoted AC and AC. If, moreover, the field isomorphism φ can ~x = ðx1; ...; xn+mÞ is an extended cluster, and xn+1; ...; xn+m are be restricted to an isomorphism of AC (or AC) and OðVÞ, we say stable variables. It is convenient to think of F as of the field of that AC (or AC)isnaturally isomorphic to OðVÞ. rational functions in n + m independent variables with rational Let f·;·g be a Poisson bracket on the ambient field F, and C be coefficients. a cluster structure in F. We say that the bracket and the In what follows, we will only deal with the case when the exchange cluster structure are compatible if, for any extended cluster matrix is skew-symmetric. In this situation the extended ex- ~x = ðx1; ...; xn+mÞ, one has change matrix can be conveniently represented by a quiver È É = ~ ; ...; + Q QðBÞ. It is a directed graph on the vertices 1 n m cor- xi; xj = ωij xi xj; [2] responding to all variables; the vertices corresponding to stable > ~ variables are called frozen.Eachentrybij 0ofthematrixB where ωij ∈ Z are constants for all i; j ∈ ½n + m.Thematrix ~x gives rise to bij edges going from the vertex i to the vertex j; each Ω = ðωijÞ is called the coefficient matrix of f·;·g (in the basis ~ ~x such edge is denoted i → j. Clearly, B can be restored uniquely ~x); clearly, Ω is skew-symmetric. The notion of compatibility from Q. extends to Poisson brackets on F C without any changes. Given a seed as above, the adjacent cluster in direction k ∈ ½n is A complete characterization of Poisson brackets compatible ~ ~ defined by with a given cluster structure C = CðBÞ in the case rank B = n is S given in ref. 2; see also ref. 4, Ch. 4. A different description of x = ðxnfx gÞ fx′g; k k k compatible Poisson brackets on F C is based on the notion of a ~x = ; ...; ′ toric action.
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