Journal of Function Spaces

Volume 2016 (2016), Article ID 9639875, 8 pages

http://dx.doi.org/10.1155/2016/9639875

## On Cluster -Algebras

Department of Mathematics and Computer Science, St. John’s University, 8000 Utopia Parkway, Queens, New York, NY 11439, USA

Received 4 February 2016; Accepted 18 May 2016

Academic Editor: Gelu Popescu

Copyright © 2016 Igor V. Nikolaev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We introduce a -algebra attached to the cluster and a quiver . If is the quiver coming from triangulation of the Riemann surface with a finite number of cusps, we prove that the primitive spectrum of times is homeomorphic to a generic subset of the Teichmüller space of surface . We conclude with an analog of the Tomita-Takesaki theory and the Connes invariant for the algebra .

#### 1. Introduction

Cluster algebras of rank are a class of commutative rings introduced by [1]. Among these algebras, one finds coordinate rings of important algebraic varieties, like the Grassmannians and Schubert varieties; cluster algebras appear in the Teichmüller theory [2]. Unlike the coordinate rings, the set of generators of cluster algebra is usually infinite and defined by induction from a* cluster * and a* quiver *; see [3] for an excellent survey; the cluster algebra is denoted by . Notice that has an additive structure of countable (unperforated) abelian group with an order satisfying the Riesz interpolation property; see Remark 5. In other words, the cluster algebra is a* dimension group* by the Effros-Handelman-Shen Theorem [4, Theorem 3.1].

The subject of our paper is an operator algebra , such that ; here is the dimension group of and is an isomorphism of the ordered abelian groups [5, Chapter 7]. is an Approximately Finite *-algebra* (-algebra) given by a Bratteli diagram derived explicitly from the pair . The -algebras were introduced and studied by [6]; we refer to as a* cluster **-algebra*.

An exact definition of can be found in Section 2.4; to give an idea, recall that the pair is called a* seed* and the cluster algebra is generated by seeds obtained via* mutation* of (and its mutants) in all* directions *, where [3, p. 5]. The mutation process can be described by an oriented regular tree ; the vertices of correspond to the seeds and the outgoing edges to the mutations in directions . The quotient of by a relation identifying equivalent seeds at the same level of is a graph with cycles. (For a quick example of such a graph, see Figure 3.) The cluster -algebra is an -algebra given by regarded as a Bratteli diagram [6].

Let be a Riemann surface of genus with cusps such that ; denote by the (decorated) Teichmüller space of , that is, a collection of all Riemann surfaces of genus with cusps endowed with the natural topology [7]. In what follows, we focus on the algebras with quivers coming from ideal triangulation of ; the corresponding cluster algebra of rank is related to the* Penner coordinates* in [2].

*Example 1. *Let be a once-punctured torus. The ideal triangulation of defines the Markov quiver (such a quiver is related to solutions in the integer numbers of the equation considered by A. A. Markov, hence the name) shown in Figure 1; see [2, Example 4.6]. The corresponding cluster -algebra of rank can be written as where is a primitive ideal of an -algebra . The unital -algebra was originally defined by [8, Section 3]; the genuine notation for such an algebra was , because ≔ free one-generator unital -group, that is, finitely piecewise affine linear continuous real-valued functions on with integer coefficients. was subsequently rediscovered after two decades by [9] and denoted by . The remarkable properties of include the following features. Every primitive ideal of is essential [10, Theorem 4.2]. is equipped with a faithful invariant tracial state [11, Theorem 3.1]. The center of coincides with the -algebra of continuous complex-valued functions on [9, p. 976]. There is an affine weak -homeomorphism of the state space of onto the space of tracial states on [10, Theorem 4.5]. Any state of has precisely one tracial extension to [12, Theorem 2.5]. The automorphism group of has precisely two connected components [10, Theorem 4.3]. The Gauss map, a Bernoulli shift for continued fractions, is generalized in [12] to the noncommutative framework of . In the light of the original definition of and the fact that the -functor preserves exact sequences (see, e.g., [4, Theorem 3.1]), the primitive spectrum of and its hull-kernel topology is widely known to the lattice-ordered group theorists and the MV-algebraists long ago before the laborious analysis in [9], where is defined in terms of the Bratteli diagram. We refer the reader to the final part of the paper [13] for a general result encompassing the characterization of the prime spectrum of . Moreover, the -algebras introduced by [14] are precisely the infinite-dimensional simple quotients of ; this fact was first proved by [8, Theorem 3.1(i)] and rediscovered independently by [9]. Summing up the above, the primitive ideals are indexed by numbers ; if is irrational, the quotient , where is the Effros-Shen algebra. In view of (1), the algebra is a noncommutative coordinate ring of the Teichmüller space . Moreover, there exists an analog of the Tomita-Takesaki theory of modular automorphisms for algebra ; see Section 4; such automorphisms correspond to the Teichmüller geodesic flow on [15]. is an ideal of for all , where . The quotient algebra can be viewed as a noncommutative coordinate ring of the Riemann surface ; in particular, the pairs are coordinates in the space . We refer the reader to [16] for a construction of the corresponding functor.