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.
Motivated by Example 1, denote by the cluster -algebra corresponding to a quiver ; let be the Tomita-Takesaki flow on ; see Section 4 for the details. Denote by the set of all primitive ideals of endowed with the Jacobson topology and let for a generic value of index . Our main result can be stated as follows.
Theorem 2. There exists a homeomorphismgiven by the formula ; the set if and only if . is an ideal of for all and the quotient algebra is a noncommutative coordinate ring of the Riemann surface .
Remark 3. Theorem 2 is valid for , that is, the Riemann surfaces with at least one cusp. This cannot be improved, since the cluster structure of algebra comes from the Ptolemy relations satisfied by the Penner coordinates; so far such coordinates are available only for the Riemann surfaces with cusps [7]. It is likely that the case has also a cluster structure; we refer the reader to [17], where a functor from the Riemann surfaces to the -algebras was constructed.
Remark 4. The braid group with admits a faithful representation by projections in the algebra ; such a construction is based on the Birman-Hilden Theorem for the braid groups. This observation and the well-known Laurent phenomenon in the cluster algebra allow generalizing the Jones and HOMFLY invariants of knots and links to an arbitrary number of variables; see [18] for the details.
The paper is organized as follows. We introduce preliminary facts and notation in Section 2. Theorem 2 is proved in Section 3. An analog of the Tomita-Takesaki theory of modular automorphisms and the Connes invariant of the cluster -algebra is constructed.
2. Notation
In this section we introduce notation and briefly review some preliminary facts. The reader is encouraged to consult [1–3, 6, 7] for the details.
2.1. Cluster Algebras
A cluster algebra of rank is a subring of the field of rational functions in variables. Such an algebra is defined by a pair , where is a cluster of variables and is a skew-symmetric integer matrix; the new cluster is obtained from by an excision of the variable and replacing it by a new variable subject to an exchange relation:Since the entries of matrix are exponents of the monomials in cluster variables, one gets a new pair , where is a skew-symmetric matrix withFor brevity, the pair is called a seed and the seed is obtained from by a mutation in the direction , where ; is involution; that is, . The matrix is called mutation finite if only finitely many new matrices can be produced from by repeated matrix mutations. The cluster algebra can be defined as the subring of generated by the union of all cluster variables obtained from the initial seed by mutations of (and its iterations) in all possible directions. We will write to denote an oriented tree whose vertices are seeds and outgoing arrows in each vertex correspond to mutations of the seed . The Laurent phenomenon proved by [1] says that , where is the ring of the Laurent polynomials in variables ; in other words, each generator of algebra can be written as a Laurent polynomial in variables with the integer coefficients.
Remark 5. The Laurent phenomenon turns the additive structure of cluster algebra into a totally ordered abelian group satisfying the Riesz interpolation property, that is, a dimension group [4, Theorem 3.1]; the abelian group with order comes from the semigroup of the Laurent polynomials with positive coefficients; see [19] for the details. A background on the partially and totally ordered, unperforated abelian groups with the Riesz interpolation property can be found in [4].
To deal with mutation formulas (3) and (4) in geometric terms, recall that a quiver is an oriented graph given by the set of vertices and the set of arrows ; an example of quiver is given in Figure 1. Let be a vertex of ; the mutated at vertex quiver has the same set of vertices as but the set of arrows is obtained by the following procedure: (i) for each subquiver , one adds a new arrow ; (ii) one reverses all arrows with source or target ; (iii) one removes the arrows in a maximal set of pairwise disjoint -cycles. The reader can verify that if one encodes a quiver with vertices by a skew-symmetric matrix with equal to the number of arrows from vertex to vertex , then mutation of seed coincides with such of the corresponding quiver . Thus, the cluster algebra is defined by a quiver ; we will denote such an algebra by .
2.2. Cluster Algebras from Riemann Surfaces
Let and be integers, such that , , and . Denote by a Riemann surface of genus with the cusp points. It is known that the fundamental domain of can be triangulated by geodesic arcs , such that the footpoints of each arc at the absolute of Lobachevsky plane coincide with a (preimage of) cusp of . If is the hyperbolic length of measured (with a sign) between two horocycles around the footpoints of , then we set ; are known to satisfy the Ptolemy relation:where are pairwise opposite sides and are the diagonals of a geodesic quadrilateral in .
Denote by the decorated Teichmüller space of , that is, the set of all complex surfaces of genus with cusps endowed with the natural topology; it is known that .
Theorem 6 (see [7]). The map on the set of geodesic arcs defining triangulation of is a homeomorphism with the image .
Remark 7. Notice that among real numbers there are only independent, since such numbers must satisfy Ptolemy relations (5).
Let be triangulation of surface by geodesic arcs ; consider a skew-symmetric matrix , where is equal to the number of triangles in with sides and in clockwise order minus the number of triangles in with sides and in the counterclockwise order. It is known that matrix is always mutation finite. The cluster algebra of rank is called associated with triangulation .
Example 8. Let be a once-punctured torus of Example 1. The triangulation of the fundamental domain of is sketched in Figure 2 in the charts and , respectively. It is easy to see that in this case with , and , where denotes a geodesic arc with the footpoints and . The Ptolemy relation (5) reduces to ; thus, . The reader is encouraged to verify that matrix has the following form:
Theorem 9 (see [2]). The cluster algebra does not depend on triangulation but only on the surface ; namely, replacement of the geodesic arc by a new geodesic arc (a flip of ) corresponds to a mutation of the seed .
Remark 10. In view of Theorems 6 and 9, corresponds to an algebra of functions on the Teichmüller space ; such an algebra is an analog of the coordinate ring of .
2.3. -Algebras
A -algebra is an algebra over with a norm and involution such that it is complete with respect to the norm and and for all . Any commutative -algebra is isomorphic to the algebra of continuous complex-valued functions on some locally compact Hausdorff space ; otherwise, represents a noncommutative topological space.
An -algebra (Approximately Finite -algebra) is defined to be the norm closure of an ascending sequence of finite-dimensional -algebras , where is the -algebra of the matrices with entries in . Here the index represents the semisimple matrix algebra . The ascending sequence mentioned above can be written aswhere are the finite-dimensional -algebras and are the homomorphisms between such algebras. The homomorphisms can be arranged into a graph as follows. Let and be the semisimple -algebras and the homomorphism. One has two sets of vertices and joined by edges whenever the summand contains copies of the summand under the embedding . As varies, one obtains an infinite graph called the Bratteli diagram of the -algebra. The matrix is known as a partial multiplicity matrix; an infinite sequence of defines a unique -algebra.
Let ; recall that by the Jacobi-Perron continued fraction of vector one understands the limit: where ; see, for example, [20]; the limit converges for a generic subset of vectors . Notice that corresponds to (a matrix form of) the regular continued fraction of ; such a fraction is always convergent. Moreover, the Jacobi-Perron fraction is finite if and only if vector , where are rational. The -algebra associated with the vector is defined by the Bratteli diagram with the partial multiplicity matrices equal to in the Jacobi-Perron fraction of ; in particular, if , coincides with the Effros-Shen algebra [14].
2.4. Cluster -Algebras
Notice that the mutation tree of a cluster algebra has grading by levels, that is, a distance from the root of . We will say that a pair of clusters and are -equivalent, if(i)and lie at the same level;(ii) and coincide modulo a cyclic permutation of variables ;(iii).
It is not hard to see that is an equivalence relation on the set of vertices of graph .
Definition 11. By a cluster -algebra , one understands an -algebra given by the Bratteli diagram of the formThe rank of is equal to such of cluster algebra .
Example 12. If is matrix (6) of Example 8, then is shown in Figure 3. (We refer the reader to Section 4 for a proof.) Notice that the graph is a part of the Bratteli diagram of the Mundici algebra ; compare [10, Figure 1].
Remark 13. It is not hard to see that is no longer a tree and is a finite graph if and only if is a finite cluster algebra.
3. Proof
Let be the rank of cluster -algebra . For the sake of clarity, we will consider the case and the general case separately.
(i) Let be the cluster -algebra of rank . In this case and either and or else . Since is trivial, we are left with , that is, the once-punctured torus .
Repeating the argument of Example 8, we get the seed , where and the skew-symmetric matrix is given by formula (6).
Let us verify that matrix is mutation finite; indeed, for each , the matrix mutation formula (4) gives us .
Therefore, the exchange relations (3) do not vary; it is verified directly that such relations have the following form:
Consider a mutation tree shown in Figure 4; the vertices of correspond to the mutations of cluster following the exchange rules (10).
The reader is encouraged to verify that modulo a cyclic permutation of variables , , and , , one obtains (resp.) the following equivalences of clusters:where ; there are no other cluster equivalences for the vertices of the same level of graph .
To determine the graph , one needs to take the quotient of by the -equivalence relations (11); since the pattern repeats for each level of , one gets the shown in Figure 3. The cluster -algebra is an -algebra with the Bratteli diagram .
Notice that the Bratteli diagram of our -algebra and such of the Mundici algebra are distinct; compare [10, Figure 1]; yet there is an obvious inclusion of one diagram into another. Namely, if one erases a “camel’s back” (i.e., the two extreme sides of the diagram) in the Bratteli diagram of , then one gets exactly the diagram in Figure 3. Formally, if is the Bratteli diagram of the Mundici algebra , the complement is a hereditary Bratteli diagram which gives rise to an ideal , such thatsee [6, Lemma 3.2]; is a primitive ideal, ibid., Theorem 3.8. (It is interesting to calculate the group in the context of the work of [13].)
On the other hand, the space (and hence ) is well understood; see, for example, [13] or [9, Proposition 7]. Namely,where is such that is the Effros-Shen algebra [14] if is an irrational number or is finite-dimensional matrix -algebra (and an extension of such by the -algebra of compact operators) if is a rational number. (Note that the third series of primitive ideals of [9, Proposition 7] correspond to the ideal .) Moreover, given the Jacobson topology on , there exists a homeomorphismdefined by the formula ; see [9, Corollary 12].
Let be the Tomita-Takesaki flow, that is, a one-parameter automorphism group of ; see Section 4. Because , the image of is correctly defined for all ; is an ideal of but not necessarily primitive. Since is nothing but (an algebraic form of) the Teichmüller geodesic flow on [15], one concludes that that the family of idealscan be taken for a coordinate system in the space . In view of (14) and , one gets the required homeomorphismsuch that the quotient algebra is a noncommutative coordinate ring of the Riemann surface .
Remark 14. The family of algebras are in general pairwise nonisomorphic. (For otherwise all ideals were primitive.) However the Grothendieck semigroups are isomorphic; see [14]; the action of is given by the following formula (see Section 4):
(ii) The general case is treated likewise. Notice that if is dimension of the space , then we have ; in particular, rank of the cluster -algebra determines completely the pair provided is a fixed constant. (If is not fixed, there is only a finite number of different pairs for given rank .)
Let be the seed given by the cluster and the skew-symmetric matrix . Since matrix comes from triangulation of the Riemann surface , is mutation finite; see [3, p. 18]; the exchange relations (3) take the following form:
One can construct the mutation tree using relations (18); the reader is encouraged to verify that is similar to the one shown in Figure 4, except for the number of the outgoing edges at each vertex that is equal to .
A tedious but straightforward calculation shows that the only equivalent clusters at the same level of are the ones at the extremities of tuples ; in other words, one gets the following system of equivalences of clusters:where .
The graph is the quotient of by the -equivalence relations (19); for such a graph is sketched in Figure 5. is an -algebra given by the Bratteli diagram .
Lemma 15. The setwhere is an -algebra associated with the convergent Jacobi-Perron continued fraction of vector ; see Section 2.3.
Proof. We adapt the argument of [9, case ] to the case . Let be dimension of the space . Roughly speaking, the Bratteli diagram of algebra can be cut in two disjoint pieces and , as it is shown by [9, Figure 7]. is a (finite or infinite) vertical strip of constant “width” , where is equal to the number of vertices cut from each level of . The reader is encouraged to verify that is exactly the Bratteli diagram of the -algebra associated with the convergent Jacobi-Perron continued fraction of a generic vector ; see Section 2.3.
On the other hand, the complement is a hereditary Bratteli diagram, which defines an ideal of algebra , such thatsee [6, Lemma 3.2]. Moreover, is a primitive ideal [6, Theorem 3.8]. (An extra care is required if is a rational vector; the complete argument can be found in [9, pp. 980–985].) Lemma 15 follows.
Lemma 16. The sequence of primitive ideals converges to in the Jacobson topology in if and only if the sequence converges to in the Euclidean space .
Proof. The proof is a straightforward adaption of the argument in [9, pp. 986–988]; we leave it as an exercise to the reader.
Let be the Tomita-Takesaki flow, that is, the group of modular automorphisms of algebra ; see Section 4. Because , the image of is correctly defined for all ; is an ideal of but not necessarily a primitive ideal. Since is an algebraic form of the Teichmüller geodesic flow on the space [15], one concludes that the family of idealscan be taken for a coordinate system in the space . In view of Lemmas 15 and 16, one gets the required homeomorphismsuch that the quotient algebra is a noncommutative coordinate ring of the Riemann surface .
Theorem 2 is proved.
4. An Analog of Modular Flow on x
4.1. Modular Automorphisms
Recall that the Ptolemy relations (5) for the Penner coordinates in the space are homogeneous; in particular, the system of such coordinates will also satisfy the Ptolemy relations. On the other hand, for the cluster -algebra the variables and one gets an obvious isomorphism for all . Since , one obtains a one-parameter group of automorphisms:By analogy with [21], we will call a Tomita-Takesaki flow on the cluster -algebra . The reader is encouraged to verify that is an algebraic form of the geodesic flow on the Teichmüller space ; see [15] for an introduction. Roughly speaking, such a flow comes from the one-parameter group of matricesacting on the space of holomorphic quadratic differentials on the Riemann surface ; the latter is known to be isomorphic to the Teichmüller space .
4.2. Connes Invariant
Recall that an analogy of the Connes invariant for a -algebra endowed with a modular automorphism group is the set [21]. The group of inner automorphisms of the space and algebra is isomorphic to the mapping class group of surface . The automorphism is called pseudo-Anosov, if , where is invariant measured foliation and is a constant called dilatation of ; is always an algebraic number of the maximal degree [22]. It is known that if is pseudo-Anosov, then there exists a trajectory of the geodesic flow and a point , such that the points and belong to [15]; is called an axis of the pseudo-Anosov automorphism . The axis can be used to calculate the Connes invariant of the cluster -algebra ; indeed, in view of formula (25) one must solve the following system of equations:for a point . Thus, coincides with the inner automorphism if and only if . Taking all pseudo-Anosov automorphisms , one gets a formula for the Connes invariant:
Remark 17. The Connes invariant (27) says that the family of cluster -algebras is an analog of the type factors of von Neumann algebras; see [21].
Disclosure
All errors and misconceptions in this paper are solely the author’s.
Competing Interests
The author declares that there are no competing interests in the paper.
Acknowledgments
It is the author’s pleasure to thank Ibrahim Assem and the SAG group of the Department of Mathematics of the University of Sherbrooke for hospitality and excellent working conditions. The author is grateful to Ibrahim Assem, Thomas Brüstle, Daniele Mundici, Ralf Schiffler, and Vasilisa Shramchenko for an introduction to the wonderland of cluster algebras and helpful correspondence.