On cluster C*-algebras

We introduce a C*-algebra A(x,Q) attached to the cluster x and a quiver Q. If Q(T) is the quiver coming from a triangulation T of the Riemann surface S with a finite number of cusps, we prove that the primitive spectrum of A(x,Q(T)) times R is homeomorphic to a generic subset of the Teichmueller space of surface S. We conclude with an analog of the Tomita-Takesaki theory and the Connes invariant T(M) for the algebra A(x,Q(T)).


Introduction
Cluster algebras of rank m are a class of commutative rings introduced by [Fomin & Zelevinsky 2002] [10]. Among these algebras one finds coordinate rings of important algebraic varieties, like the Grassmannians and Schubert varieties; cluster algebras appear in the Teichmüller theory [Fomin, Shapiro & Thurston 2008] [9]. Unlike the coordinate rings, the set of generators x i of cluster algebra is usually infinite and defined by induction from a cluster x = (x 1 , . . . , x m ) and a quiver Q, see [Williams 2014] [22] for an excellent survey; the cluster algebra is denoted by A(x, Q). Notice that the A(x, Q) has an additive structure of countable (unperforated) abelian group with an order satisfying the Riesz interpolation property; see Remark 3. In other words, the cluster algebra A(x, Q) is a dimension group by the Effros-Handelman-Shen Theorem [Effros 1981, Theorem 3.1] [7]. Figure 1: The Markov quiver Q 1,1 .
Example 1 Let S 1,1 be a once-punctured torus. The ideal triangulation of S 1,1 defines the Markov quiver 1 Q 1,1 shown in Figure 1, see [Fomin, Shapiro & Thurston 2008, Example 4.6] [9]. The corresponding cluster C * -algebra A(x, Q 1,1 ) of rank 3 can be written as: where I 0 is a primitive ideal of an AF -algebra M. The unital AF -algebra M was originally defined by [Mundici 1988, Section 3] [11]; the genuine notation for such an algebra was M 1 , because K 0 (M 1 ) = (M 1 , 1) := free one-generator unital ℓ-group, i.e. a finitely piecewise affine linear continuous real-valued functions on [0, 1] with integer coefficients. The M 1 was subsequently rediscovered after two decades by [Boca 2008] [3] and denoted by A. The remarkable properties of M 1 include the following features. Every primitive ideal of M 1 is essential [Mundici 2011, Theorem 4.2] [13]. The M 1 is equipped with a faithful invariant tracial state [Mundici 2009, Theorem 3.1] [12]. The center of M 1 coincides with the C * -algebra C  [7]), the primitive spectrum of M 1 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 [Boca 2008] [3], where M 1 is defined in terms of the Bratteli diagram. We refer the reader to the final part of a paper by [Panti 1999] [18] for a general result encompassing the characterization of the prime spectrum of (M 1 , 1) ∼ = P rim M 1 . Moreover, the AF -algebras A θ introduced by [Effros & Shen 1980] [8] are precisely the infinite-dimensional simple quotients of M 1 ; this fact was first proved by [Mundici 1988, Theorem 3.1(i)] [11] and rediscovered independently by [Boca 2008] [3]. Summing up the above, the primitive ideals I θ ⊂ M are indexed by numbers θ ∈ R; if θ is irrational, the quotient M/I θ ∼ = A θ , where A θ is the Effros-Shen algebra. In view of 1 Such a quiver is related to solutions in the integer numbers of the equation x 2 1 + x 2 2 + x 2 3 = 3x 1 x 2 x 3 considered by A. A. Markov; hence the name.
(1), the algebra M is a non-commutative coordinate ring of the Teichmüller space T 1,1 . Moreover, there exists an analog of the Tomita-Takesaki theory of modular automorphisms {σ t | t ∈ R} for algebra M, see Section 4; such automorphisms correspond to the Teichmüller geodesic flow on T 1,1 [Veech 1986] [21]. The σ t (I θ ) is an ideal of M for all t ∈ R, where σ 0 (I θ ) = I θ . The quotient algebra M/σ t (I θ ) can be viewed as a non-commutative coordinate ring of the Riemann surface S 1,1 ; in particular, the pairs (θ, t) are coordinates in the space T 1,1 ∼ = R 2 . We refer the reader to [14] for a construction of the corresponding functor.
Motivated by Example 1, denote by A(x, Q g,n ) the cluster C * -algebra corresponding to a quiver Q g,n ; let σ t : A(x, Q g,n ) → A(x, Q g,n ) be the Tomita-Takesaki flow on A(x, Q g,n ), see Section 4 for the details. Denote by P rim A(x, Q g,n ) the set of all primitive ideals of A(x, Q g,n ) endowed with the Jacobson topology and let I θ ∈ P rim A(x, Q g,n ) for a generic value of index θ ∈ R 6g−7+2n . Our main result can be stated as follows.
Theorem 1 There exists a homeomorphism given by the formula σ t (I θ ) → S g,n ; the set U = T g,n if and only if g = n = 1. The σ t (I θ ) is an ideal of A(x, Q g,n ) for all t ∈ R and the quotient algebra A(x, Q g,n )/σ t (I θ ) is a non-commutative coordinate ring of the Riemann surface S g,n .
Remark 1 Theorem 1 is valid for n ≥ 1, i.e. the Riemann surfaces with at least one cusp. This cannot be improved, since the cluster structure of algebra A(x, Q g,n ) comes from the Ptolemy relations satisfied by the Penner coordinates; so far such coordinates are available only for the Riemann surfaces with cusps [Penner 1987] [19]. It is likely, that the case n = 0 has also a cluster structure; we refer the reader to [15], where a functor from the Riemann surfaces S g,0 to the AF -algebras A(x, Q g,0 )/σ t (I θ ) was constructed.

Remark 2
The braid group B 2g+n with n ∈ {1, 2} admits a faithful representation by projections in the algebra A(x, Q g,n ); 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 K 0 (A(x, Q g,n )) allow to generalize the Jones and HOMFLY invariants of knots and links to an arbitrary number of variables, see [17] for the details.
The article is organized as follows. We introduce preliminary facts and notation in Section 2. Theorem 1 is proved in Section 3. An analog of the Tomita-Takesaki theory of modular automorphisms and the Connes invariant T (A(x, Q g,n )) of the cluster C * -algebra A(x, Q g,n ) is constructed.

Cluster algebras
A cluster algebra A of rank m is a subring of the field Q(x 1 , . . . , x m ) of rational functions in n variables. Such an algebra is defined by a pair (x, B), where x = (x 1 , . . . , x m ) is a cluster of variables and B = (b ij ) is a skew-symmetric integer matrix; the new cluster x ′ is obtained from x by an excision of the variable x k and replacing it by a new variable x ′ k subject to an exchange relation: Since the entries of matrix B are exponents of the monimials in cluster variables, one gets a new pair ( For brevity, the pair (x, B) is called a seed and the seed ( The matrix B is called mutation finite if only finitely many new matrices can be produced from B by repeated matrix mutations. The cluster algebra A(x, B) can be defined as the subring of Q(x 1 , . . . , x m ) generated by the union of all cluster variables obtained from the initial seed (x, B) by mutations of (x, B) (and its iterations) in all possible directions. We shall write − → T m to denote an oriented tree whose vertices are seeds (x ′ , B ′ ) and m outgoing arrows in each vertex correspond to mutations µ k of the seed (x ′ , B ′ ). The Laurent phenomenon proved by [Fomin & Zelevinsky 2002] [10] ] is the ring of the Laurent polynomials in variables x = (x 1 , . . . , x n ); in other words, each generator x i of algebra A(x, B) can be written as a Laurent polynomial in n variables with the integer coefficients.

Remark 3
The Laurent phenomenon turns the additive structure of cluster algebra A(x, B) into a totally ordered abelian group satisfying the Riesz interpolation property, i.e. a dimension group [Effros 1981, Theorem 3.1] [7]; the abelian group with order comes from the semigroup of the Laurent polynomials with positive coefficients, see [16] for the details. A background on the partially and totally ordered, unperforated abelian groups with the Riesz interpolation property can be found in [Effros 1981] [7].
To deal with mutation formulas (3) and (4) in geometric terms, recall that a quiver Q is an oriented graph given by the set of vertices Q 0 and the set of arrows Q 1 ; an example of quiver is given in Figure 1. Let k be a vertex of Q; the mutated at vertex k quiver µ k (Q) has the same set of vertices as Q but the set of arrows is obtained by the following procedure: (i) for each sub-quiver i → k → j one adds a new arrow i → j; (ii) one reverses all arrows with source or target k; (iii) one removes the arrows in a maximal set of pairwise disjoint 2-cycles. The reader can verify, that if one encodes a quiver Q with n vertices by a skew-symmetric matrix B(Q) = (b ij ) with b ij equal to to the number of arrows from vertex i to vertex j, then mutation µ k of seed (x, B) coincides with such of the corresponding quiver Q. Thus the cluster algebra A(x, B) is defined by a quiver Q; we shall denote such an algebra by A(x, Q).

Cluster algebras from Riemann surfaces
Let g and n be integers, such that g ≥ 0, n ≥ 1 and 2g − 2 + n > 0. Denote by S g,n a Riemann surface of genus g with the n cusp points. It is known that the fundamental domain of S g,n can be triangulated by 6g − 6 + 3n geodesic arcs γ, such that the footpoints of each arc at the absolute of Lobachevsky plane H = {x + iy ∈ C | y > 0} coincide with a (pre-image of) cusp of the S g,n . If l(γ) is the hyperbolic length of γ measured (with a sign) between two horocycles around the footpoints of γ, then we set λ(γ) = e 1 2 l(γ) ; the λ(γ) are known to satisfy the Ptolemy relation: where γ 1 , . . . , γ 4 are pairwise opposite sides and γ 5 , γ 6 are the diagonals of a geodesic quadrilateral in H. Denote by T g,n the decorated Teichmüller space of S g,n , i.e. the set of all complex surfaces of genus g with n cusps endowed with the natural topology; it is known that T g,n ∼ = R 6g−6+2n .
Theorem 2 ( [Penner 1987] [19]) The map λ on the set of 6g − 6 + 3n geodesic arcs γ i defining a triangulation of S g,n is a homeomorphism with the image T g,n .
Let T be a triangulation of surface S g,n by 6g − 6 + 3n geodesic arcs γ i ; consider a skew-symmetric matrix B T = (b ij ), where b ij is equal to the number of triangles in T with sides γ i and γ j in clockwise order minus the number of triangles in T with sides γ i and γ j in the counter-clockwise order. It is known that matrix B T is always mutation finite. The cluster algebra A(x, B T ) of rank 6g − 6 + 3n is called associated to triangulation T .
Example 2 Let S 1,1 be a once-puncuted torus of Example 1. The triangulation T of the fundamental domain (R 2 − Z 2 )/Z 2 of S 1,1 is sketched in Figure 2 in the charts R 2 and H, respectively. It is easy to see that in this case x = (x 1 , x 2 , x 3 ) with x 1 = γ 23 , x 2 = γ 34 and x 3 = γ 24 , where γ ij denotes a geodesic arc with the footpoints i and j. The Ptolemy relation (5) reduces to λ 2 (γ 23 ) + λ 2 (γ 34 ) = λ 2 (γ 24 ); thus T 1,1 ∼ = R 2 . The reader is encouraged to verify, that matrix B T has the form:

Theorem 3 ([Fomin, Shapiro & Thurston 2008] [9])
The cluster algebra A(x, B T ) does not depend on triangulation T , but only on the surface S g,n ; namely, replacement of the geodesic arc γ k by a new geodesic arc γ ′ k (a flip of γ k ) corresponds to a mutation µ k of the seed (x, B T ).
Remark 5 In view of Theorems 2 and 3, the A(x, B T ) corresponds to an algebra of functions on the Teichmüller space T g,n ; such an algebra is an analog of the coordinate ring of T g,n . Figure 2: Triangulation of the Riemann surface S 1,1 .

C * -algebras
A C * -algebra is an algebra A over C with a norm a → ||a|| and an involution a → a * such that it is complete with respect to the norm and ||ab|| ≤ ||a|| ||b|| and ||a * a|| = ||a 2 || for all a, b ∈ A. Any commutative C * -algebra is isomorphic to the algebra C 0 (X) of continuous complex-valued functions on some locally compact Hausdorff space X; otherwise, A represents a noncommutative topological space.
An AF -algebra (Approximately Finite C * -algebra) is defined to be the norm closure of an ascending sequence of finite dimensional C * -algebras M n , where M n is the C * -algebra of the n × n matrices with entries in C. Here the index n = (n 1 , . . . , n k ) represents the semi-simple matrix algebra M n = M n 1 ⊕ . . . ⊕ M n k . The ascending sequence mentioned above can be written as where M i are the finite dimensional C * -algebras and ϕ i the homomorphisms between such algebras. The homomorphisms ϕ i can be arranged into a graph as follows.
be the semisimple C * -algebras and ϕ i : M i → M i ′ the homomorphism. One has two sets of vertices V i 1 , . . . , V i k and V i ′ 1 , . . . , V i ′ k joined by b rs edges whenever the summand M ir contains b rs copies of the summand M i ′ s under the embedding ϕ i . As i varies, one obtains an infinite graph called the Bratteli diagram of the AF -algebra. The matrix B = (b rs ) is known as a partial multiplicity matrix; an infinite sequence of B i defines a unique AF -algebra.
Let θ ∈ R n−1 ; recall that by the Jacobi-Perron continued fraction of vector (1, θ) one understands the limit: , see e.g. [Bernstein 1971] [1]; the limit converges for a generic subset of vectors θ ∈ R n−1 . Notice that n = 2 corresponds to (a matrix form of) the regular continued fraction of θ; such a fraction is always convergent. Moreover, the Jacobi-Perron fraction is finite if and only if vector θ = (θ i ), where θ i are rational. The AF -algebra A θ associated to the vector (1, θ) is defined by the Bratteli diagram with the partial multiplicity matrices equal to B k in the Jacobi-Perron fraction of (1, θ); in particular, if n = 2 the A θ coincides with the Effros-Shen algebra [Effros & Shen 1980] [8].

Cluster C * -algebras
Notice that the mutation tree − → T m of a cluster algebra A(x, B) has a grading by levels, i.e. a distance from the root of − → T m . We shall say that a pair of clusters x and x ′ are ℓ-equivalent, if: (i) x and x ′ lie at the same level; (ii) x and x ′ coincide modulo a cyclic permutation of variables x i ; It is not hard to see that ℓ is an equivalence relation on the set of vertices of graph − → T m .
Definition 1 By a cluster C * -algebra A(x, B) one understands an AF -algebra given by the Bratteli diagram B(x, B) of the form: The rank of A(x, B) is equal to such of cluster algebra A(x, B).

Proof
Let m = 3(2g − 2 + n) be the rank of cluster C * -algebra A(x, Q g,n ). For the sake of clarity, we shall consider the case m = 3 and the general case m ∈ {3, 6, 9, . . .} separately.
Repeating the argument of Example 2, we get the seed (x, B T ), where x = (x 1 , x 2 , x 3 ) and the skew-symmetric matrix B T is given by formula (6).
Let us verify that matrix B T is mutation finite; indeed, for each k ∈ {1, 2, 3} the matrix mutation formula (4) gives us µ k (B T ) = −B T . Therefore, the exchange relations (3) do not vary; it is verified directly that such relations have the form: Consider a mutation tree − → T 3 shown in Figure 4; the vertices of − → T 3 correspond to the mutations of cluster x = (x 1 , x 2 , x 3 ) following the exchange rules (9).
The reader is encouraged to verify that modulo a cyclic permutation of variables one obtains (respectively) the following equivalences of clusters: Figure 4: The mutation tree.
where µ ij (x) := µ j (µ i (x)); there are no other cluster equivalences for the vertices of the same level of graph − → T 3 .
To determine the graph B(x, B T ) one needs to take the quotient of − → T 3 by the ℓ-equivalence relations (10); since the pattern repeats for each level of − → T 3 , one gets the B(x, B T ) shown in Figure 3. The cluster C * -algebra A(x, B T ) is an AF -algebra with the Bratteli diagram B(x, B T ).
Notice that the Bratteli diagram B(x, B T ) of our AF -algebra A(x, B T ) and such of the Mundici algebra M are distinct, compare [Mundici 2011, Figure 1] [13]; 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 M, then one gets exactly the diagram in Figure 3. Formally, if G is the Bratteli diagram of the Mundici algebra M, the complement G − B(x, B T ) is a hereditary Bratteli diagram which gives rise to an ideal I 0 ⊂ M, such that: see [Bratteli 1972, Lemma 3.2] [4]; the I 0 is a primitive ideal ibid., Theorem 3.8. (It is interesting to calculate the group K 0 (I 0 ) in the context of the work of [Panti 1999] [18].) On the other hand, the space P rim M (and hence P rim A(x, B T )) is well understood, see e.g. [Panti 1999] [18] or [Boca 2008, Proposition 7] [3]. Namely, where Let σ t : M/I 0 → M/I 0 be the Tomita-Takesaki flow, i.e. a one-parameter automorphism group of M/I 0 , see Section 4. Because I θ ⊂ M/I 0 , the image σ t (I θ ) of I θ is correctly defined for all t ∈ R; the σ t (I θ ) is an ideal of M/I 0 but not necessarily primitive. Since σ t is nothing but (an algebraic form of) the Teichmüller geodesic flow on T 1,1 [Veech 1986] [21], one concludes that that the family of ideals can be taken for a coordinate system in the space T 1,1 ∼ = R 2 . In view of (13) and M/I 0 ∼ = A(x, Q 1,1 ), one gets the required homeomorphism such that the quotient algebra A(x, Q 1,1 )/σ t (I θ ) is a non-commutative coordinate ring of the Riemann surface S 1,1 .

Remark 7
The family of algebras {A(x, Q 1,1 )/σ t (I θ ) | θ = Const, t ∈ R} are in general pairwise non-isomorphic. (For otherwise all ideals {σ t (I θ ) | t ∈ R} were primitive.) Yet their Grothendieck semi-groups K + 0 are, see [Effros & Shen 1980] [8]; the action of σ t is given by the formula (see Section 4): (ii) The general case m = 3k = 3(2g − 2 + n) is treated likewise. Notice that if d = 6g − 6 + 2n is dimension of the space T g,n , then we have m − d = n; in particular, rank m of the cluster C * -algebra A(x, Q g,n ) determines completely the pair (g, n) provided d is a fixed constant. (If d is not fixed, there is only a finite number of different pairs (g, n) for given rank m.) Let (x, B T ) be the seed given by the cluster x = (x 1 , . . . , x 3k ) and the skew-symmetric matrix B T . Since matrix B T comes from a triangulation of the Riemann surface S g,n , B T is mutation finite, see [Williams 2014, p.18] [22]; the exchange relations (3) take the form: One can construct the mutation tree − → T 3k using relations (17); the reader is encouraged to verify, that the − → T 3k is similar to the one shown in Figure 4, except for the number of the outgoing edges at each vertex is equal to 3k.
A tedious but straightforward calculation shows that the only equivalent clusters at the same level of − → T 3k are the ones at the extremities of tuples (x ′ 1 , . . . , x ′ 3k ); in other words, one gets the following system of equivalences of clusters: where µ ij (x) := µ j (µ i (x)). The graph B(x, B T ) is the quotient of − → T 3k by the ℓ-equivalence relations (18); for k = 2 such a graph is sketched in Figure 5. The A(x, Q g,n ) is an AF -algebra given by the Bratteli diagram B(x, B T ).

Lemma 1 The set
where A(x, Q g,n )/I θ is an AF -algebra A θ associated to the convergent Jacobi-Perron continued fraction of vector (1, θ), see Section 2.3. Proof. We adapt the argument of [Boca 2008, case k = 1] [3] to the case k ≥ 1. Let d = 6g − 6 + 2n be dimension of the space T g,n . Roughly speaking, the Bratteli diagram B(x, B T ) of algebra A(x, Q g,n ) can be cut in two disjoint pieces G θ and B(x, B T ) − G θ , as it is shown by [Boca 2008, Figure 7] [3]. The G θ is a (finite or infinite) vertical strip of constant "width" d, where d is equal to the number of vertices cut from each level of B(x, B T ). The reader is encouraged to verify, that G θ is exactly the Bratteli diagram of the AF -algebra A θ associated to the convergent Jacobi-Perron continued fraction of a generic vector (1, θ), see Section 2.3.
On the other hand, the complement B(x, B T )−G θ is a hereditary Bratteli diagram, which defines an ideal I θ of algebra A(x, Q g,n ), such that: see [Bratteli 1972 ideal. Since σ t is an algebraic form of the Teichmüller geodesic flow on the space T g,n [Veech 1986] [21], one concludes that that the family of ideals: can be taken for a coordinate system in the space T g,n ∼ = R 6g−6+2n . In view of Lemmas 1 and 2, one gets the required homeomorphism such that the quotient algebra A θ = A(x, Q g,n )/σ t (I θ ) is a non-commutative coordinate ring of the Riemann surface S g,n .
Theorem 1 is proved.
4 An analog of modular flow on A(x, Q g,n ) A. Modular automorphisms {σ t | t ∈ R}. Recall that the Ptolemy relations (5) for the Penner coordinates {λ(γ i )} in the space T g,n are homogeneous; in particular, the system {tλ(γ i ) | t ∈ R} of such coordinates will also satisfy the Ptolemy relations. On the other hand, for the cluster C * -algebra A(x, Q g,n ) the variables x i = λ(γ i ) and one gets an obvious isomorphism A(x, Q g,n ) ∼ = A(tx, Q g,n ) for all t ∈ R. Since A(tx, Q g,n ) ⊆ A(x, Q g,n ), one obtains a one-parmeter group of automorphisms: σ t : A(x, Q g,n ) −→ A(x, Q g,n ).
By analogy with [Connes 1978] [5], we shall call σ t a Tomita-Takesaki flow on the cluster C * -algebra A(x, Q g,n ). The reader is encouraged to verify, that σ t is an algebraic form of the geodesic flow T t on the Teichmüller space T g,n , see [Veech 1986] [21] for an introduction. Roughly speaking, such a flow comes from the one-parameter group of matrices acting on the space of holomorphic quadratic differentials on the Riemann surface S g,n ; the latter is known to be isomorphic to the Teichmüller space T g,n .
B. Connes invariant T (A(x, Q g,n )). Recall that an analogy of the Connes invariant T (M) for a C * -algebra M endowed with a modular automorphism group σ t is the set T (M) := {t ∈ R | σ t is inner} [Connes 1978] [5]. The group of inner automorphisms of the space T g,n and algebra A(x, Q g,n ) is isomorphic to the mapping class group Mod S g,n of surface S g,n . The automorphism φ ∈ Mod S g,n is called pseudo-Anosov, if φ(F µ ) = λ φ F µ , where F µ is an invariant measured foliation and λ φ > 1 is a constant called dilatation of φ; the λ φ is always an algebraic number of the maximal degree 6g − 6 + 2n [Thurston 1988] [20]. It is known, that if φ ∈ Mod S g,n is pseudo-Anosov then there exists a trajectory O of the geodesic flow T t and a point S g,n ∈ T g,n , such that the points S g,n and φ(S g,n ) belong to O [Veech 1986] [21]; the O is called an axis of the pseudo-Anosov automorphism φ. The axis can be used to calculate the Connes invariant T (A(x, Q g,n )) of the cluster C * -algebra A(x, Q g,n ); indeed, in view of formula (24) one must solve the following system of equations: for a point x ∈ O. Thus σ t (x) coincides with the inner automorphism φ(x) if and only if t = log λ φ . Taking all pseudo-Anosov automorphisms φ ∈ Mod S g,n , one gets a formula for the Connes invariant: T (A(x, Q g,n )) = {log λ φ | φ ∈ Mod S g,n is pseudo-Anosov}.
Remark 8 The Connes invariant (26) says that the family of cluster C *algebras A(x, Q g,n ) is an analog of the type III λ factors of von Neumann algebras, see [Connes 1978] [5].
Competing interests. The author declares that there is no conflict of interests in the paper.