# On ClusterC

Authors: Igor V. Nikolaev
Journal: Journal of Function Spaces
 (2016)

---

## Abstract

We introduce aC
⁎-algebra A
(
x
,
Q
) attached to the cluster x and a quiver Q. If Q
T is the quiver coming from 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 Teichmüller 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
).


---

## Body

## 1. Introduction

Cluster algebras of rankm 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 x
i of cluster algebra is usually infinite and defined by induction from acluster
x
=
(
x
1
,
…
,
x
m
) and aquiver
Q; see [3] for an excellent survey; the cluster algebra is denoted by A
(
x
,
Q
). Notice that A
(
x
,
Q
) 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 A
(
x
,
Q
) is adimension group by the Effros-Handelman-Shen Theorem [4, Theorem 3.1].The subject of our paper is an operator algebraA
(
x
,
Q
), such that K
0
A
x
,
Q
≅
A
(
x
,
Q
); here K
0
(
A
(
x
,
Q
)
) is the dimension group of A
(
x
,
Q
) and ≅ is an isomorphism of the ordered abelian groups [5, Chapter 7]. A
(
x
,
Q
) is an Approximately Finite C
∗
-algebra (AF-algebra) given by a Bratteli diagram derived explicitly from the pair (
x
,
Q
). The AF-algebras were introduced and studied by [6]; we refer to A
(
x
,
Q
) as acluster
C
∗
-algebra.An exact definition ofA
(
x
,
Q
) can be found in Section 2.4; to give an idea, recall that the pair (
x
,
Q
) is called aseed and the cluster algebra A
(
x
,
Q
) is generated by seeds obtained viamutation of (
x
,
Q
) (and its mutants) in alldirections
k, where 1
≤
k
≤
m [3, p. 5]. The mutation process can be described by an oriented regular tree T
→
m; the vertices of T
→
m correspond to the seeds and the outgoing edges to the mutations in directions k. The quotient B
(
x
,
Q
) of T
→
m by a relation identifying equivalent seeds at the same level of T
→
m is a graph with cycles. (For a quick example of such a graph, see Figure 3.) The cluster C
∗-algebra A
(
x
,
Q
) is an AF-algebra given by B
(
x
,
Q
) regarded as a Bratteli diagram [6].LetS
g
,
n be a Riemann surface of genus g
≥
0 with n
≥
1 cusps such that 2
g
-
2
+
n
>
0; denote by T
g
,
n
≅
R
6
g
-
6
+
2
n the (decorated) Teichmüller space of S
g
,
n, that is, a collection of all Riemann surfaces of genus g with n cusps endowed with the natural topology [7]. In what follows, we focus on the algebras A
(
x
,
Q
g
,
n
) with quivers Q
g
,
n coming from ideal triangulation of S
g
,
n; the corresponding cluster algebra A
(
x
,
Q
g
,
n
) of rank m
=
6
g
-
6
+
3
n is related to thePenner coordinates in T
g
,
n [2].Example 1.
LetS
1,1 be a once-punctured torus. The ideal triangulation of S
1,1 defines the Markov quiver (such a quiver is related to solutions in the integer numbers of the equation x
1
2
+
x
2
2
+
x
3
2
=
3
x
1
x
2
x
3 considered by A. A. Markov, hence the name)  Q
1,1 shown in Figure 1; see [2, Example 4.6]. The corresponding cluster C
∗-algebra A
(
x
,
Q
1,1
) of rank 3 can be written as (1)
A
x
,
Q
1,1
≅
M
I
0
,where I
0 is a primitive ideal of an AF-algebra M. The unital AF-algebra M was originally defined by [8, Section 3]; the genuine notation for such an algebra was M
1, because K
0
(
M
1
)
=
(
M
1
,
1
) ≔ free one-generator unital l-group, that is, finitely piecewise affine linear continuous real-valued functions on [
0,1
] with integer coefficients. M
1 was subsequently rediscovered after two decades by [9] and denoted by A. The remarkable properties of M
1 include the following features. Every primitive ideal of M
1 is essential [10, Theorem 4.2]. M
1 is equipped with a faithful invariant tracial state [11, Theorem 3.1]. The center of M
1 coincides with the C
∗-algebra C
[
0,1
] of continuous complex-valued functions on [
0,1
] [9, p. 976]. There is an affine weak ∗-homeomorphism of the state space of C
[
0,1
] onto the space of tracial states on M
1 [10, Theorem 4.5]. Any state of C
[
0,1
] has precisely one tracial extension to M
1 [12, Theorem 2.5]. The automorphism group of M
1 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 M
1. In the light of the original definition of M
1 and the fact that the K
0-functor preserves exact sequences (see, e.g., [4, Theorem 3.1]), 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 [9], where M
1 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 M
1
,
1
≅
Prim
⁡
M
1. Moreover, the AF-algebras A
θ introduced by [14] are precisely the infinite-dimensional simple quotients of M
1; this fact was first proved by [8, Theorem 3.1(i)] and rediscovered independently by [9]. 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), the algebra M is a noncommutative 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 [15]. 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 noncommutative 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 [16] for a construction of the corresponding functor.Figure 1
The Markov quiverQ
1,1.Motivated by Example1, 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 Prim
⁡
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
∈
Prim
⁡
A
(
x
,
Q
g
,
n
) for a generic value of index θ
∈
R
6
g
-
7
+
2
n. Our main result can be stated as follows.Theorem 2.
There exists a homeomorphism(2)
h
:
Prim
⁡
A
x
,
Q
g
,
n
×
R
⟶
U
⊆
T
g
,
n
∣
U
i
s
g
e
n
e
r
i
cgiven by the formula σ
t
(
I
θ
)
↦
S
g
,
n; the set U
=
T
g
,
n if and only if g
=
n
=
1. 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 noncommutative coordinate ring of the Riemann surface S
g
,
n.Remark 3.
Theorem2 is valid for n
≥
1, that is, 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 [7]. It is likely that the case n
=
0 has also a cluster structure; we refer the reader to [17], 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 4.
The braid groupB
2
g
+
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 theBirman-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 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 Section2. Theorem 2 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.


## 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

Acluster 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
i
j
) 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:(3)
x
k
x
k
′
=
∏
i
=
1
m
x
i
max
⁡
b
i
k
,
0
+
∏
i
=
1
m
x
i
max
⁡
-
b
i
k
,
0
.Since the entries of matrix B are exponents of the monomials in cluster variables, one gets a new pair x
′
,
B
′, where B
′
=
(
b
i
j
′
) is a skew-symmetric matrix with(4)
b
i
j
′
=
-
b
i
j
if
i
=
k
or
j
=
k
b
i
j
+
b
i
k
b
k
j
+
b
i
k
b
k
j
2
otherwise.For brevity, the pair (
x
,
B
) is called aseed and the seed x
′
,
B
′
≔
(
x
′
,
μ
k
(
B
)
) is obtained from (
x
,
B
) by amutation
μ
k in the direction k, where 1
≤
k
≤
m; μ
k is involution; that is, μ
k
2
=
I
d. The matrix B is calledmutation 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 will 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
′. TheLaurent phenomenon proved by [1] says that A
(
x
,
B
)
⊂
Z
[
x
±
1
], where Z
[
x
±
1
] 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 5.
The Laurent phenomenon turns the additive structure of cluster algebraA
(
x
,
B
) 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 withpositive 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 aquiver
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 subquiver 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
i
j
) with b
i
j equal 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 will denote such an algebra by A
(
x
,
Q
).


### 2.2. Cluster Algebras from Riemann Surfaces

Letg and n be integers, such that g
≥
0, n
≥
1, and 2
g
-
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 6
g
-
6
+
3
n geodesic arcs γ, such that the footpoints of each arc at the absolute of Lobachevsky plane H
=
x
+
i
y
∈
C
∣
y
>
0 coincide with a (preimage of) cusp of 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
); λ
(
γ
) are known to satisfy thePtolemy relation:(5)
λ
γ
1
λ
γ
2
+
λ
γ
3
λ
γ
4
=
λ
γ
5
λ
γ
6
,where γ
1
,
…
,
γ
4 are pairwise opposite sides and γ
5
,
γ
6 are the diagonals of a geodesic quadrilateral in H.Denote byT
g
,
n the decorated Teichmüller space of S
g
,
n, that is, 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
6
g
-
6
+
2
n.Theorem 6 (see [7]).
The mapλ on the set of 6
g
-
6
+
3
n geodesic arcs γ
i defining triangulation of S
g
,
n is a homeomorphism with the image T
g
,
n.Remark 7.
Notice that among6
g
-
6
+
3
n real numbers λ
(
γ
i
) there are only 6
g
-
6
+
2
n independent, since such numbers must satisfy n Ptolemy relations (5).LetT be triangulation of surface S
g
,
n by 6
g
-
6
+
3
n geodesic arcs γ
i; consider a skew-symmetric matrix B
T
=
(
b
i
j
), where b
i
j 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 counterclockwise order. It is known that matrix B
T is always mutation finite. The cluster algebra A
(
x
,
B
T
) of rank 6
g
-
6
+
3
n is calledassociated with triangulation T.Example 8.
LetS
1,1 be a once-punctured 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
i
j 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 following form: (6)
B
T
=
0
2
-
2
-
2
0
2
2
-
2
0
.Figure 2
Triangulation of the Riemann surfaceS
1,1.Figure 3
The Bratteli diagram of Markov’s clusterC
∗-algebra.Theorem 9 (see [2]).
The cluster algebraA
(
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 10.
In view of Theorems6 and 9, 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.


### 2.3.C
∗-Algebras

AC
∗
-algebra is an algebra A over C with a norm a
↦
a and involution a
↦
a
∗ such that it is complete with respect to the norm and a
b
≤
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.AnAF
-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 semisimple matrix algebra M
n
=
M
n
1
⊕
⋯
⊕
M
n
k. The ascending sequence mentioned above can be written as(7)
M
1
⟶
φ
1
M
2
⟶
φ
2
⋯
,where M
i are the finite-dimensional C
∗-algebras and φ
i are the homomorphisms between such algebras. The homomorphisms φ
i can be arranged into a graph as follows. Let M
i
=
M
i
1
⊕
⋯
⊕
M
i
k and M
i
′
=
M
i
1
′
⊕
⋯
⊕
M
i
k
′ 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
r
s edges whenever the summand M
i
r contains b
r
s copies of the summand M
i
s
′ under the embedding φ
i. As i varies, one obtains an infinite graph called theBratteli diagram of the AF-algebra. The matrix B
=
(
b
r
s
) is known as apartial multiplicity matrix; an infinite sequence of B
i defines a unique AF-algebra.Letθ
∈
R
n
-
1; recall that by theJacobi-Perron continued fraction of vector (
1
,
θ
) one understands the limit: (8)
1
θ
1
⋮
θ
n
-
1
=
lim
k
→
∞
⁡
0
0
⋯
0
1
1
0
⋯
0
b
1
1
⋮
⋮
⋮
⋮
0
0
⋯
1
b
n
-
1
1
⋯
0
0
⋯
0
1
1
0
⋯
0
b
1
k
⋮
⋮
⋮
⋮
0
0
⋯
1
b
n
-
1
k
0
0
⋮
1
,where b
i
(
j
)
∈
N
∪
{
0
}; see, for example, [20]; 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 isfinite if and only if vector θ
=
(
θ
i
), where θ
i arerational. The AF-algebra A
θ
associated with 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, A
θ coincides with the Effros-Shen algebra [14].


### 2.4. ClusterC
∗-Algebras

Notice that the mutation treeT
→
m of a cluster algebra A
(
x
,
B
) has grading by levels, that is, a distance from the root of T
→
m. We will say that a pair of clusters x and x
′ are l
-equivalent, if(i)
xand x
′ lie at the same level;
(ii)
x and x
′ coincide modulo a cyclic permutation of variables x
i;
(iii)
B
=
B
′.It is not hard to see thatl is an equivalence relation on the set of vertices of graph T
→
m.Definition 11.
By a clusterC
∗-algebra A
(
x
,
B
), one understands an AF-algebra given by the Bratteli diagram B
(
x
,
B
) of the form(9)
B
x
,
B
≔
T
→
m
m
o
d
l
.The rank of A
(
x
,
B
) is equal to such of cluster algebra A
(
x
,
B
).Example 12.
IfB
T is matrix (6) of Example 8, then B
(
x
,
B
T
) is shown in Figure 3. (We refer the reader to Section 4 for a proof.) Notice that the graph B
(
x
,
B
T
) is a part of the Bratteli diagram of the Mundici algebra M; compare [10, Figure 1].Remark 13.
It is not hard to see thatB
(
x
,
B
) is no longer a tree and B
(
x
,
B
) is a finite graph if and only if A
(
x
,
B
) is a finite cluster algebra.


## 2.1. Cluster Algebras

Acluster 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
i
j
) 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:(3)
x
k
x
k
′
=
∏
i
=
1
m
x
i
max
⁡
b
i
k
,
0
+
∏
i
=
1
m
x
i
max
⁡
-
b
i
k
,
0
.Since the entries of matrix B are exponents of the monomials in cluster variables, one gets a new pair x
′
,
B
′, where B
′
=
(
b
i
j
′
) is a skew-symmetric matrix with(4)
b
i
j
′
=
-
b
i
j
if
i
=
k
or
j
=
k
b
i
j
+
b
i
k
b
k
j
+
b
i
k
b
k
j
2
otherwise.For brevity, the pair (
x
,
B
) is called aseed and the seed x
′
,
B
′
≔
(
x
′
,
μ
k
(
B
)
) is obtained from (
x
,
B
) by amutation
μ
k in the direction k, where 1
≤
k
≤
m; μ
k is involution; that is, μ
k
2
=
I
d. The matrix B is calledmutation 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 will 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
′. TheLaurent phenomenon proved by [1] says that A
(
x
,
B
)
⊂
Z
[
x
±
1
], where Z
[
x
±
1
] 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 5.
The Laurent phenomenon turns the additive structure of cluster algebraA
(
x
,
B
) 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 withpositive 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 aquiver
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 subquiver 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
i
j
) with b
i
j equal 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 will denote such an algebra by A
(
x
,
Q
).


## 2.2. Cluster Algebras from Riemann Surfaces

Letg and n be integers, such that g
≥
0, n
≥
1, and 2
g
-
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 6
g
-
6
+
3
n geodesic arcs γ, such that the footpoints of each arc at the absolute of Lobachevsky plane H
=
x
+
i
y
∈
C
∣
y
>
0 coincide with a (preimage of) cusp of 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
); λ
(
γ
) are known to satisfy thePtolemy relation:(5)
λ
γ
1
λ
γ
2
+
λ
γ
3
λ
γ
4
=
λ
γ
5
λ
γ
6
,where γ
1
,
…
,
γ
4 are pairwise opposite sides and γ
5
,
γ
6 are the diagonals of a geodesic quadrilateral in H.Denote byT
g
,
n the decorated Teichmüller space of S
g
,
n, that is, 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
6
g
-
6
+
2
n.Theorem 6 (see [7]).
The mapλ on the set of 6
g
-
6
+
3
n geodesic arcs γ
i defining triangulation of S
g
,
n is a homeomorphism with the image T
g
,
n.Remark 7.
Notice that among6
g
-
6
+
3
n real numbers λ
(
γ
i
) there are only 6
g
-
6
+
2
n independent, since such numbers must satisfy n Ptolemy relations (5).LetT be triangulation of surface S
g
,
n by 6
g
-
6
+
3
n geodesic arcs γ
i; consider a skew-symmetric matrix B
T
=
(
b
i
j
), where b
i
j 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 counterclockwise order. It is known that matrix B
T is always mutation finite. The cluster algebra A
(
x
,
B
T
) of rank 6
g
-
6
+
3
n is calledassociated with triangulation T.Example 8.
LetS
1,1 be a once-punctured 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
i
j 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 following form: (6)
B
T
=
0
2
-
2
-
2
0
2
2
-
2
0
.Figure 2
Triangulation of the Riemann surfaceS
1,1.Figure 3
The Bratteli diagram of Markov’s clusterC
∗-algebra.Theorem 9 (see [2]).
The cluster algebraA
(
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 10.
In view of Theorems6 and 9, 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.


## 2.3.C
∗-Algebras

AC
∗
-algebra is an algebra A over C with a norm a
↦
a and involution a
↦
a
∗ such that it is complete with respect to the norm and a
b
≤
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.AnAF
-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 semisimple matrix algebra M
n
=
M
n
1
⊕
⋯
⊕
M
n
k. The ascending sequence mentioned above can be written as(7)
M
1
⟶
φ
1
M
2
⟶
φ
2
⋯
,where M
i are the finite-dimensional C
∗-algebras and φ
i are the homomorphisms between such algebras. The homomorphisms φ
i can be arranged into a graph as follows. Let M
i
=
M
i
1
⊕
⋯
⊕
M
i
k and M
i
′
=
M
i
1
′
⊕
⋯
⊕
M
i
k
′ 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
r
s edges whenever the summand M
i
r contains b
r
s copies of the summand M
i
s
′ under the embedding φ
i. As i varies, one obtains an infinite graph called theBratteli diagram of the AF-algebra. The matrix B
=
(
b
r
s
) is known as apartial multiplicity matrix; an infinite sequence of B
i defines a unique AF-algebra.Letθ
∈
R
n
-
1; recall that by theJacobi-Perron continued fraction of vector (
1
,
θ
) one understands the limit: (8)
1
θ
1
⋮
θ
n
-
1
=
lim
k
→
∞
⁡
0
0
⋯
0
1
1
0
⋯
0
b
1
1
⋮
⋮
⋮
⋮
0
0
⋯
1
b
n
-
1
1
⋯
0
0
⋯
0
1
1
0
⋯
0
b
1
k
⋮
⋮
⋮
⋮
0
0
⋯
1
b
n
-
1
k
0
0
⋮
1
,where b
i
(
j
)
∈
N
∪
{
0
}; see, for example, [20]; 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 isfinite if and only if vector θ
=
(
θ
i
), where θ
i arerational. The AF-algebra A
θ
associated with 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, A
θ coincides with the Effros-Shen algebra [14].


## 2.4. ClusterC
∗-Algebras

Notice that the mutation treeT
→
m of a cluster algebra A
(
x
,
B
) has grading by levels, that is, a distance from the root of T
→
m. We will say that a pair of clusters x and x
′ are l
-equivalent, if(i)
xand x
′ lie at the same level;
(ii)
x and x
′ coincide modulo a cyclic permutation of variables x
i;
(iii)
B
=
B
′.It is not hard to see thatl is an equivalence relation on the set of vertices of graph T
→
m.Definition 11.
By a clusterC
∗-algebra A
(
x
,
B
), one understands an AF-algebra given by the Bratteli diagram B
(
x
,
B
) of the form(9)
B
x
,
B
≔
T
→
m
m
o
d
l
.The rank of A
(
x
,
B
) is equal to such of cluster algebra A
(
x
,
B
).Example 12.
IfB
T is matrix (6) of Example 8, then B
(
x
,
B
T
) is shown in Figure 3. (We refer the reader to Section 4 for a proof.) Notice that the graph B
(
x
,
B
T
) is a part of the Bratteli diagram of the Mundici algebra M; compare [10, Figure 1].Remark 13.
It is not hard to see thatB
(
x
,
B
) is no longer a tree and B
(
x
,
B
) is a finite graph if and only if A
(
x
,
B
) is a finite cluster algebra.


## 3. Proof

Letm
=
3
(
2
g
-
2
+
n
) be the rank of cluster C
∗-algebra A
(
x
,
Q
g
,
n
). For the sake of clarity, we will consider the case m
=
3 and the general case m
∈
{
3,6
,
9
,
…
} separately.(i) LetA
(
x
,
B
T
) be the cluster C
∗-algebra of rank 3. In this case 2
g
-
2
+
n
=
1 and either g
=
0 and n
=
3 or else g
=
n
=
1. Since T
0,3
≅
{
p
t
} is trivial, we are left with g
=
n
=
1, that is, the once-punctured torus S
1,1.Repeating the argument of Example8, 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 matrixB
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 following form:(10)
x
1
x
1
′
=
x
2
2
+
x
3
2
,
x
2
x
2
′
=
x
1
2
+
x
3
2
,
x
3
x
3
′
=
x
1
2
+
x
2
2
.Consider a mutation treeT
→
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 (10).Figure 4
The mutation tree.The reader is encouraged to verify that modulo a cyclic permutation of variablesx
1
′
=
x
2, x
2
′
=
x
3, x
3
′
=
x
1 and x
1
′
=
x
3, x
2
′
=
x
1, x
3
′
=
x
2 one obtains (resp.) the following equivalences of clusters:(11)
μ
13
x
=
μ
21
x
,
μ
23
x
=
μ
31
x
,where μ
i
j
(
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 graphB
(
x
,
B
T
), one needs to take the quotient of T
→
3 by the l-equivalence relations (11); 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 diagramB
(
x
,
B
T
) of our AF-algebra A
(
x
,
B
T
) and such of the Mundici algebra M 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 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(12)
A
x
,
B
T
≅
M
I
0
;see [6, Lemma 3.2]; 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 [13].)On the other hand, the spacePrim
⁡
M (and hence Prim
⁡
A
(
x
,
B
T
)) is well understood; see, for example, [13] or [9, Proposition 7]. Namely,(13)
Prim
⁡
M
I
0
=
I
θ
∣
θ
∈
R
,where I
θ
⊂
M is such that M
/
I
θ
≅
A
θ is the Effros-Shen algebra [14] if θ is an irrational number or M
/
I
θ
≅
M
q is finite-dimensional matrix C
∗-algebra (and an extension of such by the C
∗-algebra of compact operators) if θ
=
p
/
q is a rational number. (Note that the third series of primitive ideals of [9, Proposition 7] correspond to the ideal I
0.) Moreover, given the Jacobson topology on Prim
⁡
M, there exists a homeomorphism(14)
h
:
Prim
⁡
M
I
0
⟶
Rdefined by the formula I
θ
↦
θ; see [9, Corollary 12].Letσ
t
:
M
/
I
0
→
M
/
I
0 be the Tomita-Takesaki flow, that is, 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; 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 [15], one concludes that that the family of ideals(15)
σ
t
I
θ
⊂
M
I
0
∣
t
∈
R
,
θ
∈
Rcan be taken for a coordinate system in the space T
1,1
≅
R
2. In view of (14) and M
/
I
0
≅
A
(
x
,
Q
1,1
), one gets the required homeomorphism(16)
h
:
Prim
⁡
A
x
,
Q
1,1
×
R
⟶
T
1,1
,such that the quotient algebra A
(
x
,
Q
1,1
)
/
σ
t
(
I
θ
) is a noncommutative coordinate ring of the Riemann surface S
1,1.Remark 14.
The family of algebras{
A
(
x
,
Q
1,1
)
/
σ
t
(
I
θ
)
∣
θ
=
Const
,
t
∈
R
} are in general pairwise nonisomorphic. (For otherwise all ideals {
σ
t
(
I
θ
)
∣
t
∈
R
} were primitive.) However the Grothendieck semigroups K
0
+ are isomorphic; see [14]; the action of σ
t is given by the following formula (see Section 4): (17)
K
0
+
A
x
,
Q
1,1
σ
t
I
θ
≅
e
t
Z
+
Z
θ
.(ii) The general casem
=
3
k
=
3
(
2
g
-
2
+
n
) is treated likewise. Notice that if d
=
6
g
-
6
+
2
n 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
3
k
) and the skew-symmetric matrix B
T. Since matrix B
T comes from triangulation of the Riemann surface S
g
,
n, B
T ismutation finite; see [3, p. 18]; the exchange relations (3) take the following form:(18)
x
1
x
1
′
=
x
2
2
+
x
3
2
+
⋯
+
x
3
k
2
,
x
2
x
2
′
=
x
1
2
+
x
3
2
+
⋯
+
x
3
k
2
,
⋮
x
3
k
x
3
k
′
=
x
1
2
+
x
2
2
+
⋯
+
x
3
k
-
1
2
.One can construct the mutation treeT
→
3
k using relations (18); the reader is encouraged to verify that T
→
3
k is similar to the one shown in Figure 4, except for the number of the outgoing edges at each vertex that is equal to 3
k.A tedious but straightforward calculation shows that the only equivalent clusters at the same level ofT
→
3
k are the ones at the extremities of tuples (
x
1
′
,
…
,
x
3
k
′
); in other words, one gets the following system of equivalences of clusters:(19)
μ
1,3
k
x
=
μ
21
x
,
μ
2,3
k
x
=
μ
31
x
,
⋮
μ
3
k
-
1,3
k
x
=
μ
3
k
,
1
x
,where μ
i
j
(
x
)
≔
μ
j
(
μ
i
(
x
)
).The graphB
(
x
,
B
T
) is the quotient of T
→
3
k by the l-equivalence relations (19); for k
=
2 such a graph is sketched in Figure 5. A
(
x
,
Q
g
,
n
) is an AF-algebra given by the Bratteli diagram B
(
x
,
B
T
).Figure 5
The Bratteli diagram of a clusterC
∗-algebra of rank 6.Lemma 15.
The set(20)
Prim
⁡
A
x
,
Q
g
,
n
=
I
θ
∣
θ
∈
R
6
g
-
7
+
2
n
i
s
g
e
n
e
r
i
c
,where A
(
x
,
Q
g
,
n
)
/
I
θ is an AF-algebra A
θ associated with the convergent Jacobi-Perron continued fraction of vector (
1
,
θ
); see Section 2.3.Proof.
We adapt the argument of [9, case k
=
1] to the case k
≥
1. Let d
=
6
g
-
6
+
2
n 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 [9, Figure 7]. 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 with the convergent Jacobi-Perron continued fraction of ageneric vector (
1
,
θ
); see Section 2.3.
On the other hand, the complementB
(
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(21)
A
x
,
Q
g
,
n
I
θ
=
A
θ
;see [6, Lemma 3.2]. Moreover, I
θ is a primitive ideal [6, Theorem 3.8]. (An extra care is required if θ
=
(
θ
i
) is a rational vector; the complete argument can be found in [9, pp. 980–985].) Lemma 15 follows.Lemma 16.
The sequence of primitive idealsI
θ
n converges to I
θ in the Jacobson topology in Prim
⁡
A
(
x
,
Q
g
,
n
) if and only if the sequence θ
n converges to θ in the Euclidean space R
6
g
-
7
+
2
n.Proof.
The proof is a straightforward adaption of the argument in [9, pp. 986–988]; we leave it as an exercise to the reader.Letσ
t
:
A
(
x
,
Q
g
,
n
)
→
A
(
x
,
Q
g
,
n
) be the Tomita-Takesaki flow, that is, the group {
σ
t
∣
t
∈
R
} of modular automorphisms of algebra A
(
x
,
Q
g
,
n
); see Section 4. Because I
θ
⊂
A
(
x
,
Q
g
,
n
), the image σ
t
(
I
θ
) of I
θ is correctly defined for all t
∈
R; σ
t
(
I
θ
) is an ideal of A
(
x
,
Q
g
,
n
) but not necessarily a primitive ideal. Since σ
t is an algebraic form of the Teichmüller geodesic flow on the space T
g
,
n [15], one concludes that the family of ideals(22)
σ
t
I
θ
⊂
A
x
,
Q
g
,
n
∣
t
∈
R
,
θ
∈
R
6
g
-
7
+
2
ncan be taken for a coordinate system in the space T
g
,
n
≅
R
6
g
-
6
+
2
n. In view of Lemmas 15 and 16, one gets the required homeomorphism(23)
h
:
Prim
⁡
A
x
,
Q
g
,
n
×
R
⟶
U
⊆
T
g
,
n
∣
U
i
s
g
e
n
e
r
i
c
,such that the quotient algebra A
θ
=
A
(
x
,
Q
g
,
n
)
/
σ
t
(
I
θ
) is a noncommutative coordinate ring of the Riemann surface S
g
,
n.Theorem2 is proved.


## 4. An Analog of Modular Flow onA
(
x
,
Q
g
,
n
)

### 4.1. 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 t
λ
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
(
t
x
,
Q
g
,
n
) for all t
∈
R. Since A
(
t
x
,
Q
g
,
n
)
⊆
A
(
x
,
Q
g
,
n
), one obtains a one-parameter group of automorphisms:(24)
σ
t
:
A
x
,
Q
g
,
n
⟶
A
x
,
Q
g
,
n
.By analogy with [21], we will call σ
t aTomita-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 thegeodesic flow
T
t on the Teichmüller space T
g
,
n; see [15] for an introduction. Roughly speaking, such a flow comes from the one-parameter group of matrices(25)
e
t
0
0
e
-
tacting 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.


### 4.2. Connes InvariantT
(
A
(
x
,
Q
g
,
n
)
)

Recall that an analogy of theConnes 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
i
s
i
n
n
e
r [21]. 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 calledpseudo-Anosov, if ϕ
(
F
μ
)
=
λ
ϕ
F
μ, where F
μ is invariant measured foliation and λ
ϕ
>
1 is a constant calleddilatation of ϕ; λ
ϕ is always an algebraic number of the maximal degree 6
g
-
6
+
2
n [22]. 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 [15]; O is called anaxis 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 (25) one must solve the following system of equations:(26)
σ
t
x
=
e
t
x
,
ϕ
x
=
λ
ϕ
x
,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:(27)
T
A
x
,
Q
g
,
n
=
log
⁡
λ
ϕ
∣
ϕ
∈
mod
⁡
S
g
,
n
i
s
p
s
e
u
d
o
-
A
n
o
s
o
v
.Remark 17.
The Connes invariant (27) says that the family of cluster C
∗-algebras A
(
x
,
Q
g
,
n
) is an analog of the type I
I
I
λ factors of von Neumann algebras; see [21].


## 4.1. 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 t
λ
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
(
t
x
,
Q
g
,
n
) for all t
∈
R. Since A
(
t
x
,
Q
g
,
n
)
⊆
A
(
x
,
Q
g
,
n
), one obtains a one-parameter group of automorphisms:(24)
σ
t
:
A
x
,
Q
g
,
n
⟶
A
x
,
Q
g
,
n
.By analogy with [21], we will call σ
t aTomita-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 thegeodesic flow
T
t on the Teichmüller space T
g
,
n; see [15] for an introduction. Roughly speaking, such a flow comes from the one-parameter group of matrices(25)
e
t
0
0
e
-
tacting 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.


## 4.2. Connes InvariantT
(
A
(
x
,
Q
g
,
n
)
)

Recall that an analogy of theConnes 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
i
s
i
n
n
e
r [21]. 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 calledpseudo-Anosov, if ϕ
(
F
μ
)
=
λ
ϕ
F
μ, where F
μ is invariant measured foliation and λ
ϕ
>
1 is a constant calleddilatation of ϕ; λ
ϕ is always an algebraic number of the maximal degree 6
g
-
6
+
2
n [22]. 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 [15]; O is called anaxis 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 (25) one must solve the following system of equations:(26)
σ
t
x
=
e
t
x
,
ϕ
x
=
λ
ϕ
x
,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:(27)
T
A
x
,
Q
g
,
n
=
log
⁡
λ
ϕ
∣
ϕ
∈
mod
⁡
S
g
,
n
i
s
p
s
e
u
d
o
-
A
n
o
s
o
v
.Remark 17.
The Connes invariant (27) says that the family of cluster C
∗-algebras A
(
x
,
Q
g
,
n
) is an analog of the type I
I
I
λ factors of von Neumann algebras; see [21].



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*Source: 9639875-2016-06-26.xml*