Research Article On Cluster -Algebras · 2019. 7. 30. · Cluster Algebras from Riemann Surfaces. Let and be integers, such that 0, 1,and2 2+ > 0. Denoteby, aRiemannsurfaceofgenus

Research ArticleOn Cluster 119862lowast-Algebras

Igor V Nikolaev

Department of Mathematics and Computer Science St Johnrsquos University 8000 Utopia Parkway Queens New York NY 11439 USA

Correspondence should be addressed to Igor V Nikolaev igorvnikolaevgmailcom

Received 4 February 2016 Accepted 18 May 2016

Academic Editor Gelu Popescu

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

We introduce a 119862lowast-algebra A(x 119876) attached to the cluster x and a quiver 119876 If 119876119879is the quiver coming from triangulation 119879 of the

Riemann surface 119878 with a finite number of cusps we prove that the primitive spectrum of A(x 119876119879) times R is homeomorphic to

a generic subset of the Teichmuller space of surface 119878 We conclude with an analog of the Tomita-Takesaki theory and the Connesinvariant 119879(M) for the algebra A(x 119876

119879)

1 Introduction

Cluster algebras of rank 119898 are a class of commutative ringsintroduced by [1] Among these algebras one finds coordinaterings of important algebraic varieties like the Grassman-nians and Schubert varieties cluster algebras appear in theTeichmuller theory [2] Unlike the coordinate rings the set ofgenerators 119909

119894of cluster algebra is usually infinite and defined

by induction from a cluster x = (1199091 119909

119898) and a quiver 119876

see [3] for an excellent survey the cluster algebra is denotedby A(x 119876) Notice that A(x 119876) has an additive structureof countable (unperforated) abelian group with an ordersatisfying the Riesz interpolation property see Remark 5 Inother words the cluster algebraA(x 119876) is a dimension groupby the Effros-Handelman-ShenTheorem [4 Theorem 31]

The subject of our paper is an operator algebra A(x 119876)such that 119870

0(A(x 119876)) cong A(x 119876) here 119870

0(A(x 119876)) is the

dimension group of A(x 119876) and cong is an isomorphism ofthe ordered abelian groups [5 Chapter 7] A(x 119876) is anApproximately Finite 119862lowast-algebra (AF-algebra) given by aBratteli diagram derived explicitly from the pair (x 119876) TheAF-algebras were introduced and studied by [6] we refer toA(x 119876) as a cluster 119862lowast-algebra

An exact definition ofA(x 119876) can be found in Section 24to give an idea recall that the pair (x 119876) is called a seed andthe cluster algebra A(x 119876) is generated by seeds obtainedvia mutation of (x 119876) (and its mutants) in all directions 119896where 1 le 119896 le 119898 [3 p 5] The mutation process canbe described by an oriented regular tree T

119898 the vertices of

T119898correspond to the seeds and the outgoing edges to the

mutations in directions 119896 The quotient B(x 119876) of T119898by a

relation identifying equivalent seeds at the same level of T119898

is a graph with cycles (For a quick example of such a graphsee Figure 3)The cluster119862lowast-algebraA(x 119876) is an AF-algebragiven byB(x 119876) regarded as a Bratteli diagram [6]

Let 119878119892119899

be a Riemann surface of genus 119892 ge 0 with119899 ge 1 cusps such that 2119892 minus 2 + 119899 gt 0 denote by 119879

119892119899cong

R6119892minus6+2119899 the (decorated) Teichmuller space of 119878119892119899 that is a

collection of all Riemann surfaces of genus 119892 with 119899 cuspsendowed with the natural topology [7] In what follows wefocus on the algebras A(x 119876

119892119899) with quivers 119876

119892119899coming

from ideal triangulation of 119878119892119899 the corresponding cluster

algebra A(x 119876119892119899) of rank 119898 = 6119892 minus 6 + 3119899 is related to the

Penner coordinates in 119879119892119899

[2]

Example 1 Let 11987811

be a once-punctured torus The idealtriangulation of 119878

11defines theMarkov quiver (such a quiver

is related to solutions in the integer numbers of the equation1199092

1+ 1199092

2+ 1199092

3= 3119909111990921199093considered by A A Markov hence

the name) 11987611

shown in Figure 1 see [2 Example 46] Thecorresponding cluster 119862lowast-algebra A(x 119876

11) of rank 3 can be

written as

A (x 11987611) cong

M

1198680

(1)

where 1198680is a primitive ideal of an AF-algebra M The

unital AF-algebraM was originally defined by [8 Section 3]

Hindawi Publishing CorporationJournal of Function SpacesVolume 2016 Article ID 9639875 8 pageshttpdxdoiorg10115520169639875

2 Journal of Function Spaces

the genuine notation for such an algebra was M1 because

1198700(M1) = (119872

1 1) fl free one-generator unital ℓ-group

that is finitely piecewise affine linear continuous real-valuedfunctions on [0 1] with integer coefficients M

1was subse-

quently rediscovered after two decades by [9] and denotedbyA The remarkable properties ofM

1include the following

features Every primitive ideal ofM1is essential [10Theorem

42]M1is equipped with a faithful invariant tracial state [11

Theorem 31]The center ofM1coincides with the119862lowast-algebra

119862[0 1] of continuous complex-valued functions on [0 1] [9p 976] There is an affine weak lowast-homeomorphism of thestate space of 119862[0 1] onto the space of tracial states on M

1

[10Theorem45] Any state of119862[0 1]has precisely one tracialextension toM

1[12Theorem 25]The automorphism group

ofM1has precisely two connected components [10Theorem

43]TheGaussmap a Bernoulli shift for continued fractionsis generalized in [12] to the noncommutative framework ofM1 In the light of the original definition of M

1and the

fact that the 1198700-functor preserves exact sequences (see eg

[4 Theorem 31]) the primitive spectrum of M1and its

hull-kernel topology is widely known to the lattice-orderedgroup theorists and the MV-algebraists long ago before thelaborious analysis in [9] where M

1is defined in terms

of the Bratteli diagram We refer the reader to the finalpart of the paper [13] for a general result encompassingthe characterization of the prime spectrum of (119872

1 1) cong

PrimM1 Moreover the AF-algebras 119860

120579introduced by [14]

are precisely the infinite-dimensional simple quotients ofM1 this fact was first proved by [8 Theorem 31(i)] and

rediscovered independently by [9] Summing up the abovethe primitive ideals 119868

120579sub M are indexed by numbers 120579 isin R

if 120579 is irrational the quotient M119868120579cong 119860120579 where 119860

120579is

the Effros-Shen algebra In view of (1) the algebra M is anoncommutative coordinate ring of the Teichmuller space11987911 Moreover there exists an analog of the Tomita-Takesaki

theory of modular automorphisms 120590119905| 119905 isin R for algebra

M see Section 4 such automorphisms correspond to theTeichmuller geodesic flow on 119879

11[15] 120590

119905(119868120579) is an ideal of

M for all 119905 isin R where 1205900(119868120579) = 119868

120579 The quotient algebra

M120590119905(119868120579) can be viewed as a noncommutative coordinate

ring of the Riemann surface 11987811 in particular the pairs (120579 119905)

are coordinates in the space 11987911cong R2 We refer the reader to

[16] for a construction of the corresponding functor

Motivated by Example 1 denote by A(x 119876119892119899) the clus-

ter 119862lowast-algebra corresponding to a quiver 119876119892119899 let 120590

119905

A(x 119876119892119899) rarr A(x 119876

119892119899) be the Tomita-Takesaki flow

on A(x 119876119892119899) see Section 4 for the details Denote by

PrimA(x 119876119892119899) the set of all primitive ideals of A(x 119876

119892119899)

endowed with the Jacobson topology and let 119868120579

isin

PrimA(x 119876119892119899) for a generic value of index 120579 isin R6119892minus7+2119899 Our

main result can be stated as follows

Theorem 2 There exists a homeomorphism

ℎPrimA (x 119876119892119899) timesR

997888rarr 119880 sube 119879119892119899| 119880 is generic

(2)

1

23

Figure 1 The Markov quiver 11987611

given by the formula 120590t(119868120579) 997891rarr 119878119892119899 the set119880 = 119879

119892119899if and only

if 119892 = 119899 = 1 120590119905(119868120579) is an ideal of A(x 119876

119892119899) for all 119905 isin R and

the quotient algebra A(x 119876119892119899)120590119905(119868120579) is a noncommutative

coordinate ring of the Riemann surface 119878119892119899

Remark 3 Theorem 2 is valid for 119899 ge 1 that is the Riemannsurfaceswith at least one cuspThis cannot be improved sincethe cluster structure of algebra A(x 119876

119892119899) comes from the

Ptolemy relations satisfied by the Penner coordinates so farsuch coordinates are available only for the Riemann surfaceswith cusps [7] It is likely that the case 119899 = 0 has also a clusterstructure we refer the reader to [17] where a functor from theRiemann surfaces 119878

1198920to the AF-algebras A(x 119876

1198920)120590119905(119868120579)

was constructed

Remark 4 The braid group 1198612119892+119899

with 119899 isin 1 2 admitsa faithful representation by projections in the algebraA(x 119876

119892119899) such a construction is based on the Birman-

Hilden Theorem for the braid groups This observation andthe well-known Laurent phenomenon in the cluster algebra1198700(A(x 119876

119892119899)) allow generalizing the Jones and HOMFLY

invariants of knots and links to an arbitrary number ofvariables see [18] for the details

The paper is organized as follows We introduce prelimi-nary facts and notation in Section 2 Theorem 2 is proved inSection 3 An analog of the Tomita-Takesaki theory of mod-ular automorphisms and the Connes invariant 119879(A(x 119876

119892119899))

of the cluster 119862lowast-algebra A(x 119876119892119899) is constructed

2 NotationIn this section we introduce notation and briefly review somepreliminary factsThe reader is encouraged to consult [1ndash3 67] for the details

21 Cluster Algebras A cluster algebra A of rank 119898 is asubring of the field Q(119909

1 119909

119898) of rational functions in 119899

variables Such an algebra is defined by a pair (x 119861) wherex = (119909

1 119909

119898) is a cluster of variables and 119861 = (119887

119894119895) is a

skew-symmetric integermatrix the new cluster x1015840 is obtainedfrom x by an excision of the variable 119909

119896and replacing it by a

new variable 1199091015840119896subject to an exchange relation

1199091198961199091015840

119896=

119898

prod

119894=1

119909

max(119887119894119896 0)119894

+

119898

prod

119894=1

119909

max(minus119887119894119896 0)119894

(3)

Journal of Function Spaces 3

Since the entries of matrix 119861 are exponents of the monomialsin cluster variables one gets a new pair (x1015840 1198611015840) where 1198611015840 =(1198871015840

119894119895) is a skew-symmetric matrix with

1198871015840

119894119895=

minus119887119894119895

if 119894 = 119896 or 119895 = 119896

119887119894119895+

1003816100381610038161003816119887119894119896

1003816100381610038161003816119887119896119895+ 119887119894119896

10038161003816100381610038161003816119887119896119895

10038161003816100381610038161003816

2

otherwise(4)

For brevity the pair (x 119861) is called a seed and the seed(x1015840 1198611015840) fl (x1015840 120583

119896(119861)) is obtained from (x 119861) by a mutation

120583119896in the direction 119896 where 1 le 119896 le 119898 120583

119896is involution

that is 1205832119896= 119868119889 The matrix 119861 is called mutation finite if

only finitely many new matrices can be produced from 119861

by repeated matrix mutations The cluster algebra A(x 119861)can be defined as the subring of Q(119909

1 119909

119898) generated by

the union of all cluster variables obtained from the initialseed (x 119861) by mutations of (x 119861) (and its iterations) in allpossible directions We will write T

119898to denote an oriented

tree whose vertices are seeds (x1015840 1198611015840) and 119898 outgoing arrowsin each vertex correspond tomutations 120583

119896of the seed (x1015840 1198611015840)

The Laurent phenomenon proved by [1] says that A(x 119861) subZ[xplusmn1] where Z[xplusmn1] is the ring of the Laurent polynomialsin variables x = (119909

1 119909

119899) in other words each generator

119909119894of algebraA(x 119861) can be written as a Laurent polynomial

in 119899 variables with the integer coefficients

Remark 5 The Laurent phenomenon turns the additivestructure of cluster algebra A(x 119861) into a totally orderedabelian group satisfying the Riesz interpolation property thatis a dimension group [4 Theorem 31] the abelian groupwith order comes from the semigroup of the Laurent poly-nomials with positive coefficients see [19] for the details Abackgroundon the partially and totally ordered unperforatedabelian groups with the Riesz interpolation property can befound in [4]

To deal with mutation formulas (3) and (4) in geometricterms recall that a quiver 119876 is an oriented graph given bythe set of vertices 119876

0and the set of arrows 119876

1 an example of

quiver is given in Figure 1 Let 119896 be a vertex of119876 the mutatedat vertex 119896 quiver 120583

119896(119876) has the same set of vertices as 119876 but

the set of arrows is obtained by the following procedure (i)for each subquiver 119894 rarr 119896 rarr 119895 one adds a new arrow 119894 rarr 119895(ii) one reverses all arrows with source or target 119896 (iii) oneremoves the arrows in a maximal set of pairwise disjoint 2-cycles The reader can verify that if one encodes a quiver 119876with 119899 vertices by a skew-symmetric matrix 119861(119876) = (119887

119894119895)with

119887119894119895equal to the number of arrows from vertex 119894 to vertex 119895

then mutation 120583119896of seed (x 119861) coincides with such of the

corresponding quiver 119876 Thus the cluster algebra A(x 119861) isdefined by a quiver 119876 we will denote such an algebra byA(x 119876)

22 Cluster Algebras from Riemann Surfaces Let 119892 and 119899 beintegers such that 119892 ge 0 119899 ge 1 and 2119892 minus 2 + 119899 gt 0Denote by 119878

119892119899a Riemann surface of genus 119892 with the 119899 cusp

points It is known that the fundamental domain of 119878119892119899

canbe triangulated by 6119892 minus 6 + 3119899 geodesic arcs 120574 such that the

footpoints of each arc at the absolute of Lobachevsky planeH = 119909 + 119894119910 isin C | 119910 gt 0 coincide with a (preimage of) cuspof 119878119892119899 If 119897(120574) is the hyperbolic length of 120574 measured (with a

sign) between two horocycles around the footpoints of 120574 thenwe set 120582(120574) = 119890(12)119897(120574) 120582(120574) are known to satisfy the Ptolemyrelation

120582 (1205741) 120582 (1205742) + 120582 (120574

3) 120582 (1205744) = 120582 (120574

5) 120582 (1205746) (5)

where 1205741 120574

4are pairwise opposite sides and 120574

5 1205746are the

diagonals of a geodesic quadrilateral in HDenote by 119879

119892119899the decorated Teichmuller space of 119878

119892119899

that is the set of all complex surfaces of genus 119892 with 119899 cuspsendowed with the natural topology it is known that 119879

119892119899cong

R6119892minus6+2119899

Theorem 6 (see [7]) The map 120582 on the set of 6119892 minus 6 + 3119899geodesic arcs 120574

119894defining triangulation of 119878

119892119899is a homeomor-

phism with the image 119879119892119899

Remark 7 Notice that among 6119892 minus 6 + 3119899 real numbers 120582(120574119894)

there are only 6119892 minus 6 + 2119899 independent since such numbersmust satisfy 119899 Ptolemy relations (5)

Let119879be triangulation of surface 119878119892119899

by 6119892minus6+3119899 geodesicarcs 120574

119894 consider a skew-symmetric matrix 119861

119879= (119887119894119895) where

119887119894119895is equal to the number of triangles in 119879 with sides 120574

119894

and 120574119895in clockwise order minus the number of triangles in

119879 with sides 120574119894and 120574

119895in the counterclockwise order It is

known that matrix 119861119879is always mutation finite The cluster

algebraA(x 119861119879) of rank 6119892 minus 6 + 3119899 is called associated with

triangulation 119879


be a once-punctured torus of Example 1The triangulation119879 of the fundamental domain (R2minusZ2)Z2of 11987811

is sketched in Figure 2 in the charts R2 and Hrespectively It is easy to see that in this case x = (119909

1 1199092 1199093)

with 1199091= 12057423 1199092= 12057434 and 119909

3= 12057424 where 120574

119894119895denotes a

geodesic arc with the footpoints 119894 and 119895 The Ptolemy relation(5) reduces to 1205822(120574

23) + 1205822

(12057434) = 120582

2

(12057424) thus 119879

11cong R2

The reader is encouraged to verify that matrix 119861119879has the

following form

119861119879= (

0 2 minus2

minus2 0 2

2 minus2 0

) (6)

Theorem 9 (see [2]) The cluster algebra A(x 119861119879) does not

depend on triangulation119879 but only on the surface 119878119892119899 namely

replacement of the geodesic arc 120574119896by a new geodesic arc 1205741015840

119896(a

flip of 120574119896) corresponds to a mutation 120583

119896of the seed (x 119861

119879)

Remark 10 In view of Theorems 6 and 9 A(x 119861119879) corre-

sponds to an algebra of functions on the Teichmuller space119879119892119899 such an algebra is an analog of the coordinate ring of

119879119892119899

23 119862lowast-Algebras A 119862lowast-algebra is an algebra119860 overC with anorm 119886 997891rarr 119886 and involution 119886 997891rarr 119886

lowast such that it is complete


1 2

23 34 4

R2

1 = infin

H

Figure 2 Triangulation of the Riemann surface 11987811

with respect to the norm and 119886119887 le 119886119887 and 119886lowast119886 = 1198862for all 119886 119887 isin 119860 Any commutative 119862lowast-algebra is isomorphicto the algebra119862

0(119883) of continuous complex-valued functions

on some locally compact Hausdorff space 119883 otherwise 119860represents a noncommutative topological space

An AF-algebra (Approximately Finite 119862lowast-algebra) is

defined to be the norm closure of an ascending sequenceof finite-dimensional 119862lowast-algebras 119872

119899 where 119872

119899is the 119862lowast-

algebra of the 119899times119899matrices with entries inC Here the index119899 = (119899

1 119899

119896) represents the semisimple matrix algebra

119872119899= 1198721198991oplus sdot sdot sdot oplus 119872

119899119896 The ascending sequence mentioned

above can be written as

1198721

1205931

997888rarr 1198722

1205932

997888rarr sdot sdot sdot (7)

where 119872119894are the finite-dimensional 119862lowast-algebras and 120593

119894are

the homomorphisms between such algebras The homomor-phisms 120593

119894can be arranged into a graph as follows Let119872

119894=

1198721198941oplus sdot sdot sdot oplus 119872

119894119896and119872

1198941015840 = 119872

1198941015840

1

oplus sdot sdot sdot oplus 1198721198941015840

119896

be the semisimple119862lowast-algebras and 120593

119894 119872119894rarr 119872

1198941015840 the homomorphism One

has two sets of vertices 1198811198941 119881

119894119896and 119881

1198941015840

1

1198811198941015840

119896

joined by119887119903119904

edges whenever the summand 119872119894119903contains 119887

119903119904copies

of the summand 1198721198941015840

119904

under the embedding 120593119894 As 119894 varies

one obtains an infinite graph called the Bratteli diagram ofthe AF-algebra The matrix 119861 = (119887

119903119904) is known as a partial

multiplicitymatrix an infinite sequence of119861119894defines a unique

AF-algebraLet 120579 isin R119899minus1 recall that by the Jacobi-Perron continued

fraction of vector (1 120579) one understands the limit

(

1

1205791

120579119899minus1

)= lim119896rarrinfin

(

(

0 0 sdot sdot sdot 0 1

1 0 sdot sdot sdot 0 119887(1)

1


119899minus1

)

)

sdot sdot sdot(

(


1 0 sdot sdot sdot 0 119887(119896)

1

0 0 sdot sdot sdot 1 119887(119896)

119899minus1

)

)

(

0

0

1

)

(8)

where 119887(119895)119894isin Ncup0 see for example [20] the limit converges

for a generic subset of vectors 120579 isin R119899minus1 Notice that 119899 =

2 corresponds to (a matrix form of) the regular continuedfraction of 120579 such a fraction is always convergent Moreover

Figure 3 The Bratteli diagram of Markovrsquos cluster 119862lowast-algebra

the Jacobi-Perron fraction is finite if and only if vector 120579 =

(120579119894) where 120579

119894are rational The AF-algebra119860

120579associated with

the vector (1 120579) is defined by the Bratteli diagram with thepartial multiplicity matrices equal to 119861

119896in the Jacobi-Perron

fraction of (1 120579) in particular if 119899 = 2119860120579coincides with the

Effros-Shen algebra [14]

24 Cluster 119862lowast-Algebras Notice that the mutation tree T119898

of a cluster algebra A(x 119861) has grading by levels that is adistance from the root of T

119898Wewill say that a pair of clusters

x and x1015840 are ℓ-equivalent if

(i) x and x1015840 lie at the same level(ii) x and x1015840 coincide modulo a cyclic permutation of

variables 119909119894

(iii) 119861 = 1198611015840

It is not hard to see that ℓ is an equivalence relation on the setof vertices of graph T

119898

Definition 11 By a cluster 119862lowast-algebra A(x 119861) one under-stands an AF-algebra given by the Bratteli diagram B(x 119861)of the form

B (x 119861) fl T119898mod ℓ (9)

The rank ofA(x 119861) is equal to such of cluster algebraA(x 119861)

Example 12 If 119861119879is matrix (6) of Example 8 then B(x 119861

119879)

is shown in Figure 3 (We refer the reader to Section 4 for aproof) Notice that the graphB(x 119861

119879) is a part of the Bratteli

diagram of the Mundici algebraM compare [10 Figure 1]

Remark 13 It is not hard to see thatB(x 119861) is no longer a treeand B(x 119861) is a finite graph if and only if A(x 119861) is a finitecluster algebra

3 Proof

Let 119898 = 3(2119892 minus 2 + 119899) be the rank of cluster 119862lowast-algebraA(x 119876

119892119899) For the sake of clarity we will consider the case

119898 = 3 and the general case119898 isin 3 6 9 separately(i) LetA(x 119861

119879) be the cluster119862lowast-algebra of rank 3 In this

case 2119892minus2+119899 = 1 and either119892 = 0 and 119899 = 3 or else119892 = 119899 = 1Since 119879

03cong 119901119905 is trivial we are left with 119892 = 119899 = 1 that is

the once-punctured torus 11987811

Repeating the argument of Example 8 we get the seed(x 119861119879) where x = (119909

1 1199092 1199093) and the skew-symmetric

matrix 119861119879is given by formula (6)


(x1 x21 x23+

x2 x3) (x1 x2 x21 x

22+

x3)( x22 x

23+

x1 x2 x3)

(x1 x2 x3)

(x1 x2 x3)(x1 x2 x3)

(x1 x2 x3)

(x1 x21 + ((x21 + x22) x3)2

x2x21 + x

22

x3)

( x22 + ((x21 + x22) x3)2

x1 x2

x21 + x

22

x3)( x

22 + x2

3

x1 x2((x22 + x

23) x1)2 + x

22

x3)

( x22 + x23

x1((x22 + x

23) x1)2 + x

23

x2 x3)

(((x21 + x23) x2)2 + x

23

x1x21 + x

23

x2 x3) (x1 x

21 + x

23

x2x21 + ((x21 + x

23) x2)2

x3)

Figure 4 The mutation tree

Let us verify that matrix 119861119879is mutation finite indeed for

each 119896 isin 1 2 3 the matrix mutation formula (4) gives us120583119896(119861119879) = minus119861

119879

Therefore the exchange relations (3) do not vary it isverified directly that such relations have the following form

11990911199091015840

1= 1199092

2+ 1199092

3

11990921199091015840

2= 1199092

1+ 1199092

3

11990931199091015840

3= 1199092

1+ 1199092

2

(10)

Consider a mutation tree T3shown in Figure 4 the

vertices of T3correspond to the mutations of cluster x =

(1199091 1199092 1199093) following the exchange rules (10)

The reader is encouraged to verify that modulo a cyclicpermutation of variables 1199091015840

1= 1199092 11990910158402= 1199093 11990910158403= 1199091and

1199091015840

1= 1199093 11990910158402= 1199091 11990910158403= 1199092one obtains (resp) the following

equivalences of clusters

12058313(x) = 120583

21(x)

12058323(x) = 120583

31(x)

(11)

where 120583119894119895(x) fl 120583

119895(120583119894(x)) there are no other cluster equiva-

lences for the vertices of the same level of graph T3

To determine the graph B(x 119861119879) one needs to take the

quotient of T3by the ℓ-equivalence relations (11) since the

pattern repeats for each level of T3 one gets the B(x 119861

119879)

shown in Figure 3 The cluster 119862lowast-algebra A(x 119861119879) is an AF-

algebra with the Bratteli diagramB(x 119861119879)

Notice that the Bratteli diagram B(x 119861119879) of our AF-

algebra A(x 119861119879) and such of the Mundici algebra M are

distinct compare [10 Figure 1] yet there is an obviousinclusion of one diagram into another Namely if one erasesa ldquocamelrsquos backrdquo (ie the two extreme sides of the diagram)in the Bratteli diagram of M then one gets exactly thediagram in Figure 3 Formally if G is the Bratteli diagramof the Mundici algebra M the complement G minus B(x 119861

119879)

is a hereditary Bratteli diagram which gives rise to an ideal1198680subM such that

A (x 119861119879) cong

M

1198680

(12)

see [6 Lemma 32] 1198680is a primitive ideal ibid Theorem 38

(It is interesting to calculate the group 1198700(1198680) in the context

of the work of [13])On the other hand the space PrimM (and hence

PrimA(x 119861119879)) is well understood see for example [13] or

[9 Proposition 7] Namely

Prim(

M

1198680

) = 119868120579| 120579 isin R (13)

where 119868120579sub M is such that M119868

120579cong 119860120579is the Effros-Shen

algebra [14] if 120579 is an irrational number or M119868120579cong 119872119902is

finite-dimensional matrix 119862lowast-algebra (and an extension ofsuch by the 119862lowast-algebra of compact operators) if 120579 = 119901119902 isa rational number (Note that the third series of primitiveideals of [9 Proposition 7] correspond to the ideal 119868

0)


Moreover given the Jacobson topology on PrimM thereexists a homeomorphism

ℎ Prim(

M

1198680

) 997888rarr R (14)

defined by the formula 119868120579997891rarr 120579 see [9 Corollary 12]

Let 120590119905 M119868

0rarr M119868

0be the Tomita-Takesaki flow

that is a one-parameter automorphism group of M1198680 see

Section 4 Because 119868120579sub M119868

0 the image 120590119905(119868

120579) of 119868

120579is

correctly defined for all 119905 isin R 120590119905(119868120579) is an ideal of M119868

0

but not necessarily primitive Since 120590119905is nothing but (an

algebraic form of) the Teichmuller geodesic flow on 11987911

[15]one concludes that that the family of ideals

120590119905(119868120579) sub

M

1198680

| 119905 isin R 120579 isin R (15)

can be taken for a coordinate system in the space 11987911cong R2

In view of (14) andM1198680cong A(x 119876

11) one gets the required

homeomorphism

ℎ PrimA (x 11987611) timesR 997888rarr 119879

11 (16)

such that the quotient algebra A(x 11987611)120590119905(119868120579) is a noncom-

mutative coordinate ring of the Riemann surface 11987811

Remark 14 The family of algebras A(x 11987611)120590119905(119868120579) | 120579 =

Const 119905 isin R are in general pairwise nonisomorphic (Forotherwise all ideals 120590

119905(119868120579) | 119905 isin R were primitive) However

theGrothendieck semigroups119870+0are isomorphic see [14] the

action of 120590119905is given by the following formula (see Section 4)

119870+

0(

A (x 11987611)

120590119905(119868120579)

) cong 119890119905

(Z + Z120579) (17)

(ii) The general case 119898 = 3119896 = 3(2119892 minus 2 + 119899) is treatedlikewise Notice that if 119889 = 6119892 minus 6 + 2119899 is dimension of thespace119879

119892119899 thenwe have119898minus119889 = 119899 in particular rank119898 of the

cluster 119862lowast-algebra A(x 119876119892119899) determines completely the pair

(119892 119899) provided 119889 is a fixed constant (If 119889 is not fixed thereis only a finite number of different pairs (119892 119899) for given rank119898)

Let (x 119861119879) be the seed given by the cluster x =

(1199091 119909

3119896) and the skew-symmetric matrix 119861

119879 Since

matrix 119861119879comes from triangulation of the Riemann surface

119878119892119899 119861119879ismutation finite see [3 p 18] the exchange relations

(3) take the following form

11990911199091015840

1= 1199092

2+ 1199092

3+ sdot sdot sdot + 119909

2

3119896

11990921199091015840

2= 1199092

1+ 1199092


2

3119896

11990931198961199091015840

3119896= 1199092

1+ 1199092


2

3119896minus1

(18)

One can construct the mutation tree T3119896

using relations(18) the reader is encouraged to verify that T

3119896is similar

middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot

Figure 5 The Bratteli diagram of a cluster 119862lowast-algebra of rank 6

to the one shown in Figure 4 except for the number of theoutgoing edges at each vertex that is equal to 3119896

A tedious but straightforward calculation shows that theonly equivalent clusters at the same level of T

3119896are the ones

at the extremities of tuples (11990910158401 119909

1015840

3119896) in other words one

gets the following system of equivalences of clusters

12058313119896

(x) = 12058321(x)

12058323119896

(x) = 12058331(x)

1205833119896minus13119896

(x) = 12058331198961

(x)

(19)

where 120583119894119895(x) fl 120583

119895(120583119894(x))

The graph B(x 119861119879) is the quotient of T

3119896by the ℓ-

equivalence relations (19) for 119896 = 2 such a graph is sketchedin Figure 5 A(x 119876

119892119899) is an AF-algebra given by the Bratteli

diagramB(x 119861119879)

Lemma 15 The set

PrimA (x 119876119892119899) = 119868

120579| 120579 isin R

6119892minus7+2119899

is generic (20)

where A(x 119876119892119899)119868120579is an AF-algebra 119860

120579associated with the

convergent Jacobi-Perron continued fraction of vector (1 120579) seeSection 23

Proof We adapt the argument of [9 case 119896 = 1] to thecase 119896 ge 1 Let 119889 = 6119892 minus 6 + 2119899 be dimension of thespace 119879

119892119899 Roughly speaking the Bratteli diagram B(x 119861

119879)

of algebraA(x 119876119892119899) can be cut in two disjoint piecesG

120579and

B(x 119861119879)minusG120579 as it is shown by [9 Figure 7]G

120579is a (finite or

infinite) vertical strip of constant ldquowidthrdquo 119889 where 119889 is equalto the number of vertices cut from each level ofB(x 119861

119879)The

reader is encouraged to verify that G120579is exactly the Bratteli

diagram of the AF-algebra119860120579associated with the convergent

Jacobi-Perron continued fraction of a generic vector (1 120579) seeSection 23

On the other hand the complement B(x 119861119879) minus G

120579is

a hereditary Bratteli diagram which defines an ideal 119868120579of

algebra A(x 119876119892119899) such that

A (x 119876119892119899)

119868120579

= 119860120579 (21)


see [6 Lemma 32] Moreover 119868120579is a primitive ideal [6

Theorem38] (An extra care is required if 120579 = (120579119894) is a rational

vector the complete argument can be found in [9 pp 980ndash985]) Lemma 15 follows

Lemma 16 The sequence of primitive ideals 119868120579119899

converges to119868120579in the Jacobson topology in PrimA(x 119876

119892119899) if and only if the

sequence 120579119899converges to 120579 in the Euclidean space R6119892minus7+2119899

Proof The proof is a straightforward adaption of the argu-ment in [9 pp 986ndash988] we leave it as an exercise to thereader

Let 120590119905 A(x 119876

119892119899) rarr A(x 119876

119892119899) be the Tomita-

Takesaki flow that is the group 120590119905| 119905 isin R of modular

automorphisms of algebra A(x 119876119892119899) see Section 4 Because

119868120579sub A(x 119876

119892119899) the image 120590119905(119868

120579) of 119868120579is correctly defined

for all 119905 isin R 120590119905(119868120579) is an ideal of A(x 119876

119892119899) but not

necessarily a primitive ideal Since 120590119905is an algebraic form

of the Teichmuller geodesic flow on the space 119879119892119899

[15] oneconcludes that the family of ideals

120590119905(119868120579) sub A (x 119876

119892119899) | 119905 isin R 120579 isin R

6119892minus7+2119899

(22)

can be taken for a coordinate system in the space 119879119892119899

cong

R6119892minus6+2119899 In view of Lemmas 15 and 16 one gets the requiredhomeomorphism

ℎ PrimA (x 119876119892119899) timesR


(23)

such that the quotient algebra 119860120579= A(x 119876

119892119899)120590119905(119868120579) is a

noncommutative coordinate ring of theRiemann surface 119878119892119899

Theorem 2 is proved

4 An Analog of Modular Flow on A(x 119876119892119899)

41 Modular Automorphisms 120590119905| 119905 isin R Recall that the

Ptolemy relations (5) for the Penner coordinates 120582(120574119894) in

the space 119879119892119899

are homogeneous in particular the system119905120582(120574119894) | 119905 isin R of such coordinates will also satisfy the

Ptolemy relations On the other hand for the cluster 119862lowast-algebra A(x 119876

119892119899) the variables 119909

119894= 120582(120574

119894) and one gets an

obvious isomorphism A(x 119876119892119899) cong A(119905x 119876

119892119899) for all 119905 isin R

Since A(119905x 119876119892119899) sube A(x 119876

119892119899) one obtains a one-parameter

group of automorphisms

120590119905 A (x 119876

119892119899) 997888rarr A (x 119876

119892119899) (24)

By analogy with [21] we will call 120590119905a Tomita-Takesaki flow on

the cluster119862lowast-algebraA(x 119876119892119899)The reader is encouraged to

verify that120590119905is an algebraic formof the geodesic flow119879119905 on the

Teichmuller space 119879119892119899 see [15] for an introduction Roughly

speaking such a flow comes from the one-parameter groupof matrices

(

119890119905

0

0 119890minus119905

) (25)

acting on the space of holomorphic quadratic differentials ontheRiemann surface 119878

119892119899 the latter is known to be isomorphic

to the Teichmuller space 119879119892119899

42 Connes Invariant119879(A(x 119876119892119899)) Recall that an analogy of

theConnes invariant 119879(M) for a119862lowast-algebraM endowedwithamodular automorphism group120590

119905is the set119879(M) fl 119905 isin R |

120590119905is inner [21] The group of inner automorphisms of the

space 119879119892119899

and algebra A(x 119876119892119899) is isomorphic to the map-

ping class group mod 119878119892119899

of surface 119878119892119899 The automorphism

120601 isin mod119878119892119899

is called pseudo-Anosov if 120601(F120583) = 120582

120601F120583

where F120583is invariant measured foliation and 120582

120601gt 1 is

a constant called dilatation of 120601 120582120601is always an algebraic

number of the maximal degree 6119892 minus 6 + 2119899 [22] It is knownthat if 120601 isin mod119878

119892119899is pseudo-Anosov then there exists

a trajectory O of the geodesic flow 119879119905 and a point 119878

119892119899isin

119879119892119899 such that the points 119878

119892119899and 120601(119878

119892119899) belong to O [15]

O is called an axis of the pseudo-Anosov automorphism120601 The axis can be used to calculate the Connes invariant119879(A(x 119876

119892119899)) of the cluster 119862lowast-algebra A(x 119876

119892119899) indeed in

view of formula (25) one must solve the following system ofequations

120590119905(119909) = 119890

119905

119909

120601 (119909) = 120582120601119909

(26)

for a point 119909 isin O Thus 120590119905(119909) coincides with the inner

automorphism 120601(119909) if and only if 119905 = log 120582120601 Taking all

pseudo-Anosov automorphisms 120601 isin mod119878119892119899 one gets a

formula for the Connes invariant

119879 (A (x 119876119892119899))

= log 120582120601| 120601 isin mod 119878

119892119899is pseudo-Anosov

(27)

Remark 17 The Connes invariant (27) says that the family ofcluster 119862lowast-algebras A(x 119876

119892119899) is an analog of the type III

120582

factors of von Neumann algebras see [21]

Disclosure

All errors and misconceptions in this paper are solely theauthorrsquos

Competing Interests

The author declares that there are no competing interests inthe paper

Acknowledgments

It is the authorrsquos pleasure to thank Ibrahim Assem andthe SAG group of the Department of Mathematics of theUniversity of Sherbrooke for hospitality and excellent work-ing conditions The author is grateful to Ibrahim AssemThomas Brustle Daniele Mundici Ralf Schiffler and VasilisaShramchenko for an introduction to the wonderland ofcluster algebras and helpful correspondence


References

[1] S Fomin and A Zelevinsky ldquoCluster algebras I foundationsrdquoJournal of the American Mathematical Society vol 15 no 2 pp497ndash529 2002

[2] S Fomin M Shapiro and D Thurston ldquoCluster algebras andtriangulated surfaces I Cluster complexesrdquo Acta Mathematicavol 201 no 1 pp 83ndash146 2008

[3] L K Williams ldquoCluster algebras an introductionrdquo Bulletin ofthe American Mathematical Society vol 51 no 1 pp 1ndash26 2014

[4] E G Effros Dimensions and Clowast-Algebras vol 46 of CBMSRegional Conference Series in Mathematics Conference Boardof the Mathematical Sciences 1981

[5] B Blackadar K-Theory for Operator Algebras vol 5 MSRIPublications Springer 1986

[6] O Bratteli ldquoInductive limits of finite dimensional Clowast-algebrasrdquoTransactions of the American Mathematical Society vol 171 pp195ndash234 1972

[7] R C Penner ldquoThe decorated Teichmuller space of puncturedsurfacesrdquoCommunications inMathematical Physics vol 113 no2 pp 299ndash339 1987

[8] D Mundici ldquoFarey stellar subdivisions ultrasimplicial groupsand119870

0of AF119862-algebrasrdquoAdvances inMathematics vol 68 no

1 pp 23ndash39 1988[9] F P Boca ldquoAnAF algebra associatedwith the Farey tessellationrdquo

Canadian Journal of Mathematics vol 60 no 5 pp 975ndash10002008

[10] D Mundici ldquoRevisiting the Farey AF-algebrardquoMilan Journal ofMathematics vol 79 no 2 pp 643ndash656 2011

[11] D Mundici ldquoRecognizing the Farey-Stern-Brocot AF algebraDedicated to the memory of Renato Caccioppolirdquo RendicontiLincei-Matematica e Applicazioni vol 20 pp 327ndash338 2009

[12] C Eckhardt ldquoA noncommutative Gauss maprdquo MathematicaScandinavica vol 108 no 2 pp 233ndash250 2011

[13] G Panti ldquoPrime ideals in free 119897 -groups and free vector latticesrdquoJournal of Algebra vol 219 no 1 pp 173ndash200 1999

[14] E G Effros and C L Shen ldquoApproximately finite 119862-algebrasand continued fractionsrdquo Indiana University Mathematics Jour-nal vol 29 no 2 pp 191ndash204 1980

[15] W A Veech ldquoThe Teichmuller geodesic flowrdquo Annals ofMathematics Second Series vol 124 no 3 pp 441ndash530 1986

[16] I VNikolaev ldquoOn aTeichmuller functor between the categoriesof complex tori and the Effros-Shen algebrasrdquoNew York Journalof Mathematics vol 15 pp 125ndash132 2009

[17] I Nikolaev ldquoRiemann surfaces and AF-algebrasrdquo Annals ofFunctional Analysis vol 7 no 2 pp 371ndash380 2016

[18] I Nikolaev ldquoCluster Clowast-algebras and knot polynomialsrdquo httparxivorgabs160301180

[19] I Nikolaev ldquoK-theory of cluster Clowast-algebrasrdquo httparxivorgabs151200276

[20] L Bernstein The Jacobi-Perron AlgorithmmdashIts Theory andApplication vol 207 of Lecture Notes in Mathematics SpringerBerlin Germany 1971

[21] A Connes ldquoVon Neumann algebrasrdquo in Proceedings of theInternational Congress of Mathematicians (ICM rsquo78) pp 97ndash109Helsinki Finland 1978

[22] W P Thurston ldquoOn the geometry and dynamics of diffeo-morphisms of surfacesrdquo Bulletin of the American MathematicalSociety vol 19 no 2 pp 417ndash431 1988

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the genuine notation for such an algebra was M1 because

1198700(M1) = (119872

1 1) fl free one-generator unital ℓ-group

that is finitely piecewise affine linear continuous real-valuedfunctions on [0 1] with integer coefficients M

1was subse-

quently rediscovered after two decades by [9] and denotedbyA The remarkable properties ofM

1include the following

features Every primitive ideal ofM1is essential [10Theorem

42]M1is equipped with a faithful invariant tracial state [11

Theorem 31]The center ofM1coincides with the119862lowast-algebra

119862[0 1] of continuous complex-valued functions on [0 1] [9p 976] There is an affine weak lowast-homeomorphism of thestate space of 119862[0 1] onto the space of tracial states on M

1

[10Theorem45] Any state of119862[0 1]has precisely one tracialextension toM

1[12Theorem 25]The automorphism group

ofM1has precisely two connected components [10Theorem

43]TheGaussmap a Bernoulli shift for continued fractionsis generalized in [12] to the noncommutative framework ofM1 In the light of the original definition of M

1and the

fact that the 1198700-functor preserves exact sequences (see eg

[4 Theorem 31]) the primitive spectrum of M1and its

hull-kernel topology is widely known to the lattice-orderedgroup theorists and the MV-algebraists long ago before thelaborious analysis in [9] where M

1is defined in terms

of the Bratteli diagram We refer the reader to the finalpart of the paper [13] for a general result encompassingthe characterization of the prime spectrum of (119872

1 1) cong

PrimM1 Moreover the AF-algebras 119860

120579introduced by [14]

are precisely the infinite-dimensional simple quotients ofM1 this fact was first proved by [8 Theorem 31(i)] and

rediscovered independently by [9] Summing up the abovethe primitive ideals 119868

120579sub M are indexed by numbers 120579 isin R

if 120579 is irrational the quotient M119868120579cong 119860120579 where 119860

120579is

the Effros-Shen algebra In view of (1) the algebra M is anoncommutative coordinate ring of the Teichmuller space11987911 Moreover there exists an analog of the Tomita-Takesaki

theory of modular automorphisms 120590119905| 119905 isin R for algebra

M see Section 4 such automorphisms correspond to theTeichmuller geodesic flow on 119879

11[15] 120590

119905(119868120579) is an ideal of

M for all 119905 isin R where 1205900(119868120579) = 119868

120579 The quotient algebra

M120590119905(119868120579) can be viewed as a noncommutative coordinate

ring of the Riemann surface 11987811 in particular the pairs (120579 119905)

are coordinates in the space 11987911cong R2 We refer the reader to

[16] for a construction of the corresponding functor

Motivated by Example 1 denote by A(x 119876119892119899) the clus-

ter 119862lowast-algebra corresponding to a quiver 119876119892119899 let 120590

119905

A(x 119876119892119899) rarr A(x 119876

119892119899) be the Tomita-Takesaki flow

on A(x 119876119892119899) see Section 4 for the details Denote by

PrimA(x 119876119892119899) the set of all primitive ideals of A(x 119876

119892119899)

endowed with the Jacobson topology and let 119868120579

isin

PrimA(x 119876119892119899) for a generic value of index 120579 isin R6119892minus7+2119899 Our

main result can be stated as follows

Theorem 2 There exists a homeomorphism

ℎPrimA (x 119876119892119899) timesR


(2)

1

23

Figure 1 The Markov quiver 11987611

given by the formula 120590t(119868120579) 997891rarr 119878119892119899 the set119880 = 119879

119892119899if and only

if 119892 = 119899 = 1 120590119905(119868120579) is an ideal of A(x 119876

119892119899) for all 119905 isin R and

the quotient algebra A(x 119876119892119899)120590119905(119868120579) is a noncommutative

coordinate ring of the Riemann surface 119878119892119899

Remark 3 Theorem 2 is valid for 119899 ge 1 that is the Riemannsurfaceswith at least one cuspThis cannot be improved sincethe cluster structure of algebra A(x 119876

119892119899) comes from the

Ptolemy relations satisfied by the Penner coordinates so farsuch coordinates are available only for the Riemann surfaceswith cusps [7] It is likely that the case 119899 = 0 has also a clusterstructure we refer the reader to [17] where a functor from theRiemann surfaces 119878

1198920to the AF-algebras A(x 119876

1198920)120590119905(119868120579)

was constructed

Remark 4 The braid group 1198612119892+119899

with 119899 isin 1 2 admitsa faithful representation by projections in the algebraA(x 119876

119892119899) such a construction is based on the Birman-

Hilden Theorem for the braid groups This observation andthe well-known Laurent phenomenon in the cluster algebra1198700(A(x 119876

119892119899)) allow generalizing the Jones and HOMFLY

invariants of knots and links to an arbitrary number ofvariables see [18] for the details

The paper is organized as follows We introduce prelimi-nary facts and notation in Section 2 Theorem 2 is proved inSection 3 An analog of the Tomita-Takesaki theory of mod-ular automorphisms and the Connes invariant 119879(A(x 119876

119892119899))

of the cluster 119862lowast-algebra A(x 119876119892119899) is constructed

2 NotationIn this section we introduce notation and briefly review somepreliminary factsThe reader is encouraged to consult [1ndash3 67] for the details

21 Cluster Algebras A cluster algebra A of rank 119898 is asubring of the field Q(119909

1 119909

119898) of rational functions in 119899

variables Such an algebra is defined by a pair (x 119861) wherex = (119909

1 119909

119898) is a cluster of variables and 119861 = (119887

119894119895) is a

skew-symmetric integermatrix the new cluster x1015840 is obtainedfrom x by an excision of the variable 119909

119896and replacing it by a

new variable 1199091015840119896subject to an exchange relation

1199091198961199091015840

119896=

119898

prod

119894=1

119909

max(119887119894119896 0)119894

+

119898

prod

119894=1

119909

max(minus119887119894119896 0)119894

(3)




1198871015840

119894119895=

minus119887119894119895

if 119894 = 119896 or 119895 = 119896

119887119894119895+

1003816100381610038161003816119887119894119896

1003816100381610038161003816119887119896119895+ 119887119894119896

10038161003816100381610038161003816119887119896119895

10038161003816100381610038161003816

2

otherwise(4)




119896is involution




1 119909





119896of the seed (x1015840 1198611015840)


1 119909







1 an example of




119894119895)with










120582 (1205741) 120582 (1205742) + 120582 (120574

3) 120582 (1205744) = 120582 (120574

5) 120582 (1205746) (5)

where 1205741 120574


5 1205746are the



119892119899


119892119899cong

R6119892minus6+2119899










119879= (119887119894119895) where


119894


119879 with sides 120574119894and 120574








1 1199092 1199093)

with 1199091= 12057423 1199092= 12057434 and 119909

3= 12057424 where 120574

119894119895denotes a


23) + 1205822

(12057434) = 120582

2

(12057424) thus 119879

11cong R2


following form

119861119879= (

0 2 minus2

minus2 0 2

2 minus2 0

) (6)




119896(a


119896of the seed (x 119861

119879)



119879119892119899




1 2

23 34 4

R2

1 = infin

H







119899 where 119872



1 119899





1198721

1205931

997888rarr 1198722

1205932



119894are



119894=


119894119896and119872

1198941015840 = 119872

1198941015840

1


119896


119894 119872119894rarr 119872



119894119896and 119881

1198941015840

1

1198811198941015840

119896

joined by119887119903119904


119903119904copies

of the summand 1198721198941015840

119904







(

1

1205791

120579119899minus1


(

(



1


119899minus1

)

)

sdot sdot sdot(

(


1 0 sdot sdot sdot 0 119887(119896)

1

0 0 sdot sdot sdot 1 119887(119896)

119899minus1

)

)

(

0

0

1

)

(8)






(120579119894) where 120579













(iii) 119861 = 1198611015840


119898


B (x 119861) fl T119898mod ℓ (9)



119879)





3 Proof












(x1 x21 x23+

x2 x3) (x1 x2 x21 x

22+

x3)( x22 x

23+

x1 x2 x3)

(x1 x2 x3)

(x1 x2 x3)(x1 x2 x3)

(x1 x2 x3)

(x1 x21 + ((x21 + x22) x3)2

x2x21 + x

22

x3)

( x22 + ((x21 + x22) x3)2

x1 x2

x21 + x

22

x3)( x

22 + x2

3

x1 x2((x22 + x

23) x1)2 + x

22

x3)

( x22 + x23

x1((x22 + x

23) x1)2 + x

23

x2 x3)

(((x21 + x23) x2)2 + x

23

x1x21 + x

23

x2 x3) (x1 x

21 + x

23

x2x21 + ((x21 + x

23) x2)2

x3)




119879


11990911199091015840

1= 1199092

2+ 1199092

3

11990921199091015840

2= 1199092

1+ 1199092

3

11990931199091015840

3= 1199092

1+ 1199092

2

(10)





1= 1199092 11990910158402= 1199093 11990910158403= 1199091and

1199091015840



12058313(x) = 120583

21(x)

12058323(x) = 120583

31(x)

(11)

where 120583119894119895(x) fl 120583






119879)






119879)


A (x 119861119879) cong

M

1198680

(12)






Prim(

M

1198680

) = 119868120579| 120579 isin R (13)





0)



ℎ Prim(

M

1198680

) 997888rarr R (14)


Let 120590119905 M119868

0rarr M119868




0 the image 120590119905(119868

120579) of 119868

120579is


0




120590119905(119868120579) sub

M

1198680

| 119905 isin R 120579 isin R (15)




homeomorphism


11 (16)








119870+

0(

A (x 11987611)

120590119905(119868120579)

) cong 119890119905

(Z + Z120579) (17)






(1199091 119909


119879 Since




11990911199091015840

1= 1199092

2+ 1199092


2

3119896

11990921199091015840

2= 1199092

1+ 1199092


2

3119896

11990931198961199091015840

3119896= 1199092

1+ 1199092


2

3119896minus1

(18)



3119896is similar





3119896are the ones


1015840



12058313119896

(x) = 12058321(x)

12058323119896

(x) = 12058331(x)

1205833119896minus13119896

(x) = 12058331198961

(x)

(19)

where 120583119894119895(x) fl 120583

119895(120583119894(x))


3119896by the ℓ-



diagramB(x 119861119879)

Lemma 15 The set

PrimA (x 119876119892119899) = 119868

120579| 120579 isin R

6119892minus7+2119899

is generic (20)






119879)


120579and




119879)The





120579is



A (x 119876119892119899)

119868120579

= 119860120579 (21)










Let 120590119905 A(x 119876

119892119899) rarr A(x 119876

119892119899) be the Tomita-



119868120579sub A(x 119876

119892119899) the image 120590119905(119868



119892119899) but not




120590119905(119868120579) sub A (x 119876

119892119899) | 119905 isin R 120579 isin R

6119892minus7+2119899

(22)


cong


ℎ PrimA (x 119876119892119899) timesR


(23)


119892119899)120590119905(119868120579) is a


Theorem 2 is proved




the space 119879119892119899



119892119899) the variables 119909

119894= 120582(120574



119892119899) for all 119905 isin R

Since A(119905x 119876119892119899) sube A(x 119876



120590119905 A (x 119876

119892119899) 997888rarr A (x 119876

119892119899) (24)






(

119890119905

0

0 119890minus119905

) (25)








space 119879119892119899




120601 isin mod119878119892119899


120601F120583


120601gt 1 is





119892119899isin


119892119899and 120601(119878

119892119899) belong to O [15]



119892119899) indeed in


120590119905(119909) = 119890

119905

119909

120601 (119909) = 120582120601119909

(26)





119879 (A (x 119876119892119899))

= log 120582120601| 120601 isin mod 119878


(27)



120582


Disclosure


Competing Interests


Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198871015840

119894119895=

minus119887119894119895

if 119894 = 119896 or 119895 = 119896

119887119894119895+

1003816100381610038161003816119887119894119896

1003816100381610038161003816119887119896119895+ 119887119894119896

10038161003816100381610038161003816119887119896119895

10038161003816100381610038161003816

2

otherwise(4)




119896is involution




1 119909





119896of the seed (x1015840 1198611015840)


1 119909







1 an example of




119894119895)with










120582 (1205741) 120582 (1205742) + 120582 (120574

3) 120582 (1205744) = 120582 (120574

5) 120582 (1205746) (5)

where 1205741 120574


5 1205746are the



119892119899


119892119899cong

R6119892minus6+2119899










119879= (119887119894119895) where


119894


119879 with sides 120574119894and 120574








1 1199092 1199093)

with 1199091= 12057423 1199092= 12057434 and 119909

3= 12057424 where 120574

119894119895denotes a


23) + 1205822

(12057434) = 120582

2

(12057424) thus 119879

11cong R2


following form

119861119879= (

0 2 minus2

minus2 0 2

2 minus2 0

) (6)




119896(a


119896of the seed (x 119861

119879)



119879119892119899




1 2

23 34 4

R2

1 = infin

H







119899 where 119872



1 119899





1198721

1205931

997888rarr 1198722

1205932



119894are



119894=


119894119896and119872

1198941015840 = 119872

1198941015840

1


119896


119894 119872119894rarr 119872



119894119896and 119881

1198941015840

1

1198811198941015840

119896

joined by119887119903119904


119903119904copies

of the summand 1198721198941015840

119904







(

1

1205791

120579119899minus1


(

(



1


119899minus1

)

)

sdot sdot sdot(

(


1 0 sdot sdot sdot 0 119887(119896)

1

0 0 sdot sdot sdot 1 119887(119896)

119899minus1

)

)

(

0

0

1

)

(8)






(120579119894) where 120579













(iii) 119861 = 1198611015840


119898


B (x 119861) fl T119898mod ℓ (9)



119879)





3 Proof












(x1 x21 x23+

x2 x3) (x1 x2 x21 x

22+

x3)( x22 x

23+

x1 x2 x3)

(x1 x2 x3)

(x1 x2 x3)(x1 x2 x3)

(x1 x2 x3)

(x1 x21 + ((x21 + x22) x3)2

x2x21 + x

22

x3)

( x22 + ((x21 + x22) x3)2

x1 x2

x21 + x

22

x3)( x

22 + x2

3

x1 x2((x22 + x

23) x1)2 + x

22

x3)

( x22 + x23

x1((x22 + x

23) x1)2 + x

23

x2 x3)

(((x21 + x23) x2)2 + x

23

x1x21 + x

23

x2 x3) (x1 x

21 + x

23

x2x21 + ((x21 + x

23) x2)2

x3)




119879


11990911199091015840

1= 1199092

2+ 1199092

3

11990921199091015840

2= 1199092

1+ 1199092

3

11990931199091015840

3= 1199092

1+ 1199092

2

(10)





1= 1199092 11990910158402= 1199093 11990910158403= 1199091and

1199091015840



12058313(x) = 120583

21(x)

12058323(x) = 120583

31(x)

(11)

where 120583119894119895(x) fl 120583






119879)






119879)


A (x 119861119879) cong

M

1198680

(12)






Prim(

M

1198680

) = 119868120579| 120579 isin R (13)





0)



ℎ Prim(

M

1198680

) 997888rarr R (14)


Let 120590119905 M119868

0rarr M119868




0 the image 120590119905(119868

120579) of 119868

120579is


0




120590119905(119868120579) sub

M

1198680

| 119905 isin R 120579 isin R (15)




homeomorphism


11 (16)








119870+

0(

A (x 11987611)

120590119905(119868120579)

) cong 119890119905

(Z + Z120579) (17)






(1199091 119909


119879 Since




11990911199091015840

1= 1199092

2+ 1199092


2

3119896

11990921199091015840

2= 1199092

1+ 1199092


2

3119896

11990931198961199091015840

3119896= 1199092

1+ 1199092


2

3119896minus1

(18)



3119896is similar





3119896are the ones


1015840



12058313119896

(x) = 12058321(x)

12058323119896

(x) = 12058331(x)

1205833119896minus13119896

(x) = 12058331198961

(x)

(19)

where 120583119894119895(x) fl 120583

119895(120583119894(x))


3119896by the ℓ-



diagramB(x 119861119879)

Lemma 15 The set

PrimA (x 119876119892119899) = 119868

120579| 120579 isin R

6119892minus7+2119899

is generic (20)






119879)


120579and




119879)The





120579is



A (x 119876119892119899)

119868120579

= 119860120579 (21)










Let 120590119905 A(x 119876

119892119899) rarr A(x 119876

119892119899) be the Tomita-



119868120579sub A(x 119876

119892119899) the image 120590119905(119868



119892119899) but not




120590119905(119868120579) sub A (x 119876

119892119899) | 119905 isin R 120579 isin R

6119892minus7+2119899

(22)


cong


ℎ PrimA (x 119876119892119899) timesR


(23)


119892119899)120590119905(119868120579) is a


Theorem 2 is proved




the space 119879119892119899



119892119899) the variables 119909

119894= 120582(120574



119892119899) for all 119905 isin R

Since A(119905x 119876119892119899) sube A(x 119876



120590119905 A (x 119876

119892119899) 997888rarr A (x 119876

119892119899) (24)






(

119890119905

0

0 119890minus119905

) (25)








space 119879119892119899




120601 isin mod119878119892119899


120601F120583


120601gt 1 is





119892119899isin


119892119899and 120601(119878

119892119899) belong to O [15]



119892119899) indeed in


120590119905(119909) = 119890

119905

119909

120601 (119909) = 120582120601119909

(26)





119879 (A (x 119876119892119899))

= log 120582120601| 120601 isin mod 119878


(27)



120582


Disclosure


Competing Interests


Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2

23 34 4

R2

1 = infin

H







119899 where 119872



1 119899





1198721

1205931

997888rarr 1198722

1205932



119894are



119894=


119894119896and119872

1198941015840 = 119872

1198941015840

1


119896


119894 119872119894rarr 119872



119894119896and 119881

1198941015840

1

1198811198941015840

119896

joined by119887119903119904


119903119904copies

of the summand 1198721198941015840

119904







(

1

1205791

120579119899minus1


(

(



1


119899minus1

)

)

sdot sdot sdot(

(


1 0 sdot sdot sdot 0 119887(119896)

1

0 0 sdot sdot sdot 1 119887(119896)

119899minus1

)

)

(

0

0

1

)

(8)






(120579119894) where 120579













(iii) 119861 = 1198611015840


119898


B (x 119861) fl T119898mod ℓ (9)



119879)





3 Proof












(x1 x21 x23+

x2 x3) (x1 x2 x21 x

22+

x3)( x22 x

23+

x1 x2 x3)

(x1 x2 x3)

(x1 x2 x3)(x1 x2 x3)

(x1 x2 x3)

(x1 x21 + ((x21 + x22) x3)2

x2x21 + x

22

x3)

( x22 + ((x21 + x22) x3)2

x1 x2

x21 + x

22

x3)( x

22 + x2

3

x1 x2((x22 + x

23) x1)2 + x

22

x3)

( x22 + x23

x1((x22 + x

23) x1)2 + x

23

x2 x3)

(((x21 + x23) x2)2 + x

23

x1x21 + x

23

x2 x3) (x1 x

21 + x

23

x2x21 + ((x21 + x

23) x2)2

x3)




119879


11990911199091015840

1= 1199092

2+ 1199092

3

11990921199091015840

2= 1199092

1+ 1199092

3

11990931199091015840

3= 1199092

1+ 1199092

2

(10)





1= 1199092 11990910158402= 1199093 11990910158403= 1199091and

1199091015840



12058313(x) = 120583

21(x)

12058323(x) = 120583

31(x)

(11)

where 120583119894119895(x) fl 120583






119879)






119879)


A (x 119861119879) cong

M

1198680

(12)






Prim(

M

1198680

) = 119868120579| 120579 isin R (13)





0)



ℎ Prim(

M

1198680

) 997888rarr R (14)


Let 120590119905 M119868

0rarr M119868




0 the image 120590119905(119868

120579) of 119868

120579is


0




120590119905(119868120579) sub

M

1198680

| 119905 isin R 120579 isin R (15)




homeomorphism


11 (16)








119870+

0(

A (x 11987611)

120590119905(119868120579)

) cong 119890119905

(Z + Z120579) (17)






(1199091 119909


119879 Since




11990911199091015840

1= 1199092

2+ 1199092


2

3119896

11990921199091015840

2= 1199092

1+ 1199092


2

3119896

11990931198961199091015840

3119896= 1199092

1+ 1199092


2

3119896minus1

(18)



3119896is similar





3119896are the ones


1015840



12058313119896

(x) = 12058321(x)

12058323119896

(x) = 12058331(x)

1205833119896minus13119896

(x) = 12058331198961

(x)

(19)

where 120583119894119895(x) fl 120583

119895(120583119894(x))


3119896by the ℓ-



diagramB(x 119861119879)

Lemma 15 The set

PrimA (x 119876119892119899) = 119868

120579| 120579 isin R

6119892minus7+2119899

is generic (20)






119879)


120579and




119879)The





120579is



A (x 119876119892119899)

119868120579

= 119860120579 (21)










Let 120590119905 A(x 119876

119892119899) rarr A(x 119876

119892119899) be the Tomita-



119868120579sub A(x 119876

119892119899) the image 120590119905(119868



119892119899) but not




120590119905(119868120579) sub A (x 119876

119892119899) | 119905 isin R 120579 isin R

6119892minus7+2119899

(22)


cong


ℎ PrimA (x 119876119892119899) timesR


(23)


119892119899)120590119905(119868120579) is a


Theorem 2 is proved




the space 119879119892119899



119892119899) the variables 119909

119894= 120582(120574



119892119899) for all 119905 isin R

Since A(119905x 119876119892119899) sube A(x 119876



120590119905 A (x 119876

119892119899) 997888rarr A (x 119876

119892119899) (24)






(

119890119905

0

0 119890minus119905

) (25)








space 119879119892119899




120601 isin mod119878119892119899


120601F120583


120601gt 1 is





119892119899isin


119892119899and 120601(119878

119892119899) belong to O [15]



119892119899) indeed in


120590119905(119909) = 119890

119905

119909

120601 (119909) = 120582120601119909

(26)





119879 (A (x 119876119892119899))

= log 120582120601| 120601 isin mod 119878


(27)



120582


Disclosure


Competing Interests


Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(x1 x21 x23+

x2 x3) (x1 x2 x21 x

22+

x3)( x22 x

23+

x1 x2 x3)

(x1 x2 x3)

(x1 x2 x3)(x1 x2 x3)

(x1 x2 x3)

(x1 x21 + ((x21 + x22) x3)2

x2x21 + x

22

x3)

( x22 + ((x21 + x22) x3)2

x1 x2

x21 + x

22

x3)( x

22 + x2

3

x1 x2((x22 + x

23) x1)2 + x

22

x3)

( x22 + x23

x1((x22 + x

23) x1)2 + x

23

x2 x3)

(((x21 + x23) x2)2 + x

23

x1x21 + x

23

x2 x3) (x1 x

21 + x

23

x2x21 + ((x21 + x

23) x2)2

x3)




119879


11990911199091015840

1= 1199092

2+ 1199092

3

11990921199091015840

2= 1199092

1+ 1199092

3

11990931199091015840

3= 1199092

1+ 1199092

2

(10)





1= 1199092 11990910158402= 1199093 11990910158403= 1199091and

1199091015840



12058313(x) = 120583

21(x)

12058323(x) = 120583

31(x)

(11)

where 120583119894119895(x) fl 120583






119879)






119879)


A (x 119861119879) cong

M

1198680

(12)






Prim(

M

1198680

) = 119868120579| 120579 isin R (13)





0)



ℎ Prim(

M

1198680

) 997888rarr R (14)


Let 120590119905 M119868

0rarr M119868




0 the image 120590119905(119868

120579) of 119868

120579is


0




120590119905(119868120579) sub

M

1198680

| 119905 isin R 120579 isin R (15)




homeomorphism


11 (16)








119870+

0(

A (x 11987611)

120590119905(119868120579)

) cong 119890119905

(Z + Z120579) (17)






(1199091 119909


119879 Since




11990911199091015840

1= 1199092

2+ 1199092


2

3119896

11990921199091015840

2= 1199092

1+ 1199092


2

3119896

11990931198961199091015840

3119896= 1199092

1+ 1199092


2

3119896minus1

(18)



3119896is similar





3119896are the ones


1015840



12058313119896

(x) = 12058321(x)

12058323119896

(x) = 12058331(x)

1205833119896minus13119896

(x) = 12058331198961

(x)

(19)

where 120583119894119895(x) fl 120583

119895(120583119894(x))


3119896by the ℓ-



diagramB(x 119861119879)

Lemma 15 The set

PrimA (x 119876119892119899) = 119868

120579| 120579 isin R

6119892minus7+2119899

is generic (20)






119879)


120579and




119879)The





120579is



A (x 119876119892119899)

119868120579

= 119860120579 (21)










Let 120590119905 A(x 119876

119892119899) rarr A(x 119876

119892119899) be the Tomita-



119868120579sub A(x 119876

119892119899) the image 120590119905(119868



119892119899) but not




120590119905(119868120579) sub A (x 119876

119892119899) | 119905 isin R 120579 isin R

6119892minus7+2119899

(22)


cong


ℎ PrimA (x 119876119892119899) timesR


(23)


119892119899)120590119905(119868120579) is a


Theorem 2 is proved




the space 119879119892119899



119892119899) the variables 119909

119894= 120582(120574



119892119899) for all 119905 isin R

Since A(119905x 119876119892119899) sube A(x 119876



120590119905 A (x 119876

119892119899) 997888rarr A (x 119876

119892119899) (24)






(

119890119905

0

0 119890minus119905

) (25)








space 119879119892119899




120601 isin mod119878119892119899


120601F120583


120601gt 1 is





119892119899isin


119892119899and 120601(119878

119892119899) belong to O [15]



119892119899) indeed in


120590119905(119909) = 119890

119905

119909

120601 (119909) = 120582120601119909

(26)





119879 (A (x 119876119892119899))

= log 120582120601| 120601 isin mod 119878


(27)



120582


Disclosure


Competing Interests


Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











ℎ Prim(

M

1198680

) 997888rarr R (14)


Let 120590119905 M119868

0rarr M119868




0 the image 120590119905(119868

120579) of 119868

120579is


0




120590119905(119868120579) sub

M

1198680

| 119905 isin R 120579 isin R (15)




homeomorphism


11 (16)








119870+

0(

A (x 11987611)

120590119905(119868120579)

) cong 119890119905

(Z + Z120579) (17)






(1199091 119909


119879 Since




11990911199091015840

1= 1199092

2+ 1199092


2

3119896

11990921199091015840

2= 1199092

1+ 1199092


2

3119896

11990931198961199091015840

3119896= 1199092

1+ 1199092


2

3119896minus1

(18)



3119896is similar





3119896are the ones


1015840



12058313119896

(x) = 12058321(x)

12058323119896

(x) = 12058331(x)

1205833119896minus13119896

(x) = 12058331198961

(x)

(19)

where 120583119894119895(x) fl 120583

119895(120583119894(x))


3119896by the ℓ-



diagramB(x 119861119879)

Lemma 15 The set

PrimA (x 119876119892119899) = 119868

120579| 120579 isin R

6119892minus7+2119899

is generic (20)






119879)


120579and




119879)The





120579is



A (x 119876119892119899)

119868120579

= 119860120579 (21)










Let 120590119905 A(x 119876

119892119899) rarr A(x 119876

119892119899) be the Tomita-



119868120579sub A(x 119876

119892119899) the image 120590119905(119868



119892119899) but not




120590119905(119868120579) sub A (x 119876

119892119899) | 119905 isin R 120579 isin R

6119892minus7+2119899

(22)


cong


ℎ PrimA (x 119876119892119899) timesR


(23)


119892119899)120590119905(119868120579) is a


Theorem 2 is proved




the space 119879119892119899



119892119899) the variables 119909

119894= 120582(120574



119892119899) for all 119905 isin R

Since A(119905x 119876119892119899) sube A(x 119876



120590119905 A (x 119876

119892119899) 997888rarr A (x 119876

119892119899) (24)






(

119890119905

0

0 119890minus119905

) (25)








space 119879119892119899




120601 isin mod119878119892119899


120601F120583


120601gt 1 is





119892119899isin


119892119899and 120601(119878

119892119899) belong to O [15]



119892119899) indeed in


120590119905(119909) = 119890

119905

119909

120601 (119909) = 120582120601119909

(26)





119879 (A (x 119876119892119899))

= log 120582120601| 120601 isin mod 119878


(27)



120582


Disclosure


Competing Interests


Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















Let 120590119905 A(x 119876

119892119899) rarr A(x 119876

119892119899) be the Tomita-



119868120579sub A(x 119876

119892119899) the image 120590119905(119868



119892119899) but not




120590119905(119868120579) sub A (x 119876

119892119899) | 119905 isin R 120579 isin R

6119892minus7+2119899

(22)


cong


ℎ PrimA (x 119876119892119899) timesR


(23)


119892119899)120590119905(119868120579) is a


Theorem 2 is proved




the space 119879119892119899



119892119899) the variables 119909

119894= 120582(120574



119892119899) for all 119905 isin R

Since A(119905x 119876119892119899) sube A(x 119876



120590119905 A (x 119876

119892119899) 997888rarr A (x 119876

119892119899) (24)






(

119890119905

0

0 119890minus119905

) (25)








space 119879119892119899




120601 isin mod119878119892119899


120601F120583


120601gt 1 is





119892119899isin


119892119899and 120601(119878

119892119899) belong to O [15]



119892119899) indeed in


120590119905(119909) = 119890

119905

119909

120601 (119909) = 120582120601119909

(26)





119879 (A (x 119876119892119899))

= log 120582120601| 120601 isin mod 119878


(27)



120582


Disclosure


Competing Interests


Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article On Cluster -Algebras · 2019. 7. 30. · Cluster Algebras from Riemann Surfaces. Let and be integers, such that 0, 1,and2 2+ > 0. Denoteby, aRiemannsurfaceofgenus