Quantum cluster algebras and the dual canonical basis [Elektronische Ressource] / Philipp Lampe

Philipp Lampe

Description

QuantumclusteralgebrasandthedualcanonicalbasisDissertationzurErlangungeinesDoktortitels(Dr. rer. nat.)derMathematisch NaturwissenschaftlichenFakult at¨derRheinischenFriedrich WilhelmsUniversit at¨ BonnvorgelegtvonPhilippLampeausRheineBonn,2010AngefertigtmitGenehmigungderMathematisch-NaturwissenschaftlichenFakultat¨ derRheinischenFriedrich-Wilhelms-Universitat¨ BonnErstgutachter: Prof. Dr. JanSchroer¨Zweitgutachterin: Prof. Dr. CatharinaStroppelTagderPromotion: 28.02.2011JahrderVerof¨ fentlichung: 20112ZusammenfassungSeiQ ein Dynkinkocher¨ vom TypA mit alternierender Orientierung oder der Kronec kerkocher¨ . Wir betrachten die direkte Summe M der unzerlegbaren, injektiven Mo ¨ ¨duln uber der Wegealgebra des Kochers und ihrer Auslander Reiten Verschiebungen.Sei g die zu Q assoziierte komplexe Liealgebra. Sie besitzt eine Dreieckszerlegungg = n ⊕ h⊕ n. Hierbei ist n eine maximal nilpotente Unterliealgebra von g. Zu−M gehort¨ in naturlicher¨ Weise ein Element w in der Weylgruppe des gleichen Typs.DieDimensionsvektorenderunzerlegbaren,injektivenModulnundihrerVerschiebun gen entsprechen nach dem Satz von Gabriel bzw. nach dem Satz von Kac positivenWurzelnimWurzelsystemderLiealgebrag. IhreAnordnungerbringteinenreduziertenAusdruckfur¨ dasWeylgruppenelementw.

Sujets

• Mathematics

Informations

• Publié par: rheinische_friedrich-wilhelms-universitat_bonn
• Publié le: 01 janvier 2011
• Nombre de lectures: 25
• Langue: Deutsch

Extrait

Quantum cluster algebras and the dual canonical basis

Dissertation

zur

Erlangung eines Doktortitels (Dr. rer. nat.)

der

Mathematisch-NaturwissenschaftlichenFakult¨at

der

RheinischenFriedrich-WilhelmsUniversit¨atBonn

vorgelegt von

Philipp Lampe

aus

Rheine

Bonn, 2010

Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakulta¨t der Rheinischen Friedrich-Wilhelms-Universita¨t Bonn

Erstgutachter: Prof. Dr. Jan Schro¨ er Zweitgutachterin: Prof. Dr. Catharina Stroppel

Tag der Promotion: 28.02.2011

Jahr der Vero¨ ffentlichung: 2011

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Zusammenfassung SeiQ vom Typ cher ein Dynkinko¨ Amit alternierender Orientierung oder der Kronec- kerk¨ocher.WirbetrachtendiedirekteSummeMder unzerlegbaren, injektiven Mo- duln ¨ uberderWegealgebradesKo ¨ chersundihrerAuslander-Reiten-Verschiebungen. Seigdie zuQassoziierte komplexe Liealgebra. Sie besitzt eine Dreieckszerlegung g=n − ⊕h⊕n. Hierbei istneine maximal nilpotente Unterliealgebra vong. Zu M in natu¨geho¨ rt Weise ein Element rlicherwin der Weylgruppe des gleichen Typs. Die Dimensionsvektoren der unzerlegbaren, injektiven Moduln und ihrer Verschiebun- gen entsprechen nach dem Satz von Gabriel bzw. nach dem Satz von Kac positiven Wurzeln im Wurzelsystem der Liealgebrag. Ihre Anordnung erbringt einen reduzierten Ausdruck fu¨  r das Weylgruppenelementw. Die vorliegende Arbeit erbringt den Nachweis, dass geschickt gewa¨hlte Erzeuger der zuw Unteralgebrageho¨ rigenU q + (w)der quantisierten universellen einhu¨ Alge- llenden braU q (n)nkdeertw,ennn ¨ o,retsulCreippurgsiinengen,sogenanntehcu ¨ eblrpaepdnMe so dassU q + (w)ureiruktuantnerqeitSdteraClusrtenisie.t¨hlaarreglbe Aus der Konstruktion der quantisierten Clusteralgebrenstruktur aufU q + (w)und der Wahl der Erzeuger ergeben sich folgende Eigenschaften: Erstens: Die quantisierten Clustervariablen stimmen jeweils bis auf eine Potenz des Deformationsparametersqeinem Element im Dualen der von Lusztig definiertenmit kanonischen Basis unter Kashiwaras Bilinearform u¨ berein. Zweitens: Geiß-Leclerc-Schro¨ er haben fu¨ r ein derartigesweine azyklische Cluster- algebra konstruiert. Sie wird realisiert als Unteralgbera der graduiert dualen Hopfal- gebraderuniverselleneinh¨ullendenAlgebraU(n). Unsere quantisierte Clusteralge- bra degeneriert zu Geiß-Leclerc-Schro¨ ers Clusteralgebra im klassischen Limesq= 1 Clusteralgebra hat eingefrorene hnliche  die quantisierte als auch die gewo¨ . Sowohl sowie mutierbare Clustervariablen. Drittens:BestimmteElementeinderdualenkanonischenBasiserf¨ullennennerfreie Rekursionsgleichungen. Die Rekursionsgleichungen implizieren die Austauschrelatio- nen fu¨ r quantisierte Clusteralgebren. DieArbeitistinzweiTeilegegliedert.DerersteTeilbehandeltdenFalldesDynkink¨o- chers vom TypAder zweite Teil behandelt den Kron-mit alternierender Orientierung, ¨ eckerkocher. Nach einem Satz von Lusztig besitzt die AlgebraU q + (w) in beiden Fa¨llen einePoincar´e-Birkhoff-Witt-Basis,diesichausdemreduziertenAusdruckf¨urwergibt. Die duale kanonische Basis kann mit Hilfe der Poincare´-Birkhoff-Witt-Basis u¨  ber eine Invarianz- und eine Gittereigenschaft charakterisiert werden. Wirbeschreibenzun¨achstdieBegradigungsrelationenderErzeugervonU q + (w)aus derPoincar ´ e-Birkhoff-Witt-Basis.SodannleitenwirnennerfreieRekursionsgleichun- gen fur bestimmte Elemente in der dualen kanonischen Basis her. Die Nennerfreiheit ¨ erlaubt es, die Invarianz- und die Gittereigenschaft nachzuweisen. Es folgen die Aus- tauschrelationen der quantisierten Clusteralgebra.

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Contents Summary 1 Introduction 2 Quantum cluster algebras of type A 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Representation theory of the quiver of type A and cluster algebras . . 2.3 The preprojective algebra and rigid modules . . . . . . . . . . . . . . 2.4 Notations from Lie theory . . . . . . . . . . . . . . . . . . . . . . . 2.5 The cluster algebra attached to the terminal module . . . . . . . . . . 2.6 The description of cluster variables . . . . . . . . . . . . . . . . . . . 2.7 Definition of the quantized enveloping algebra . . . . . . . . . . . . . 2.8 The subalgebraU v + (w)aninPohedtrkhoff-Wcar´e-Bi.....tibtsasi 2.9 The quantum shuffle algebra and Euler numbers . . . . . . . . . . . . 2.10 The straightening relations for the generators ofU v + (w). . . . . . . . 2.11ThedualPoincar´e-Birkhoff-Wittbasis................. 2.12 The dual canonical basis . . . . . . . . . . . . . . . . . . . . . . . . 2.13 The quantum cluster algebra structure induced by the dual canonical basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 A quantum cluster algebra of Kronecker type 3.1 Representation theory of the Kronecker quiver . . . . . . . . . . . . . 3.2 The preprojective algebra . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Theδ-functions and the cluster algebra structure . . . . . . . . . . . . 3.4 Mutations of rigid modules . . . . . . . . . . . . . . . . . . . . . . . 3.5 Bases of the cluster algebraA(C M ). . . . . . . . . . . . . . . . . . 3.6 The quantized universal enveloping algebraU q (g)of typeA ) 1 (1 . . . . 3.7 The Poincare´-Birkhoff-Witt basis . . . . . . . . . . . . . . . . . . . . 3.8 The derivation of the straightening relations . . . . . . . . . . . . . . 3.9 The dual canonical basis . . . . . . . . . . . . . . . . . . . . . . . . 3.10 A recursion for dual canonical basis elements . . . . . . . . . . . . . 3.11 The quantized version of the explicit formula for cluster variables . . 3.12 The quasi-commutativity of adjacent quantized cluster variables and the quantum exchange relation . . . . . . . . . . . . . . . . . . . . . 3.13 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography

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3 5 10 10 11 13 18 20 21 25 28 30 33 38 40 44 53 53 54 54 57 59 59 61 63 65 67 71 74 75 77

1 Introduction Cluster algebrasare commutative algebras created in 2000 by Fomin-Zelevinsky [14] in the hope to obtain a combinatorial description of the dual of Lusztig’scanonical basisof a quantum group. A cluster algebra of rankn(for some natural numbern) is a subalgebra of the field Q(x 1  . . .  x n )of rational functions innvariables. Its generators are calledcluster variables. Each cluster variable belongs to several overlappingclusters. Every cluster, and hence every cluster variable, is obtained from an initial cluster by a sequence of mutations. Every mutation replaces an element in a cluster by an explicitly defined rational function in the variables of that cluster. A cluster together with the exchange matrix that describes the mutation rule is called aseed. We refer to Fomin-Zelevinsky [14] for definitions and to Fomin-Zelevinsky [16] for a good survey about cluster alge- bras. A cluster algebra is said to be of finite type if it exhibits only finitely many seeds. Fomin-Zelevinsky [15] classified cluster algebras of finite type. They are parametrized by the same Cartan-Killing types as semisimple Lie algebras. The classification of cluster algebras of finite type indicates a strong connection to Lie theory, and in fact it quickly turned out that Fomin-Zelevinsky’s theory of cluster algebras has many in- teresting applications and coheres with various mathematical objects. Let us mention the representation theory of quivers and finite-dimensional algebras, the representation theory of preprojective algebras, root systems of Kac-Moody algebras, Calabi-Yau cat- egories, quantum groups, and Lusztig’s canonical basis of universal enveloping alge- bras. A momentous step in the development was thecategorificationof acyclic clus- ter algebras bycluster categories. Cluster categories were defined by Buan-Marsh- Reineke-Reiten-Todorov [4]. The cluster categoryC Q associated with a quiverQis an orbit category of the bounded derived category of the category of representations of a quiver. Keller [29] proved that cluster categories are triangulated categories. Key ingredients for the verification of the categorification of acyclic cluster algebras by cluster categories are due to Geiß-Leclerc-Schro¨ er [23, 24] and Caldero-Keller [8, 9]. The process of mutation in the cluster algebra resembles tilting in the cluster category. Hence, we obtain a link between quiver representations and triangulated categories on one side and a large class of cluster algebras on the other side. Furthermore, Geiß-Leclerc-Schro¨ er [21] provided a categorification of cluster al- gebras byKac-Moody groupsundunipotent cells. In this construction the categorified cluster algebras are not necessarily acyclic. Letgbe a Kac-Moody Lie algebra and let g=n − ⊕h⊕nbe its triangular decomposition. Geiß-Leclerc-Schroer’s construction ¨ is related to preprojective algebrasΛassociated with quiversQ. Buan-Iyama-Reiten- Scott [3] attached to every elementwin the Weyl group of corresponding type a sub- categoryC w ⊂mod(λ). The categoryC w is a Frobenius category, so we can construct its stable categoryC w  Geiß-Leclerc-. It is a Calabi-Yau category of dimension two. Sch ¨  [21] prove that the categoriesC w  endow the Theycategorify cluster algebras. roer coordinate ringC[N(w)]of the unipotent groupN(w)with the structure of a cluster algebra. Here,Ndenotes the pro-unipotent pro-group associated with the completion n b andN(w) =N∩(w −1 N − w). The coordinate ringC[N(w)]is naturally isomorphic to a subalgebra of the graded dualU(n) g ∗ r of the universal enveloping algebra ofn. All cluster monomials lie in the dual semicanonical basis. In this thesis we transfer to the quantized setup and investigate quantum cluster algebra structures on subalgebras of thequantizeduniversal enveloping algebraU q (n) 5

ofnattached to Weyl group elementsw. Let us mention that cluster algebras also gained popularity in other branches of mathematics, for example Poisson geometry, see Gekhtman-Shapiro-Vainshtein, Te- ich ¨ ullertheory,seeFock-Goncharov[13],combinatorics,seeMusiker-Propp[45], m integrable systems, see Fomin-Zelevinsky [18], etc. Now we give a more detailed description of the structure and the results of the thesis. LetQ= (Q 0  Q 1 )be anacyclic quiver, i.e., a directed graph without oriented cycles whose set of vertices is denoted byQ 0 whose set of arrows is denoted byand Q 1 . The thesis focuses on two examples, namelyalternating quivers of typeAand the Kronecker quiver.

Figure 1: Alternating quivers of typeA 3 andA 5 and the Kronecker quiver Letk Abe a field.representationMofQis defined to be a collectionM= ((V i ) i∈Q 0 (φ a ) a∈Q 1 )ofk-vector spacesV i associated with every vertexiandk-linear mapsφ a :V i →V j associated with every arrowa:i→jinQ 1 . AmorphismF:M→ NfromMto another representationN= ((W i ) i∈Q 0 (ψ a ) a∈Q 1 )is a collection ofk- linear mapsF i :V i →W i such that the diagram V i φ a // V j F i F j    W i ψ a // W j commutes for everya:i→jinQ 1 . The finite-dimensional representations ofQ together with their morphisms form an abelian categoryrep k (Q) particular, there. In is a direct sumM⊕Nfor every two representationsMandN. The categoryrep k (Q)is equivalent to the category mod(kQ), wherekQis thepath algebraof the quiver, given as follows: As a vector spacekQis generated by all paths in the quiver (including a path of length zero for everyi∈Q 0  ). Theproduct of two paths is the concatenation of paths if possible and zero otherwise. The vectordim(M) = (dimV i ) i∈Q 0 ∈N Q 0 is called thedimension vectorofM. The quiverQis calledrepresentation-finiteifQadmits only finitely many (iso- morphism classes of) indecomposable representations. Gabriel’s theorem [19] asserts thatQis representation-finite if and only ifQis an orientation of aDynkin diagram of typeA,D, orE example, the alternating quiver of type. ForA 3 form above is representation-finite. It admits six (isomorphism classes of) indecomposable represen- tations. TheAuslander-Reiten quiverin Figure 2 epitomizes the categoryrep k (Q). Representations are displayed by their dimension vectors. On the other hand the Kronecker quiver is representation-infinite. But it is atame quiver. Albeit there are infinitely many (isomorphism classes of) indecomposablekQ- modules, the indecomposablekQ-modules can be classified. The three kinds of inde- composables are calledpreprojective,preinjective, andregular part of the prein-. A 6

Figure 2: The Auslander-Reiten quiver of typeA 3 jective component of the Auslander-Reiten quiver of the Kronecker quiver is shown in Figure 3. We are interested in the injective modulesI i (fori∈Q 0 ) and theirAuslander- Reiten translatesτ(I i )(fori∈Q 0 ). PutQ 0 ={12 . . .  n}. Thus, we consider2n modules. In the examples of Figures 2 and 3 these are all indecomposable modules in the Auslander-Reiten quiver of typeA 3 and the four modules at rightmost position in the preinjective component of the Auslander-Reiten quiver of the Kronecker quiver. The direct sumMof the injective modules and their Auslander-Reiten translates is aterminalkQLeß-erclScc- ¨ ohr2[reW.]1thtiehdolum-fGeinseoheseeint2nmod- ules we associate an elementwin theWeyl groupof theKac-Moody Lie algebraof corresponding type together with a reduced expression for it.

Figure 3: A part of the preinjective component of the AR quiver of the Kronecker quiver Geiß-Leclerc-Schro¨ er  [21] attached towacluster algebraA(w)of rankn. For the terminal moduleMrfbamotevocl’ser¨orteuseLlcie-ßcSrhre-citiaheindofGlsee algebraA(w)containsnmutatableandnfrozencluster variables. the case of InA n there are only finitely many cluster variables. In the particular example ofA 3 from above the exchange graph is aStasheff polyhedron. The cluster algebra attached toM in the Kronecker case is generated by two frozen cluster variablesp 0  p 1 and a sequence (x n ) n∈Z  with an initial clusterof mutatable cluster variables. Starting(x 0  x 1  p 0  p 1 ) we get all further cluster variables by a sequence of mutations.

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If we putp 0 =p 1 = 1, then the exchange relation which allows to switch between adjacent clusters becomesx k+1 x k−1 =x k 2 + 1fork∈Z. If we furthermore specialize x 0 =x 1 = 1, then we getx 2 = 2,x 3 = 5,x 4 = 13,x 5 = 34,x 6 = 89, etc. Every term in the sequence is a natural number. (In fact, the sequence is every other Fibonacci number.) The integrality is an instance of the Fomin-Zelevinsky’sLaurent phenomenon [14]: Every cluster variable is a Laurent polynomial inx 0 andx 1 . Caldero-Zelevinsky [10] gave an explicit formula for the cluster variables in terms of binomial coefficients by interpreting coefficients as Euler characteristics of quiver Grassmannians arising in the Caldero-Chapoton map [7]. A monomial in the cluster variable of a single cluster is calledcluster monomial. The representation theory of the path algebrakQis closely related to the represen- tation theory of the correspondingpreprojective algebraΛ [48] proved that. Ringel the categorymod(Λ)is isomorphic to a category calledC(1 τ) objects in the. The categoryC(1 τ)are pairs(X f)consisting of akQ-moduleXand akQ-module ho- momorphismf:X→τ(X)fromXto its translateτ(X); morphisms inC(1 τ)from a pair(X f)to a pair(Y g)are given by akQ-module homomorphismh:X→Y for which the diagram h X // Y f g τ(  X) τ(h)  // τ(Y) commutes. The algebraΛis finite-dimensional if and only ifQis an orientation of a Dynkin diagram, see Reiten [47, Theorem 2.2a]. To construct the cluster algebraA(w)ch-Srcleec-LißGe,verydtoecaeha]tt[r12 ¨ reo terminalCQ-moduleMa natural subcategoryC M ⊆nil(Λ)of nilpotentΛ-modules. The categoryC M is a Frobenius category. The stable categoryC M is triangulated by a theoremofHappel[26,Section2.6].Geiß-Leclerc-Schr¨oer[21,Theorem11.1]showed that ifM=I⊕τ(I)whereIis the direct sum of all indecomposable injective repre- sentations, then there is an equivalence of triangulated categoriesC M ' C Q between C M and the cluster categoryC Q as defined by Buan-Marsh-Reineke-Reiten-Todorov [4] to be the orbit categoryD b (mod(kQ))/τ D−1 ◦[1]. Geiß-Leclerc-Schr¨oer[21,Section4]implementedtheclusteralgebraA(w)as a subalgebra of the graded dual of the universal enveloping algebraU(n)of the maxi- mal nilpotent subalgebranof the symmetric Kac-Moody Lie algebragattached to the quiverQ, i.e.,A(C M )⊆U(n) g ∗ r osla]12[devorpc-erclLeer ¨ ohrSc,Gerß-eiMo.ovre that all cluster monomials are in the dual of Lusztig’s semicanonical basis. There is an isomorphism betweenU(n)and an algebraMofC-valued functions onΛ. We refer to [21] for a precise definition ofM. It is generated by functionsd i that map aΛ-module Xto the Euler characteristic of the flag variety ofXof typei elements. Prominent inA(C M )are (under the described isomorphism) theδ-functions of certain rigidΛ- modules. Additionally, there is an isomorphismA(C M )'C[N(w)]whereC[N(w)] is the coordinate ring of the unipotent subgroupN(w)attached to the adaptable Weyl group elementwofM. Therefore, we may call theC M acategorificationof the cluster algebraA(C M ). We transfer to the quantized setup. Thequantized universal enveloping algebra U q (n)is a self-dualHopf algebra. Following Lusztig [42] we attach towa subalgebra U q + (w)ofU q (n) subalgebra is generated by. The2nelements that satisfy straight- ening relations; it degenerates to a commutative algebra in the classical limitq= 1. 8

The generators are constructed via Lusztig’sT algebra-automorphisms. TheU q + (w) possesses four distinguished bases, aPoincare´-Birkhoff-Witt basis, acanonical basis, and their duals. The thesis concerns the dual of Lusztig’s canonical basis of the subal- gebraU q + (w)under Kashiwara’s bilinear form [28]. The dual canonical basis elements canbedescribedaslinearcombinationsofdualPoincar´e-Birkhoff-Wittbasiselements satisfying a lattice property and an invariance property. It is conjectured (see for example [33]) that the quantized coordinate ringC q [N(w)] isquantum cluster algebraA q (w)in the sense of Berenstein-Zelevinsky [6] and that the setM q of all quantum cluster monomials, taken up to powers ofq, is a subset of the dual canonical basisB ∗ , i.e., the following diagram commutes: A q (w) // C q [N(w)] ⊂U q (n) ∗ gr OOOO

 ?    ? M q   // B ∗ The thesis is divided into two parts. The first part concerns alternating quivers of typeA n  In, the second part concerns the Kronecker quiver. both cases, we prove recursions for dual canonical basis elements inU q + (w) recursions imply quan-. The tum exchange relations so that the integral form ofU q + (w)becomes (after extending coefficients) a quantum cluster algebraA q (w). It follows that the quantum cluster vari- ables are, up to a power ofq, elements in the dual of Lusztig’s canonical basis under Kashiwara’s bilinear form. The proof relies on the exact form of the straightening relations. In the caseA n , the description of the straightening relations features (besides Lusztig’sT-automorphisms) Leclerc’s embedding [36] ofU q (n)in thequantum shuffle algebra. The straightening relations for the Kronecker case are due to Leclerc [37]. The exact form of the straight- ening relations enables us to verify that recursively defined variables satisfy the lattice property and the invariance property of the dual canonical basis. In the case of the Kronecker quiver, we give explicit formulae for the quantum clus- ter variables that quantize Caldero-Zelevinsky’s formulae [10] for the ordinary cluster variables. In this case we also provide formulae for expansions of products of dual canonical basis elements. Acknowledgements:eoswhr,uperomysfulthank¨reoSnhc,raJivosmhIarteaytil encouragement, supervision and support from the preliminary to the concluding level enabled me to develop an understanding of the subject. I am also grateful to Bernard Leclerc for invaluable discussions. I would like to thank the Bonn International Grad- uate School in Mathematics (BIGS) for financial support.

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2 Quantum cluster algebras of type A 2.1 Introduction Cluster algebrasare commutative rings defined by Fomin-Zelevinsky [14] to inves- tigatetotal positivityandcanonical bases study of cluster algebras promptly. The extended over various branches of mathematics. One of the two original motivations, namely the connection between cluster algebras and canonical bases, has only been observed in a few cases. The passage from cluster algebras to canonical bases features Berenstein-Zelevinsky’squantum cluster algebras[6]. To give a more detailed description of this connection we introduce the following notations from Lie theory: Letgbe acomplex Kac-Moody Lie algebrawithCartan matrixC. It admits a triangular decompositiong=n − ⊕h⊕n. There exist quantiza- tions of the universal enveloping algebras ofgandn, calledU v (g)andU v (n), respec- tively. With every Weyl group elementw∈WLusztig [42] associates a subalgebra U v + (w)⊂U v (n). Lusztig’s construction [42] ofU v + (w)involves the evaluation ofT- automorphisms at initial subsequences of a reduced expressioni= (i r  . . .  i 1 )forw. According to Lusztig [42]U v + (w)possesses several bases: For every reduced expres- sioniofwthere is aisastb ´ r-eiBkrohffW-tiPoinca. Furthermore, there is thecanonical basis. It is conjectured that the integral form of the subalgebraU v + (w)⊂U v (n)is (after extending coefficients) a quantum cluster algebra. The conjecture has only been verified in very few cases, see Berenstein-Zelevinsky [5] for typeA 2 andA 3 , and the author [34] for an example of Kronecker type. There- fore, particular instances are worthwile. In this note we focus on typeA n (for a natural numbern) and a particular Weyl group elementwof length2n. We are going to prove thatU v + (w)carries indeed a quantum cluster algebra structure. In this caseg=sl n+1 is thecomplex semi-simple Lie algebraof traceless(n+ 1)×(n+ 1)matrices andn consists of all strictly upper triangular matrices. The topic is truely linked with therepresentation theory of quivers. The caseg= sl n+1 is related toDynkin quiversof typeA n . We choose a particular orientation: Let Q= (Q 0  Q 1 )be a Dynkin quiver of typeA n with an alternating orientation beginning with a source. We denote the set of vertices byQ 0 ={12 . . .  n}. Figure 4 illustrates the examplen= 13. The choice of the orientation matches the choice of the Weyl group elementw. The reduced expression ofw(that is used to constructU v + (w)) and its initial subsequences (that are used to construct the generatorsU v + (w)) are related to the indecomposable injective modules over thepath algebraofQand theirAuslander- Reiten translates, respectively.

Figure 4: The quiverQof typeA 13 We denote the resulting quantum cluster algebra byA v (w). It is a deformation of the cluster algebraA(w) [21] attached to Geiß-Leclerc-Schro¨ er w Geiß-Leclerc-. In Schro¨ er’s setting, the cluster variables areδ-functions of rigid modules over thepre- projective algebraofQ. The cluster algebraA(w), just as the quantum cluster algebra A v (w), is of typeA n  cluster contains. Everynfrozen andnmutatable cluster vari- 10