AUSLANDER ALGEBRAS AND INITIAL SEEDS FOR CLUSTER ALGEBRAS

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1 AUSLANDER ALGEBRAS AND INITIAL SEEDS FOR CLUSTER ALGEBRAS CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER Abstract. Let Q be a Dynkin quiver and Π the corresponding set of positive roots. For the preprojective algebra Λ associated to Q we produce a rigid Λ-module I Q with r = Π pairwise non-isomorphic indecomposable direct summands by pushing the injective modules of the Auslander algebra of kq to Λ. If N is a maximal unipotent subgroup of a complex simply connected simple Lie group of type Q, then the coordinate ring C[N] is an upper cluster algebra. We show that the elements of the dual semicanonical basis which correspond to the indecomposable direct summands of I Q coincide with certain generalized minors which form an initial cluster for C[N], and that the corresponding exchange matrix of this cluster can be read from the Gabriel quiver of End Λ(I Q). Finally, we exploit the fact that the categories of injective modules over Λ and over its covering e Λ are triangulated in order to show several interesting identities in the respective stable module categories. Introduction Let k be a field and Q be a Dynkin quiver. So the underlying graph Q of Q is a simply laced Dynkin diagram. We produce for the preprojective algebra Λ over k associated to Q a module I Q by pushing a minimal injective cogenerator over the Auslander algebra Γ Q of kq to Λ -mod. It is easy to see that I Q decomposes into r = Π pairwise non-isomorphic direct summands. We show that I Q is a rigid module module, i.e. Ext Λ (I Q, I Q ) =. Moreover, the Gabriel quiver ǍQ of End Λ (I Q ) op is obtained from the Auslander-Reiten quiver A Q of kq by inserting an extra arrow x τx for each non-projective vertex x. In [9] we have shown that if M = m i= M i for pairwise non-isomorphic indecomposable Λ-modules M i, then Ext Λ (M, M) = implies m r. So our result shows that this maximum is assumed for each Dynkin quiver. By [6, Theorem.] we conclude that I Q is a maximal -orthogonal Λ-module and thus End Λ (I Q ) is a higher Auslander algebra in the sense of Iyama []. It follows that each rigid module M as above can be completed to a rigid module with r pairwise non-isomorphic indecomposable direct summands. Note, that for the proof of the main result of [6] it is essential that the quiver ǍQ has no loops. Let now G be a complex simply connected simple Lie group of type Q with N G a maximal unipotent subgroup. Choose i = (i, i,..., i r ) a reduced expression for the longest element w of the Weyl group W of G. It follows from [4] that the coordinate ring C[N] is an (upper) cluster algebra. Associated to i one obtains an initial seed ( (j, i) j=,,...,r, B(i) ), where the (j, i) are certain generalized minors, and the exchange matrix B(i) is obtained naturally from the quiver ǍQ described above. Next, i provides us with a convenient labelling of the indecomposable direct summands of I Q, that is, I Q = r j= I(j, i). Clearly the I(j, i) are rigid. Thus, if we restrict to the Date: June 6, 5. Mathematics Subject Classification. 4M99, 6D7, 6E, 6G, 6G7, 7B37, G4. Ch.G. acknowledges support from DGAPA grant IN4-3. B.L. is grateful to the GDR 43 and the GDR 49 for their support. J.S. was supported by a research fellowship from the DFG.

2 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER special case k = C the Λ-modules I(j, i) serve as natural labels for elements ρ I(j,i) of the dual of Lusztig s semicanonical basis. This is a natural basis for C[N] and we show that ρ I(j,i) = (j, i).. Main results.. We say that a quiver Q is a Dynkin quiver if its underlying graph Q is a Dynkin diagram of type A, D, E. For a field k we consider the path category k[q] (or kq for short). The category kq -mod of finitely presented k-functors M: kq k -mod is equivalent to the category of finitely presented left modules over the corresponding path algebra which we denote by some abuse also by kq. For a quiver Q we consider the double Q which is obtained from Q by adding a new arrow i a j for each arrow i a j in Q. The preprojective algebra Λ is the quotient of the path algebra kq by the ideal generated by the elements ρ q = a a aa for q Q, a Q t(a)=q a Q h(a)=q see also.. In what follows, Q will always be a connected Dynkin quiver. This implies that Λ is a finite-dimensional selfinjective algebra, which depends only on Q. Like kq, the algebra Λ can also be considered as a k-category. We have the universal covering F : Λ Λ where Λ is the path category k[zq] modulo the usual mesh relations. The fundamental group Z of Λ acts on Λ via the translation τ. Associated to F we have the push-down functor F λ : Λ -mod Λ -mod, see.3. In ZQ we find the Auslander-Reiten quiver A Q of kq as a full convex subquiver. The Auslander category Γ Q is the full subcategory of Λ which has the vertices of A Q as objects. Denote the inclusion of Γ Q into Λ by J. There is a natural equivalence R Q from Γ Q to the category of indecomposable kq-modules, kq -ind. We say that an object x Obj(Γ Q ) is projective if τx Obj(Γ Q ), dually x is injective if τ x Obj(Γ Q ), see.4. Associated to Γ Q we consider the N -graded category ˇΓ Q. It has the same objects as Γ Q but the morphisms of degree i are given by ˇΓ Q,i (x, y) = Γ Q (τ i x, y) if τ i x Obj(Γ Q ). We equip ˇΓ Q with the natural composition. The Gabriel quiver ǍQ of ˇΓ Q is obtained from A Q by inserting an additional (degree ) arrow x τx for each non-projective x Obj(Γ Q ), see Start modules. Let us write D for the usual duality Hom k (, k). We denote by J the functor which considers a Γ Q -module (trivially) as a Λ-module. Thus if apply the functor F λ J to the injective Γ Q -module DΓ Q (, x) we obtain a Λ-module. We call I Q := F λ J (DΓ Q (, x)). x Obj(Γ Q ) the start module for Λ associated to Q. Note that F λ J (DΓ Q (, x)) is isomorphic to a submodule of F λ J (DΓ Q (, τ x)) if x Obj(Γ Q ) is not injective, and F λ J (DΓ Q (, x)) is an injective Λ-module if x Obj(Γ Q ) is injective, see.4 and 3.. Consider E = x,y Obj(ΓQ )ˇΓ Q (x, y) as a (graded) associative k-algebra with multiplication induced from the composition of morphisms. Theorem. Let Λ be the preprojective algebra associated to a Dynkin quiver Q. Then E is isomorphic to End Λ (I Q ) op. In particular, the Gabriel quiver of End Λ (I Q ) op is identified with ǍQ as described above in.. Moreover I Q is rigid in the sense that Ext Λ (I Q, I Q ) =.

3 AUSLANDER ALGEBRAS AND INITIAL SEEDS FOR CLUSTER ALGEBRAS 3 The proof of this result is prepared in 3.4, 3.5, 3.6 and finished in Reduced expressions. Let π : Obj(Γ Q ) Q denote the map induced by the composition F J. We call a total ordering x() < x() <... < x(r) of the objects of Γ Q adapted (to Q) if Γ Q (x(i), x(j)) = for i < j. It is easy to find such orderings given that the quiver A Q is directed. We call a vertex i Q a source in Q if no arrow ends at i. In this case we denote by s i (Q) the quiver which is obtained from Q by reversing each arrow starting in i. Let i = (i, i,..., i r ) be a reduced expression for the longest element w W, that is w = s i s i s ir where r = Π. We say that i is adapted to Q if i is a source in Q and i k+ is a source in s ik s i (Q) if k < r. This is dual to the original definition in [6]. In this situation i := (π(x()),..., π(x(r))) is a reduced expression for the longest element w of the Weyl group W associated to Q, see.4 below. In fact, the adapted orderings of Obj(Γ Q ) correspond in this way bijectively to the reduced expressions for w which are adapted to Q, see [3, Theorem.5]. We set I(j, i) := F λ J DΓ Q (, x(j)) for j r. Thus, i provides us with a convenient way of labelling the direct summands of I Q..4. Let now g be a complex simple Lie algebra of type Q with the usual Serre generators e i, h i, f i for i Q. Thus the h i form a basis of the abelian subalgebra h and the e i resp. f i generate maximal nilpotent subalgebras n resp. n. The simple roots α i form a basis of the dual space h such that α i (h j ) = a i,j where (a i,j ) i,j Q is the Cartan matrix of Q. The fundamental weights (ϖ i ) i Q are the basis of h dual to the basis (h i ) i Q of h. The Weyl group W is the subgroup of GL(h ) which is generated by the reflections s i for i Q such that s i (α) = α α(h i ) α i for α h. This is a finite reflection group. Let now G be a complex simply connected simple algebraic group with Lie(G) = g. It has maximal unipotent subgroups N resp. N with Lie(N) = n resp. Lie(N ) = n, and the maximal torus H has Lie(H) = h. Moreover, we have standard embeddings ϕ i : SL G such that Moreover set exp(tf i ) = ϕ i ( t ) and exp(te i) = ϕ i ( t ) for i Q. η i (t) := ϕ i ( t t ) H for i Q and t C. Next, recall that N G (H)/H is canonically isomorphic to the Weyl group W defined above. In fact, it is possible to choose representatives w N G (H) for the elements w W such that s i = exp(f i ) exp( e i ) exp(f i ), uv = ū w if l(uv) = l(u) + l(v). We identify the weight lattice P = i Q Zϖ i with the group of multiplicative characters of H in such a way that η i (t) ϖ j = t δ i,j for i, j Q. If we write? λ for the character of H corresponding to the weight λ it follows that h w(λ) = ( w h w) λ for h H, λ P, w W.

4 4 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER.5. Cluster algebras. The coordinate ring of the affine base space C[N \G] consists of the functions f C[G] which are invariant under N, i.e. f(g) = f(ng) for all g G and n N. Now C[N \G] is naturally a G-module via gf(x) = f(xg) for g, x G. It is well-known that each irreducible highest weight G-module L(λ) can be realized as a direct summand of C[N \G] by taking L(λ) = {f C[N \G] f(hg) = h λ f(g) for h H, g G}. For each L(λ) we choose a highest weight vector u λ which we normalize by the condition u λ ( G ) =. Following [5] we define for each fundamental weight ϖ i generalized minors ϖi,w(ϖ i ) := w u ϖi L(ϖ i ) for any w W. In [4] it is shown in particular that the coordinate ring of the double Bruhat cell G e,w = B (B w B ) has the structure of an (upper) cluster algebra. Here, B and B are opposite Borel subgroups of G with B N and B N. For a reduced expression i = (i, i,..., i r ) for w which is adapted to Q and k [ n, ] [, r] we set v >k := s ir s ir s ik+ if k and v >k := w if k. Then, following [4] set (k, i) := ϖik,v >k (ϖ ik ) where we take i k = k for k [ n, ]. The (k, i) for k [ n, ] [, r] form an initial cluster for C[G e,w ]. There is also an easy algorithm to calculate from i the corresponding exchange matrix. Now set e(i) := {i [, r] x(i) Obj(Γ Q ) non-projective}. There is a closely related upper cluster algebra structure on the coordinate ring C[N], whose initial cluster is given by the restrictions to N of the functions (k, i) with k [ n, ] e(i) (see 4.4). The quiver associated to the corresponding exchange matrix is obtained from ǍQ by removing the arrows between injective vertices. See Section 4 for a more detailed discussion..6. Semicanonical basis. In [7] Lusztig introduces the semicanonical basis of the enveloping algebra U(n). Its elements are labelled naturally by the irreducible components of the preprojective varieties Λ v for v N Q. In [8] we started the study of the dual semicanonical basis which can be regarded as a basis of C[N]. In particular, we found that to (the isoclass of) a rigid Λ-module M there corresponds naturally an element ρ M of this basis. If we set { i if R Q (x(j)) θ : [, r] [ n, ] e(i), j = DkQ(, i), k if τ x(j) = x(k), we can state our second main result precisely: Theorem. For j [, r] we have (j, i) := (θ(j), i) = ρ I(j,i). The proof of this result is done after some preparation in Section Dualities. Denote by? the involution on Q with a a and a a for all a Q. This induces an anti-automorphism of Λ which we also denote by?. Thus we have a self-duality S : Λ -mod Λ -mod with SM(?) = DM(? ). Let us define for a reduced expression i = (i,..., i r ) for w which is adapted to Q r P(j, i) := F λ J Γ Q (x(j), ) and P Q := P(j, i). j=

5 AUSLANDER ALGEBRAS AND INITIAL SEEDS FOR CLUSTER ALGEBRAS 5 Then it is easy to see that P Q = S IQ op. Thus, Ext Λ (P Q, P Q ) = and End Λ (P Q ) = End Λ (I Q op) op. Now, Ǎ Q op is the Gabriel quiver of End Λ (I Q op) op, see Theorem, and it is not hard to see that ǍQop may be identified with Ǎop Q (recall that the same happens for Auslander-Reiten quivers: A Q op = A op Q ). So End Λ(I Q op) = End Λ (P Q ) op and End Λ (I Q ) op have the same Gabriel quiver ǍQ, but they are usually not isomorphic. On the other hand, S induces the canonical anti-automorphism? on Lusztig s algebra of constructible functions M = U(n) which commutes with the comultiplication, see [7, Section 3.4]. This yields by duality an automorphism ω of the coordinate ring C[N] which anti-commutes with the comultiplication. The corresponding anti-automorphism of N leaves the one-parameter subgroups C N, t exp(te i ) invariant. Note that i := (µ(i r ), µ(i r ),..., µ(i )) is a reduced expression for w which is adapted to Q op, see.3 for the definition of µ. We leave it to the reader to verify the following: see also.6. ρ P(j,i) = ω(ρ I(r+ j,i )) for j r,.8. Triangulated structures. In the appendix (Section 7) we point out that the category of injective Λ-modules is triangulated. This implies that the category of injective Λ-modules is also triangulated. This fact helps us to explain the unusual symmetries in the stable module categories over both categories by an old result of Freyd. Among other useful formulas we can recover the famous 6-periodicity for modules over the preprojective algebra.. The universal cover of a preprojective algebra.. Quiver categories. Let Q = (Q, Q, t, h) be a quiver with vertices Q, arrows Q and t, h: Q Q such that we have a: t(a) h(a) for each arrow a Q. A path in Q is a sequence of arrows a n a n a such that t(a i+ ) = h(a i ) for i =,,..., n. On Z Q we have the Ringel bilinear form v, w = i Q v(i)w(i) a Q v(t(a))w(h(a)). Let k be a field. Since we need to consider infinite coverings of preprojective algebras we have to consider k-categories rather than k-algebras. If Q is a quiver we denote by k[q] = kq the k-category which has Q as objects and with the morphism space kq(p, q) having the paths from p to q as a basis. The composition is naturally induced from the concatenation of paths... Conventions. If D is a k-category we denote by D -mod the category of finitely presented (covariant) k-functors D k -mod. These functors are also called left modules, see for example [5, Section.] for more details. Let G be a group of k-automorphisms of D which acts from the left on D. Then G acts naturally from the right on D -mod. If g G and M is a left D-module, then we write M g ( ) := M(g ) for the twisted module. For example, if x D we get for the projective module D(x, ) an isomorphism D g (x, ) = D(gx, ).

6 6 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER.3. Construction of Λ. Let Q = (Q, Q, t, h) be a (connected) Dynkin quiver, so the underlying graph Q is one of the following: A n : 3 (n ) (n ) n (n ) D n : 3 (n ) n 4 n E n : 3 (n ) (n ) n=6,7,8 n Define a (possibly trivial) involution µ on the vertices of Q by n + q in case A n, n q in case D n and (n odd and q n ), µ(q) = 6 q in case E 6 and q 5, q otherwise. Following [3] let ZQ be the quiver with vertices ZQ = Z Q and arrows ZQ = Z {Q Q } where t(i, a) = (i +, t(a)), t(i, a ) = (i, h(a)), h(i, α) = (i, h(a)), h(i, a ) = (i, t(a)). We think of Q op as a subquiver of ZQ via the embedding q (, q). The quiver ZQ admits a translation automorphism τ induced by τ(i, q) = (i +, q). Moreover we have a Nakayama permutation ˆν of ZQ. In order to define it we need the following auxiliary function: For any vertex q denote by l(q) the number of arrows pointing towards on the unique walk from to q. Now, ˆν is defined on the vertices by (p + q + l(µ(q)) l(q), µ(q)) in case A n, (p + n + l(µ(q)) l(q), µ(q)) in case D n, (p + q + + l(µ(q)) l(q), µ(q)) in case E 6 and q 5, ˆν(p, q) = (p + 5, µ(q)) in case E 6 and q = 6, (p + 8, µ(q)) in case E 7, (p + 4, µ(q)) in case E 8. So, ˆν is a translation reflection stabilizing the middle line in case A n and a translation in the cases D k, E 7 and E 8. At the beginning of 6. we show as an example part of ZQ for a quiver of type D 5, together with the action of ˆν. We consider the mesh ideal I in the category k[zq] which is generated by the elements (.) (i, a ) (i, a) (i, a) (i +, a ) for all (i, q) ZQ. a Q t(a)=q a Q h(a)=q Note that Λ := k[zq]/i is independent of the orientation of Q (up to relabelling of the vertices), and that I is invariant under the quiver automorphisms τ and ˆν. Thus we will use the same names for the induced automorphisms of Λ. Note further that the commuting automorphisms τ and ˆν of Λ are determined up to isomorphism by their effect on objects.

7 AUSLANDER ALGEBRAS AND INITIAL SEEDS FOR CLUSTER ALGEBRAS 7 The action of the group Z via τ on Λ induces self-equivalences? (i) of Λ -mod with M (i) ( ) := M(τ i ) for i Z. Moreover we obtain the covering functor F : Λ Λ which sends (i, q) to q. Associated to F we have the push-down F λ : Λ -mod Λ -mod with and the obvious effect on morphisms. (F λ M)(q) = i Z M(i, q).4. Auslander category. We define a function N : Q N by the property τ N(q) (, q) = ˆν(, µ(q)). This is well-defined by the construction of ˆν since µ is an involution on Q, see.3. The function N depends on the orientation of Q in case A n, D k+ and E 6. In any case we have N(q) + N(µ(q)) = h(q), where h(q) denotes the Coxeter number of Q. Define now Γ Q as the full subcategory of Λ which has the objects of the form (i, q) = τ i N(q)ˆν(, µ(q)) with q Q and i N(q). In other words, we take the objects which lie between the two copies of Q op in ZQ which are obtained via q (, q) resp. q ˆν(, q). This is the Auslander category of kq. Note that Γ Q depends on the orientation of Q. By construction we have a full embedding ι = ι Q : kq op Γ Q induced by q (, q). Moreover, Γ Q is canonically equivalent to the category of indecomposable kq-modules via the functor (.) R Q : Γ Q kq -ind, x Γ Q (ˆνι, x), where Γ Q (ˆνι, x) = Γ Q (, x) ˆνι: kq op k -mod is a contravariant functor which we have to interpret as a left kq-module. For example, R Q (, q) = DkQ(, q) and R Q (ˆν(, q)) = kq(q, ). Thus the Gabriel quiver of Γ Q (which is the full subquiver of ZQ with the vertices from Γ Q ) is the Auslander-Reiten quiver A Q of kq. Similarly, since we consider left modules, one obtains an equivalence Γ Q -inj kq op -mod = (kq -mod) op, I I ˆνι. Here, Γ Q -inj denotes the category of injective left Γ Q -modules. Note that the q Q parametrize the indecomposable projective-injective Γ Q -modules, namely we have DΓ Q (, (, q)) = Γ Q (ˆν(, q), ). Recall, that as an Auslander category Γ Q has dominant dimension at least [, VI.5]. This means by duality that each (indecomposable) injective Γ Q -module DΓ Q (, x) has a projective presentation P,x P,x DΓ Q (, x) with P,x and P,x also injective. 3. Start modules 3.. Adjoint functors. Let J : Γ Q Λ be the full embedding of locally bounded categories. Since Γ Q is convex in Λ we get an exact functor extension by { J : Γ Q -mod Λ M(x) if x Γ Q, -mod with (J M)(x) = else. For a Γ Q -module N and x Γ Q we have natural isomorphisms Hom ΓQ (N, DΓ Q (, x)) = DN(x) = Hom eλ (J N, D Λ(, x)).

8 8 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER We conclude that J has a right adjoint J ρ which is defined by being left exact and { J ρ DΓ Q (, x) if x Obj(Γ Q ), D Λ(, x) = else. The adjunction morphism ι M : J J ρ M M is injective, its co-kernel is co-generated by y Obj( e Λ)\Obj(Γ Q ) M(y). 3.. Lemma. Let x, y Obj(Γ Q ) and i Z. Then Hom eλ (J DΓ Q (, x), J DΓ (i) Q (, y)) = { Γ Q (τ i y, x) if i and τ i y Obj(Γ Q ), else. In the first case we have more generally J ρ (J DΓ (i) Q (, y)) = DΓ Q (, τ i y). Note, that J DΓ(, y) is the injective Γ Q -module whith socle concentrated in y, but seen as Λ-module, thus we may apply to it the translation functor? (i), see.3. Proof: The Λ-module M = J DΓ (i) Q (, y) has simple socle concentrated in the onedimensional space M(τ i y). If τ i y Obj(Γ Q ) then this does not belong to the support of J DΓ Q (, x). On the other hand, if i <, it is sufficient to show that there are no maps from an induced projective-injective module to M since Γ Q has dominant dimension, see.4. Now, = M(ˆν(, q)) = Hom eλ ( Λ(ˆν(, q), ), M) = Hom eλ (J Γ Q (ˆν(, q), ), M) with the first identity holding for i <. Thus, let i and τ i y Γ Q. In this case we have in Λ -mod an injective presentation M D Λ(, τ i x) j D Λ(, y j ) m(j) for certain y j Obj( Λ) \ Obj(Γ Q ). Our claim follows now from the construction of J ρ A graded category. We construct the N -graded category ˇΓ Q. It has the same objects as Γ Q, but the homogenous components are ˇΓ Q,i (x, y) := Γ Q (τ i x, y) if τ i x Obj(Γ Q ) and ˇΓ Q,i (x, ) = otherwise. The natural composition is given by { ˇΓ Q,j (y, z) ˇΓ Q,i (x, y) ˇΓ ψ (τ j φ) if τ i+j x Obj(Γ Q ), Q,i+j (x, z), (ψ φ), else. By construction, each morphism in ˇΓ Q, (x, ) factors through τx ˇΓ Q, (x, τx) if τx Obj(Γ Q ), otherwise ˇΓ Q, (x, ) =. Moreover we have τ n x τ x τx = τ n x ˇΓ Q,n (x, τ n x) if τ n x Obj(Γ Q ), where we consider on the left hand side τ i x ˇΓ Q, (τ i x, τ i x). Now, ˇΓ Q can be described easily by a graded quiver ǍQ. It has the same vertices and degree arrows as the Auslander- Reiten quiver A Q of kq (i.e. the quiver of Γ Q ) moreover there is a degree arrow t x : x τx for each x Obj(Γ Q ) with τx Obj(Γ Q ). The degree relations are the mesh relations for Γ Q. Moreover, each degree arrow a: x y with y not projective gives rise to a degree relation t y a (τa)t x. This has to be interpreted as a zero-relation if x is projective. A nice way to remember these relations is the following: For each arrow between to vertices which are not both injective there is a (generic) homogeneous length relation in the opposite direction, see also 6..

9 AUSLANDER ALGEBRAS AND INITIAL SEEDS FOR CLUSTER ALGEBRAS Dynkin quivers. Let us collect some basic facts about the representation theory of a Dynkin quiver Q. Define i q := dim DkQ(, q) and p q := dim kq(q, ) for q Q, the dimension vectors of the indecomposable injective and projective kq-modules, respectively. We have then for i N(q) (3.) Φ i i q = dim(τ i QDkQ(, q)) = dim(τ i N(q) Q kq(q, )) = Φ i N(q) p µ(q), where Φ denotes the Coxeter transformation and τ Q the Auslander-Reiten translate in kq -mod. Next, if, denotes the Ringel bilinear form of kq we have dim M, dim N = dim Hom Q (M, N) dim Ext Q (M, N), v, w = Φv, Φw = w, Φv, v, i q = v(q) thus v, q Q i q = v, where v := q Q v(q), see for example [8,.4] Lemma. Let τ i (, p) = (i, p) and τ j ˆν(, q) belong to Obj(Γ Q ), then (Φ i+j i p )(q) if i + j N(p), dim Γ Q (τ j ˆν(, q), τ i (, p)) = (Φ i j p q )(p) if i + j N(µ(q)), if i + j > min{n(p), N(µ(q))}. Note that the three cases are not exclusive, however they cover obviously all possibilities for (i, j) [, N(p)] [, N(µ(q))]. Proof: In the first case we have (3.) dim Γ Q (τ j ˆν(, q), τ i (, p)) = dim Γ Q (ˆν(, q), τ i+j (, q)). On the other hand, the equivalence R Q from Γ Q to the category of indecomposable kqmodules commutes with translations R Q (τ i (, q)) = τ i QDkQ(, q) for i N(q). Thus (3.) is equal to dim Hom Q (kq(q, ), τ i+j Q DkQ(, p)) = dim Hom Q(τ j Q kq(q, ), τ QDkQ(, i p)). Our claim follows now from (3.) since for a finite-dimensional kq-module M we have that Hom Q (kq(q, ), M) = M(q) = D Hom Q (M, DkQ(, q)). The second case is treated similarly. Finally we have Γ Q (τ j ˆν(, q), τ i (, p)) = Λ(τ j ˆν(, q), τ i (, p)) = Λ(ˆν(, q), τ i+j N(p)ˆν(, µ(p))). The last term vanishes obviously for i + j > N(p). Γ Q (τ j ˆν(, q), τ i (, p)) = for i + j > N(µ(q)). A similar argument shows that Lemma. If N(p) N(q) > j holds for some p, q Q, then Φ j i p, i q =. Proof: Since DkQ(, q) is injective we have Φ j i p, i q = dim Hom Q (τ j Q DkQ(, p), DkQ(, q)) = dim Λ(τ j (, p), (, q)). On the other hand for N(p) j > N(q) there is no path from τ j (, p) = (N(p) j, µ(p)) to (, q) in ZQ.

10 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER 3.5. Proposition. With the notation of. and 3.4 we have (3.3) (dim I Q )(µ(p)) = for p Q, and (3.4) dim E = q Q N(q) i= Proof: For (3.3) we observe first that q Q N(q) (i + )(Φ i i q )(p) i= (( N(q) + ) ( i + N(q) )) Φ i i q. dim F λ J DΓ Q (, (j, q))(µ(p)) = (Φ i i q )(p) for (j, q) Obj(Γ Q ) by 3.4. and the definition of the push-down F λ. Now (3.3) follows from the definition of I Q. For (3.4) we observe first that dim F λ J Γ Q (τ j ˆν(, q), ) = i=j N(µ(q)) k=j Φ k p q for τ j ˆν(, q) Obj(Γ Q ) again by 3.4. and the definition of the push-down. Now, by construction of E we have dim E = N(µ(q)) i dim F λ J Γ Q (τ j ˆν(, q), ) q Q = q Q = q Q i= N(µ(q)) i= N(q) i= i j= j= i j= N(µ(q)) k=i N(q) j k=i Φ k p q Φ k i q = q Q N(q) i= N(q)+ j=i j Φ i i q Proposition. Let Q be a Dynkin quiver. Then for v = N(q) q Q d= (i + )Φi i q holds v, v = N(p) (( ) ( )) N(p) + d + Φ d i p. p Q j= Proof: For convenience let us write N(p, q) := {,,..., N(p)} {,,..., N(q)} and N(p, q, ) := {(i, j) N(p, q) i j}, similarly N(p, q, <) := N(p, q) \ N(p, q, ). Now we have v, v = (i + )(j + ) Φ i i p, Φ j i q p,q Q = p,q Q p,q Q (i,j) N(p,q) (i,j) N(p,q, ) (i,j) N(p,q,<) (i + )(j + ) Φ i i p, Φ j i q (i + )(j + ) Φ j i q, Φ i+ i p

11 AUSLANDER ALGEBRAS AND INITIAL SEEDS FOR CLUSTER ALGEBRAS = p,q Q p,q Q (i,j) N(p,q, ) N(p) i=n(q)+ (i + ) Φ i i p, Φ j i q (i + )(N(q) + ) Φ i i p, Φ N(q)+ i q here, the second sum vanishes by Lemma 3.4., thus from 3.4 = p,q Q = p,q Q = p Q p,q Q (i,j) N(p,q, ) N(p) d= N(p) d= N(p) d= N(p) (i + ) Φ i j i p, i q min{n(p),n(q)+d} k=d (k + ) Φ d i p, k=d N(p) k=n(p)+d (k + ) Φ d i p, i q q Q i q (k + ) Φ d i p, i q. Here again, the second sum vanishes by Lemma Finally, Φ d i p, q Q i q = Φ d i p as observed at the beginning of 3.4, so our claim follows Proof of Theorem. The first claim follows directly from the construction of ˇΓ Q, Lemma 3. and the fact that Hom Λ (F λ N, F λ N) = i Z Hom eλ (N, N (i) ). For the second claim we have to show by [8, Lemma ] that dim I Q, dim I Q = dim E. This follows from the dimension formulas (3.3) and (3.4) of Proposition 3.5 together with Proposition The cluster algebras C[G e,w ] and C[N] In the next two sections we will use the setup from.4 and.5. following result, see for instance [, Section 4.4.3] We will need the 4.. Lemma. Let i = (i, i,..., i m ) be a reduced expression for some element w W, and L(λ) an irreducible representation of highest weight λ for G. If u λ is a highest weight vector for L(λ) then we have wu λ = f (bm) i m f (b ) i f (b ) i (u λ ) and f im ( wu λ ) =. Here b := λ(h i ), b k := (s ik s i s i (λ))(h ik ) for k m and f (b k) i k := b k! f b k i k U(g). For v L(λ), we write fj max (v) := (/m!)fj m(v) where m = max{p f p j (v) }. With this notation we could restate the equality of the lemma as wu λ = f max i m fi max fi max (u λ ).

12 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER 4.. The group G has Bruhat decompositions with respect to B and B, namely G = BūB = B vb. u W The intersection of two cells G u,v = (BūB) (B vb ) is called a double Bruhat cell. In particular taking u = e, the unit in W, and v = w we obtain G e,w = B (B w B ), the intersection of B with the big cell relative to B. By [4, Proposition.8], G e,w consists of all elements x of B such that ϖi,w (ϖ i )(x) for every i. This is a Zariski open subset of B, hence an algebraic variety of dimension n+r, where r = Π is the number of positive roots associated to the Dynkin type of G. Moreover, we see that the algebra of regular functions C[G e,w ] is obtained from C[B] by adjoining formal inverses to the functions ϖi,w (ϖ i ). On the other hand, N can be described as the subvariety of B given by the equations v W ϖi,ϖ i (x) =, ( i n). Hence the algebra C[N] is the quotient of C[B] by the ideal generated by the elements ( ϖi,ϖ i ) i=,,...,n In [], Fomin and Zelevinsky have introduced a transcendence basis F (i) of the field of rational functions C(G e,w ) consisting of certain generalized minors. In [4] Berenstein, Fomin and Zelevinsky have shown that each F (i) can be taken as the initial cluster for a natural cluster algebra structure on the ring C[G e,w ]. We are now going to recall their construction We add n additional letters i n,..., i at the beginning of i, where i j = j, and obtain an (r + n)-tuple (i n,..., i, i,..., i r ) = ( n,...,, i,..., i r ). For k [ n, ] [, r] let { k + r + if i l i k for all l > k, = min{l l > k and i l = i k } otherwise. Then k is called i-exchangeable if k and k + are in [, r]. Let e(i) [, r] be the set of i- exchangeable elements. One easily checks that e(i) contains r n elements. More precisely, the set of indices i k for k [, r] e(i) is exactly [, n] Next, one defines a quiver Ãi with set of vertices [ n, ] [, r]. Assume that k and l are vertices such that the following hold: k < l; {k, l} e(i). There is an arrow k l in Ãi if and only if k + = l, and there is an arrow l k if and only if l < k + < l + and a ik, i l =. Here, (a ij ) i,j n denotes the Cartan matrix of the root system of G. By definition these are all the arrows of Ãi Remark. If i is a reduced expression for w which is adapted to a Dynkin quiver Q, then it is easy to obtain Ãi from the Auslander-Reiten quiver A Q. See the examples in 6.5 and 6.6.

13 AUSLANDER ALGEBRAS AND INITIAL SEEDS FOR CLUSTER ALGEBRAS Now define an (r + n) (r n)-matrix B(i) = (b kl ) as follows. The columns of B(i) are indexed by the elements in e(i), and the rows by [ n, ] [, r]. Set if there is an arrow k l in Ãi, b kl = if there is an arrow l k in Ãi, otherwise For k [ n, ] [, r] one defines a generalized minor (k, i) as follows. For k [, r] set v >k = s ir s ir s ik+ and for k [ n, ] put v >k = w. Then define (k, i) = ϖ ik, v >k (ϖ ik ). Since s j (ϖ i ) = ϖ i for j i, it is easy to see that if k [, r] is not exchangeable then (k, i) = ϖik, ϖ ik. On the other hand for i [ n, ] we have ( i, i) = ϖi, w (ϖ i ). It is known [, Theorem.] that this collection of n + r minors is a transcendence basis of the field C(G e,w ), for any reduced expression i of w. By 4., we see that if we remove from this collection the n minors ϖi, ϖ i we obtain a transcendence basis of the field C(N) Let F be the field of rational functions over C in n + r independent variables x = (x n,..., x, x,..., x r ). Let A(i) C denote the upper cluster algebra associated to the seed ( x, B(i)), a subalgebra of F (see [4, Definition.6]). Here the non-exchangeable indices in [ n, ] [, r] label the generators of the coefficient group (see [4,.]). Berenstein, Fomin and Zelevinsky then show that the isomorphism of fields ϕ i from F to C(G e,w ) defined by ϕ i (x k ) = (k, i), (k [ n, ] [, r]), restricts to an algebra isomorphism A(i) C C[G e,w ], see [4, Theorem.]. Note that by varying the reduced expression i we obtain a priori several cluster algebra structures on C[G e,w ], but according to [4, Remark.4] all these structures coincide and give rise to the same cluster variables and clusters. Note also that in type A n, the upper cluster algebra A(i) C coincides with the cluster algebra A(i) C, see [4, Remark.8] Let x be the subset of x obtained by removing the variables indexed by the n nonexchangeable elements in [, r]. Let F be the field of rational functions over C in the r variables of x. Finally, let B(i) be the matrix obtained from B(i) by removing the rows labelled by the n non-exchangeable elements in [, r], and let A(i) C denote the upper cluster algebra associated to the seed ( x, B(i) ), a subalgebra of F. By 4., we see that the isomorphism of fields ϕ i : F C(N) defined by ϕ i (x k) = (k, i), (x k x ), restricts to an algebra isomorphism A(i) C C[N]. Clearly the same remarks as in apply to the cluster algebra structures of C[N].

14 4 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER 4.5. We shall now describe in representation theoretic terms the restriction to N of the regular function (k, i). Let L(ϖ i ) be the fundamental irreducible g-module with highest weight ϖ i. Fix a highest weight vector u ϖi. It is well-known that L(ϖ i ) can be realized in a canonical way as a subspace of the vector space C[N] by restricting the summand L(ϖ i ) of C[N \G] to C[N]. Thus, the highest weight vector u ϖi becomes identified with the constant regular function. Using this identification we obtain the following lemma: Lemma. For k [ n, ] [, r] we have { f max i (k, i) = r fi max r fi max k+ (u ϖik ) fi max r fi max (u ϖ ik f max i r if k [, r], ) if k [ n, ]. In particular for k [ n, ], the minor ( k, i) = ϖk,w (ϖ k ) is a lowest weight vector of L(ϖ k ). Proof: This follows from [5, p. 5 5] and 4. by restricting regular functions on G to the subgroup N, see also [6, p. 3]. 5. Cluster variables and semicanonical basis 5.. Let Q be a Dynkin quiver such that Q is the diagram of G. In this section we prove that if i is a reduced expression for w which is adapted to Q, the minors (k, i) C[N] coincide with certain dual semicanonical basis vectors coming from the injective modules of the Auslander algebra of CQ. In particular, this shows that the set of minors { (k, i)} depends only on Q, not on the choice of a particular expression i adapted to Q. 5.. Recall that by pushing down the injective modules of the Auslander algebra of CQ we obtain a set of indecomposable rigid modules I(j, i) over the preprojective algebra Λ, see. and.3. Since these modules are rigid, they have an open orbit in their module variety, and the closure of this orbit is an irreducible component. Therefore the module I(j, i) can be used to label an element ρ I(j,i) of the dual semicanonical basis (see [7], [8, Section 7.]) The proof of Theorem will make use of certain results of [7] that we shall now recall. Let I i denote the injective envelope of the simple Λ-module S i with dimension vector e i, thus I i = DΛ(, i). In [7] the fundamental g-module L(ϖ i ) was realized in terms of the lattice of submodules of I i. This goes as follows. Let M denote Lusztig s algebra of constructible functions on the varieties of finitedimensional Λ-modules, and let M be its graded Hopf dual, an algebra isomorphic to C[N]. For a finite-dimensional Λ-module X, let δ X denote the linear form on M obtained by evaluation at X. Then in the identification C[N] = M, the subspace L(ϖ i ) gets identified to the subspace of M spanned by the linear forms δ X where X runs over the lattice of submodules of I i, and one has explicit formulas for the action of the Chevalley generators of g on each vector δ X [7, Theorem 3]. In particular δ Ii = ρ Ii is a lowest weight vector of L(ϖ i ), and δ, where means the zero submodule of I i, is a highest weight vector. Let X be a submodule of I i. We have a short exact sequence of Λ-modules X I i p Y, where Y is determined up to isomorphism by the isomorphism class of X, see [7, Lemma ]. For j [, n] let m j denote the multiplicity of S j in the socle of Y, and let X j be the unique submodule of I i such that X X j I i and X j /X is isomorphic to S m j j. Thus X j is the pullback of p and the inclusion of S m j j into Y.

15 AUSLANDER ALGEBRAS AND INITIAL SEEDS FOR CLUSTER ALGEBRAS Lemma. With the above notation, we have f max j (δ X ) = f (m j) j (δ X ) = δ Xj. Proof: By [7, Theorem 3 (ii)], we have that fj k (δ X ) = δ X(k) f=(x=x() X(k)) where the integral is over the variety of flags f of submodules of I i such that X(s)/X(s ) is isomorphic to S j for all < s k. For k > m j this variety is empty by definition of m j, hence f k j (δ X) =. For k = m j, all flags f have their last step equal to X j. Moreover since X j /X = S m j j this variety is isomorphic to the variety of complete flags in C m j, whose Euler characteristic is m j!. Hence f m j j (δ X ) = m j!δ Xj, as claimed Lemma. Let k [ n, ] e(i) and j = θ (k). Then, if i = i k, the module I(j, i) is a submodule of I i and in L(ϖ i ) there holds { f max i δ I(j,i) = r fi max r fi max k+ (δ ) if k e(i), fi max r fi max (δ ) if k [ n, ]. f max i r Proof: Recall that we defined the function θ in.5. If k = i [ n, ], then I(j, i) = I i. On the other hand in this case the product fi max r fi max r f max maps δ to the lowest weight vector of L(ϖ i ), that is, to δ Ii, as required. If k e(i), then R Q (x(j)) belongs to the τ-orbit of R Q (x(k)) by definition of θ, therefore to the τ-orbit of DkQ(, i) see.3. It follows that I(j, i) is a submodule of I i, see.. More precisely, j = θ (k) = min{l [k +, r] i l = i}. We conclude that in L(ϖ i ) holds fi max j fi max k+ (δ ) = δ by 5.4, since the socle of I i is S i. Now consider for l [j, r] the Λ-submodule Ĩx(j)( l, i) of D Λ(, x(j)) with Ĩ x(j) ( l, i)(x(m)) := { D Λ(x(m), x(j)) if m l, else. With I x(j) ( l, i) := F λ Ĩ x(j) we see that I x(j) ( r, i) = I(j, i) is a submodule of I i since F λ D Λ(, x(j)) = I i. Using Lemma 5.4 we conclude that in L(ϖ i ) we have fi max l (δ Ix(j) ( l,i)) = δ Ix(j) ( l,i) for l [j, r]. This yields that δ I(j,i) is the extremal vector of L(ϖ i ) with weight s ir s ir s ik+ (ϖ i ), and the lemma follows. We can now finish the proof of Theorem. Using Lemma 4.5. and Lemma 5.5 we obtain that (k, i) = δ I(j,i). But since I(j, i) is rigid, its orbit is open and we have δ I(j,i) = ρ I(j,i). 6. Examples Our running example will be the Dynkin quiver Q of type D 5 with the following orientation: i

16 6 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER 6.. Γ Q for D 5. We show the quiver A Q of Γ Q : ˆν(,5) ˆν(,) ˆν(,3) ˆν(,) (3,4) ˆν(,4) (,) (,3) (,) (,4) (,5) (,) (,3) (,) (,4) (,5) (,) (,3) The isomorphism classes of the indecomposable representations of Q correspond to the vertices of A Q. Each such representation is uniquely determined by its dimension vector: The dimension vector of the injective Γ Q -module DΓ Q (, (, q)) is given by the q-components of the corresponding dimension vectors. The dimension vector of DΓ Q (, τ j (, q)) is obtained from this by translation and cut-off. Here we show for example dim DΓ Q (, (, 3)) and dim DΓ Q (, τ(, 3)): (,) (,4) 6.. The category ˇΓ Q. We display here the quiver of the (graded) category ˇΓ Q associated to a quiver of type D 5 with the same orientation as above. The arrows of degree one are dotted. (4,4) (3,) (3,3) (3,) (3,4) (,5) (,) (,3) (,) (,4) (,5) (,) (,3) (,) (,4) (,5) (,) (,3) Recall that also the relations are easy to read off: For each arrow α: x y we have the corresponding mesh relation (.) from y to x if α is of degree. Otherwise, if x is not an injective vertex, (i.e. here if x {(, ), (, 3), (, 5)}) there are one or two paths of length (and degree ) from y to x. The first case occurs when y is a projective vertex (,) (,4)

17 AUSLANDER ALGEBRAS AND INITIAL SEEDS FOR CLUSTER ALGEBRAS 7 (i.e. here if y {(3, ), (3, 3), (, 5)}) and the unique path of length from y to x is a zero-relation, otherwise the two paths form a commutativity relation A projective-injective ˇΓ Q -module. We display for ˇΓ Q as in 6. the projective module ˇΓ Q ((, 5), ). Since (, 5) is an injective vertex of A Q we have ˇΓ Q ((, 5), ) = DˇΓ Q (, (, µ(5)). Moreover, the projective modules ˇΓ Q ((j, 5), ) for j are easily found as submodules of ˇΓ Q ((, 5), ). Each entry (n, q) represents a basis vector which corresponds to a (graded) simple composition factor of this type. The arrows indicate as usual the action of ˇΓ Q. (,5) (,3) (,5) (,) (,4) (,3) (,) (,3) (,3) (,5) (,) (,4) (,) (,3) (,) (,3) (,5) (,) (,4) (,) (,4) 6.4. Dimensions. We include the result of some calculations of d(q) := dim End Λ (I Q ) for specific orientations. This can be done quite easily on a computer using the formula (3.4). A n : ( ) ( ) ( ) ( ) n n n n (n ) n d(a n ) = n, D n : E n : (n ) (n ) n 4 3 n (n ) 5 ( ) n d(d n ) = ( n 4 ) if n = 6, d(e n ) = 33 if n = 7, 74 if n = 8. ( n 3 ) + ( n ),

18 8 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER 6.5. An adapted ordering on Obj(Γ Q ): x(9) x() x(8) x(7) x(6) x(5) x(3) x(4) x() x() x() x(8) x(9) x(7) x(6) x(5) x(3) x(4) According to.3 we obtain for w the following reduced expression which is adapted to Q: i = (4,,, 3, 5, 4,,, 3, 5, 4,,, 3, 5, 4,, 3, 4, ) 6.6. The quiver Ãi. For the adapted expression i from 6.5 we obtain à i = The exchangeable vertices are {,, 3, 4, 5, 6, 7, 8, 9,,,, 3, 4, 6} x() x() 4 7. Appendix: Using the triangulated structure 7.. Proposition (Freyd/Krause). [4, Appendix B] Let D be a triangulated k-category with suspension functor Σ. (a) The category D -mod is a Frobenius category. Thus the stable category D-mod is triangulated with suspension functor Ω D, the inverse of Heller s loop functor. (b) In D-mod we have a functorial isomorphism M Σ = Ω 3 D M. 7.. Remarks. (a) We may consider D as a D-D-bimodule, i.e. a functor D op D k -mod. Similarly, DD with DD(a, b) := Hom k (D(b, a), k) is a D-D-bimodule. (b) Suppose that D admits Auslander-Reiten triangles with translate τ : D D. In this case we set ν := Στ. Then the Auslander-Reiten formula D(x, Σy) = Hom k (D(y, τx), k) may be interpreted as an isomorphism of bimodules (7.) DD = D ν where D ν (a, b) := D(a, νb). We conclude that N M := DD D M = D ν D M = M ν is a Nakayama functor for D -mod and ν the corresponding Nakayama automorphism for D, see for example [3, Section ]. In this situation we will write M (i) := M τ i. (c) If moreover D is locally bounded, then D-mod admits Auslander-Reiten triangles with translation τ D = Ω DN. With Proposition 7. we obtain the functorial isomorphisms (7.) M ( ) = τd Ω D M and τ 3 D M = M Στ 3 (in D-mod).

19 AUSLANDER ALGEBRAS AND INITIAL SEEDS FOR CLUSTER ALGEBRAS 9 In fact, we have τ D Ω D = Ω 3 N =? Σν =? τ and τ 3 D = Ω6 D N 3 =? Σ ν 3 =? Στ 3. (d) If we consider in this context right modules (i.e. contravariant functors), we get in mod-d a functorial isomorphism M Σ = Ω 3 D M and consequently M () = τd Ω D M and τ 3 DM = M Στ 3 (in mod-d) Derived categories. It follows from Happel s description [, I.5.6] of the derived category D b (kq op ) := D b (kq op -mod) that we have a natural equivalence Λ -inj = D b (kq op ). In particular, Λ -inj is a triangulated category which admits Auslander-Reiten triangles. The suspension functor resp. the Auslander-Reiten translate are ΣI = I ˆντ resp. τi = I τ, see.. In our situation, these functors are up to isomorphism determined by their effect on objects. We conclude that we have an isomorphism of bimodules (7.3) Λˆν = D Λ, see 7. (b). In order to state our next result we introduce Γ Q, the full subcategory of the Auslander category Γ Q which contains all objects except those of the form ˆν(, q) for q Q. We call Γ Q the stable Auslander category of kq Proposition. The category Λ is isomorphic to the repetitive category Γ Q of the stable Auslander category of kq. Proof: Recall that by definition the objects of Γ Q are of the form (z, x) with z Z and x Obj(Γ Q ) and we have Γ Q (x, x ) if z = z, Γ Q ((z, x), (z, x )) = DΓ Q (x, x ) if z = z, else. Here we define the dual Γ Q -Γ Q -bimodule DΓ Q by DΓ Q (x, y) := Hom k (Γ Q (y, x), k), compare 7. (a). Now, by the bimodule isomorphism (7.3) we see that the assignation induces an isomorphism Γ Q Λ. (z, (i, q)) τ iˆν z (, q) 7.5. Conclusions. (a) In our situation we note that in D b (kq) = D b (kq op ) we have an isomorphism of functors (7.4) Σ = τ h(q) where h(q) is the Coxeter number of Q. In fact, both functors coincide on objects as one easily verifies in Λ. In our quiver situation this is sufficient. From (7.) we obtain immediately the remarkable functorial isomorphism (7.5) τ 6 e Λ = M (h(q) 6) in Λ-mod = D b (kq)-mod. (b) The action of the infinite cyclic group τ on Λ provides us with a Galois covering F : Λ Λ,

20 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER see for example [4]. We conclude that Λ -inj is the orbit category of Λ -inj modulo the induced action of τ. Now, Λ -inj = D b (kq) is a triangulated category, and the hypothesis of [3] are obviously fulfilled. Thus Λ -inj is also a triangulated category with Auslander- Reiten triangles and the corresponding translation τ is the identity. Note moreover that the induced suspension is isomorphic to the corresponding Nakayama automorphism, i.e. Σ = ν. By applying (7.) to D = (Λ -inj) op we conclude that τ 3 ΛM = M ν and τ 6 ΛM = M holds functorially in Λ-mod. In case ν is just a translation, see.3, we even have τ 3 Λ M = M. This is our interpretation of the proof for the 6-periodicity of τ Λ in []. Even more directly by (7.) we conclude that τ Λ Ω Λ M = M functorially. This means that the triangulated category Λ-mod is of Calabi-Yau dimension. (c) Since Λ = Γ Q we have by Happel s Theorem [, II.4] Λ-mod = D b (Γ Q -mod) as triangulated categories. If we consider the push-down functor F λ : Λ-mod Λ-mod associated to the Galois covering F : Λ Λ the subcategory of Λ-modules of the first kind (i.e. the subcategory of objects which are isomorphic to a push-down) is equivalent to the orbit category D b (Γ Q -mod)/ τσ via the above identifications and (7.), see []. Here, Σ resp. τ are the suspension resp. the Auslander-Reiten translation in D b (Γ Q -mod). Now, for a Dynkin quiver Q with Coxeter number h(q) 6 the algebras Γ Q are (quasi-) tilted. Thus, in these cases we find Λ-mod = D b (Γ Q )/ Στ is a cluster category in the sense of [7] Remark. Let H be a (basic, connected) finite-dimensional hereditary k-algebra of finite representation type. Then H is a species of type A G in the sense of Dlab and Ringel, see [9]. In this case we may also study the stable Auslander category Γ H. The same argument as in 7.3 and 7.4 shows that Γ H -inj = D b (H op ). The Auslander-Reiten translate in D b (H op ) induces an automorphism τ of Γ H. Thus we may consider the Galois covering Γ H Γ H / τ. So we are tempted to consider Γ H / τ as the preprojective algebra of H. However, τ is now in general not determined by its effect on objects since Out k (H) = Aut k (H)/ Inn(H) is possibly non-trivial. Thus we are (for the moment) unable to compare Γ H / τ with the possible choices for the preprojective algebra of H in the sense of Dlab and Ringel []. Anyway, if we denote by H the (unoriented) diagram of H then we find the list which we present in Figure of interest due to its similarity with the cluster types of C[N]. In the case of Coxeter number c( H ) = 6 we display the diagram of a canonical tubular algebra following Lenzing [5] and the corresponding extended affine root system in the sense of Saito [9]. In the case of G there are two different diagrams of canonical algebras which produce derived equivalent algebras (for adequate choices of bimodules), the corresponding two root systems are isomorphic up to the marking. Note moreover, that in this case our previous calculations (7.5) predict that in the stable module category of the repetitive algebra Γ H -mod the Auslander-Reiten translate should be 6-periodic. This is in fact the case for all (tubular) algebras which are derived equivalent to a canonical algebra with a diagram from our list.

21 AUSLANDER ALGEBRAS AND INITIAL SEEDS FOR CLUSTER ALGEBRAS c( H ) H Γ H is tilted of 3 A A 4 A 3 A 3 B B 5 A 4 D 6 Figure. c( H ) H Γ H is tilted of root system 6 A 5 Ẽ (,) 8 (,) (,) B 3 F (,) 4 (,) (,) C 3 F (,) 4 D 4 Ẽ (,) 6 (3,) (,3) G G (3,) (,3) (3,) G (,3) Acknowledgement. We like to thank Henning Krause for drawing our attention to the result on functor categories over triangulated categories. We also thank Markus Reineke for pointing out the reference for the material in.3. References [] M. Auslander and I. Reiten, DTr-periodic modules and functors, Representation theory of algebras (Cocoyoc, 994), CMS Conf. Proc., vol. 8, Amer. Math. Soc., Providence, RI, 996, pp [] M. Auslander, I. Reiten, and S. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 997, Corrected reprint of the 995 original. [3] R. Bédard, On the spanning vectors of Lusztig cones, Represent. Theory 4 (), (electronic). [4] A. Berenstein, S. Fomin, and A. Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 6 (5), no., 5. [5] A. Berenstein and A. Zelevinsky, Total positivity in Schubert varieties, Comment. Math. Helv. 7 (997), no., [6], Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math. 43 (), no., [7] A. Buan, R. Marsh, M. Reineke, I. Reiten, and G. Todorov, Tilting theory and cluster combinatorics, 36 pages, to appear in Adv. Math., arxiv:math.rt/454. [8] W. Crawley-Boevey, On the exceptional fibres of Kleinian singularities, Amer. J. Math. (), no. 5, [9] V. Dlab and C.M. Ringel, Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc. 6 (976), no. 73, v+57. [], The preprojective algebra of a modulated graph, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 979), Lecture Notes in Math., vol. 83, Springer, Berlin, 98, pp [] P. Dowbor and A. Skowroński, Galois coverings of representation-infinite algebras, Comment. Math. Helv. 6 (987), no.,

22 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER [] S. Fomin and A. Zelevinsky, Double Bruhat cells and total positivity, J. Amer. Math. Soc. (999), no., [3] P. Gabriel, Auslander-Reiten sequences and representation-finite algebras, Representation theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 979), Springer, Berlin, 98, pp. 7. [4], The universal cover of a representation-finite algebra, Representations of algebras (Puebla, 98), Springer, Berlin-New York, 98, pp [5] P. Gabriel and A. V. Roĭter, Representations of finite-dimensional algebras, Algebra, VIII, Springer, Berlin, 99, With a chapter by B. Keller, pp. 77. [6] Ch. Geiß, B. Leclerc, and J. Schröer, Rigid modules over preprojective algebras, 3 pages, arxiv:math.rt/5334. [7], Verma modules and preprojective algebras, 4 pages, arxiv:math.rt/43. [8], Semicanonical bases and preprojective algebras, Ann. Sci. École Norm. Sup. 38 (5), 93 53, arxiv:math.rt/4448. [9] Ch. Geiss and J. Schröer, Extension-orthogonal components of preprojective varieties, Trans. Amer. Math. Soc. 357 (5), no. 5, (electronic). [] D. Happel, Triangulated categories in the representation theory of finite dimensional algebras, Cambridge University Press, 988, LMS Lecture Note Series 9. [] O. Iyama, Higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories, 4 pages, arxiv:math.rt/475. [] A. Joseph, Quantum groups and their primitive ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 995. [3] B. Keller, On triangulated orbit categories, pages, arxiv:math.rt/5334. [4] H. Krause, Cohomological quotients and smashing localizations, Amer. J. Math., to appear, 3. [5] H. Lenzing, A K-theoretic study of canonical algebras, Representation theory of algebras (Cocoyoc, 994), CMS Conf. Proc., vol. 8, Amer. Math. Soc., Providence, RI, 996, pp [6] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (99), no., [7], Semicanonical bases arising from enveloping algebras, Adv. Math. 5 (), no., [8] C.M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Math., vol. 99, Springer- Verlag, Berlin-New York, 984. [9] K. Saito, Extended affine root systems. I. Coxeter transformations, Publ. Res. Inst. Math. Sci. (985), no., Instituto de Matemáticas, UNAM, Ciudad Universitaria, 45 México D.F., Mexico address: christof@math.unam.mx Laboratoire LMNO, Université de Caen, F-43 Caen Cedex, France address: leclerc@math.unicaen.fr Department of Pure Mathematics, University of Leeds, Leeds LS 9JT, UK address: jschroer@maths.leeds.ac.uk