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arXiv:1304

.0667v2

[math.RT]15Apr2013

CLASSIFYING -TILTING MODULES OVER PREPROJECTIVEALGEBRAS OF DYNKIN TYPE

YUYA MIZUNO

Abstract. We study support -tilting modules over preprojective algebras of Dynkintype. We classify basic support -tilting modules by giving a bijection with elementsin the corresponding Weyl groups. Moreover we show that they are in bijection withthe set of torsion classes, the set of torsion-free classes and other important objects inrepresentation theory of preprojective algebras. We also study g-matrices of support -tilting modules, which show terms of minimal projective presentations of indecomposabledirect summands. We give an explicit description of g-matrices and prove that cones

given by g-matrices coincide with chambers of the associated root systems.

Contents

Introduction 11. Preliminaries 41.1. Preprojective algebras 41.2. Support -tilting modules 41.3. Torsion classes 62. Support -tilting ideals and the Weyl group 7

2.1. Support -tilting ideals 72.2. Bijection between support -tilting modules and the Weyl group 122.3. Partial orders of support -tilting modules and the Weyl group 153. g-matrices and cones 174. Further connections 22References 24

Introduction

Preprojective algebras first appeared in the work of Gelfand-Ponomarev [GP]. Sincethen, they have been one of the important objects in the representation theory of algebras.

Preprojective algebras appear also in many branches of mathematics. One of the notableexamples is the study of quantum groups, where preprojective algebras play central roles inLusztigs Lagrangian construction of semicanonical basis [L1, L2] and subsequent Geiss-Leclerc-Schroers construction of cluster monomials of certain types of cluster algebras[GLS1, GLS3]. In this paper we study preprojective algebras from a new perspective.

In [BIRS] (see also [IR]), the authors studied preprojective algebras of non-Dynkinquivers via tilting theory. As an application, they succeeded to construct a large class of2-Calabi-Yau categories which have close connections with cluster algebras. On the otherhand, in Dynkin cases (i.e. the underlying graph of a quiver is An (n 1), Dn (n 4) and

The author is supported by Grant-in-Aid for JSPS Fellowships No.23.5593.

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En (n = 6, 7, 8)), the preprojective algebras are selfinjective, so that all tilting modules aretrivial (i.e. projective). In this paper, instead of tilting modules, we will study support

-tilting modules (Definition 1.3) over the preprojective algebras.The notion of support -tilting modules was introduced in [AIR], as a generalizationof tilting modules. It is shown that there are deep connections between support -tiltingmodules, torsion classes, silting complexes and cluster tilting objects. Support -tiltingmodules also have nicer mutation properties than tilting modules (Definition-Theorem1.5). Moreover, certain support -tilting modules over selfinjective algebras provide tilt-ing complexes [M]. It is therefore fruitful to investigate these remarkable modules forpreprojective algebras and they are also interesting in their own right. To explain ourresults, we give the following set-up.

Let Q be a finite connected acyclic quiver with the set Q0 = {1, . . . , n} of vertices, the preprojective algebra of Q and Ii the two-sided ideal of generated by 1 ei, whereei is an idempotent of corresponding to i Q0. If Q is non-Dynkin, then Ii is a tilting

module. These ideals Ii are quite useful as shown in [IR, BIRS, AIRT]. On the otherhand, if Q is Dynkin, then Ii is never a tilting module. Instead, we show that Ii is asupport -tilting module (Lemma 2.3) and this observation is crucial in this paper. Westudy ideals of of the form Ii1Ii2 Iik for some k 0 and i1, . . . , ik Q0, and denoteby I1, . . . , I n the set of such ideals.

Our main goal is to give the following bijections which provide a complete descriptionof support -tilting modules over preprojective algebras of Dynkin type.

Theorem 0.1 (Theorem 2.21). LetQ be a Dynkin quiver with the set Q0 = {1, . . . , n} ofvertices and the preprojective algebra of Q. There are bijections between the followingobjects.

(a) The Weyl group WQ associated with Q.

(b) The set I1, . . . , I n.(c) The set s-tilt of isomorphism classes of basic support -tilting -modules.(d) The set s-tiltop of isomorphism classes of basic support -tilting op-modules.

It is known that support -tilting modules have a natural structure of partially orderedset (Definition 1.4). On the other hand, it is well-known that the Weyl group also has astructure of partial orders. We relate partial orders of these two objects as follows.

Theorem 0.2 (Theorem 2.30). The bijection WQ s-tilt in Theorem 0.1 gives anisomorphism of partially ordered sets (WQ, L) and (s-tilt, )

op, where L is the leftorder on WQ (Definition 2.25).

We also study the g-matrix (Definition 3.1) of support -tilting module T, which shows

terms of minimal projective presentations of each indecomposable direct summand ofT. By g-matrices, we define cones, which admit geometric realizations of the simplicialcomplex of support -tilting modules. We give a complete description of g-matrices interms of reflection transformations and show that their cones coincide with chambers ofthe associated root system. The results are summarized as follows, where e1, . . . , en is thecanonical basis ofZn, C0 := {a1e1 + + anen | ai R>0} and

: WQ GLn(Z) is thecontragradient of the geometric representation (Definition 3.5).

Theorem 0.3 (Theorem 3.9). Let Q be a Dynkin quiver with the set Q0 = {1, . . . , n} ofvertices and be the preprojective algebra of Q.

(a) The set of g-matrices of support -tilting -modules coincides with (WQ).
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CLASSIFYING -TILTING MODULES OVER PREPROJECTIVE ALGEBRAS OF DYNKIN TYPE 3

(b) For any w WQ, we have

C(w) = w(C0),

where C(w) denotes the cone of the support -tilting module Iw (Definition 3.1)and w :=

(w). In particular, cones of basic support -tilting -modules givechambers of the associated root system.

The above results give several important applications. Among others we conclude thatthere are only finitely many basic support -tilting -modules. This fact yields bijectionsbetween basic support -tilting -modules, torsion classes in mod and torsion-free classesin mod by a result in [IJ]. This seems to be interesting by itself since almost all prepro-jective algebras are of wild representation type. Then we can give complete descriptionsof these two classes in terms of WQ (Proposition 4.2).

As another application by combining with results in [AIR], [KY, BY] and [ORT], weextend bijections of Theorem 0.1 as follows.

Corollary 0.4 (Corollary 4.1). Objects of Theorem 0.1 are in bijection with the followingobjects.

(e) The set of torsion classes in mod.(f) The set of torsion-free classes in mod.(g) The set of isomorphism classes of basic two-term silting complexes in Kb(proj).(h) The set of intermediate bounded co-t-structures in Kb(proj) with respect to the

standard co-t-structure.(i) The set of intermediate bounded t-structures in Db(mod) with length heart with

respect to the standard t-structure.(j) The set of isomorphism classes of two-term simple-minded collections inDb(mod).

(k) The set of quotient closed subcategories in modKQ.

(l) The set of subclosed subcategories inmodKQ.

We now describe the organization of this paper.In section 1, we recall definitions of preprojective algebras, support -tilting modules

and torsion classes.In section 2.1, we introduce support -tilting ideals (Definition 2.1), which play key

roles in this paper. Moreover, we prove that any element of I1, . . . , I n is a basic support-tilting ideal (Theorem 2.2). In section 2.2, we prove that WQ correspond bijectively withsupport -tilting modules (Theorem 2.21). We give a description of mutations of support-tilting modules in terms of WQ. We also determine the annihilators of all support -tilting modules. In section 2.3, we compare partial orders of s-tilt and WQ. We give aclose connection between a partial order of WQ and that of s-tilt (Theorem 2.30).

In section 3, we deal with g-matrices of support -tilting modules. We give an explicitdescription of them in terms of linear transformations. Furthermore, we study cones ofsupport -tilting modules and show that they coincide with chambers of the associatedroot system.

In section 4, we show Corollary 0.4. In particular, we give complete descriptions oftorsion classes and torsion-free classes (Proposition 4.2). Furthermore, we explain a rela-tionship between our results and other works.

Notations. Let K be an algebraically closed field and we denote by D := HomK(, K).By a finite dimensional algebra , we mean a basic finite dimensional algebra over K. Bya module, we mean a right module unless stated otherwise. We denote by mod the
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category of finitely generated -modules and by proj the category of finitely generatedprojective -modules. The composition gf means first f, then g. We denote by Q0 the

set of vertices and by Q1 the set of arrows of a quiver Q. For a quiver a Q1, we denoteby s(a) and e(a) the start and end vertices of a respectively. For X mod, we denoteby SubX (respectively, FacX) the subcategory of mod whose objects are submodules(respectively, factor modules) of finite direct sums of copies of X. We denote by addMthe subcategory ofmod consisting of direct summands of finite direct sums of copies ofM mod.

Acknowledgement. First and foremost, the author would like to thank Osamu Iyamafor his support and patient guidance. He would like to express his gratitude to SteffenOppermann, who kindly explain results of his paper. He is grateful to Joseph Grant,Laurent Demonet and Dong Yang for answering questions and helpful comments. Hethanks Kota Yamaura, Takahide Adachi and Gustavo Jasso for their help and stimulating

discussions.

1. Preliminaries

1.1. Preprojective algebras. In this subsection, we recall definitions and some proper-ties of preprojective algebras. We refer to [BGL, DR1, DR2, Ri] for basic properties andbackground information.

Definition 1.1. Let Q be a finite connected acyclic quiver with vertices Q0 = {1, . . . , n}.The preprojective algebra associated to Q is the algebra

= KQ/

aQ1

(aa aa)

where Q is the double quiver of Q, which is obtained from Q by adding for each arrowa : i j in Q1 an arrow a

: i j pointing in the opposite direction.

If Q is a non-Dynkin quiver, the bounded derived category Db(fd) of the categoryfd of finite dimensional modules are 2-Calabi-Yau (2-CY for short), that is, there is afunctorial isomorphism

DHomDb(fd)(X, Y)= HomDb(fd )(Y, X[2]).

We refer to [AR1, Boc, CB] and see [GLS2] for the detailed proofs. On the other hand,the algebra is finite-dimensional selfinjective if Q is a Dynkin quiver.

In this paper, we use the two-sided ideal Ii of generated by 1 ei, where ei is aprimitive idempotent of for i Q0. It is easy to see that the ideal has the following

property, where we denote by Si := /Ii.

Lemma 1.2. Let Q be a finite connected acyclic quiver, the preprojective algebra ofQ and X mod. Then XIi is maximal amongst submodules Y of X such that anycomposition factor of X/Y is isomorphic to a simple module Si.

1.2. Support -tilting modules. We give definitions of support -tilting modules. Werefer to [AIR] for the details about support -tilting modules, and see [ARS] for definitionsand properties of the functor .

Let be a finite dimensional algebra.

Definition 1.3. (a) We call X in mod -rigid if Hom(X,X) = 0.

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(b) We call X in mod -tilting (respectively, almost complete -tilting) if X is -rigid and |X| = || (respectively, |X| = || 1), where |X| denotes the number of

non-isomorphic indecomposable direct summands of X.(c) We call X in mod support -tilting if there exists an idempotent e of such thatX is a -tilting (/e)-module.

(d) We call a pair (X, P) of X mod and P proj -rigid if X is -rigid andHom(P, X) = 0.

(e) We call a -rigid pair (X, P) a support -tilting (respectively, almost completesupport -tilting) pair if |X| + |P| = || (respectively, |X| + |P| = || 1).

Note that (X, P) is a -rigid (respectively, support -tilting) pair for if and onlyif X is a -rigid (respectively, -tilting) (/e)-module, where e is an idempotent of such that addP = adde [AIR, Proposition 2.3]. Moreover, if (X, P) and (X, P) aresupport -tilting pairs for , then we get addP = addP. Thus, support -tilting moduleX determines support -tilting pair (X, P) uniquely and we can identify support -tiltingmodules with support -tilting pairs.

We call (X, P) basic if X and P are basic, and we say that (X, P) is a direct summandof (X, P) if X is a direct summand of X and P is a direct summand of P. We denoteby s-tilt the set of isomorphism classes of basic support -tilting -modules.

Next we see some properties of support -tilting modules. The set of support -tiltingmodules has a natural partial order defined as follows.

Definition 1.4. Let be a finite dimensional algebra. For T, T s-tilt, we write

T T

ifFacT FacT. Then gives a partial order on s-tilt [AIR, Theorem 2.18].

Then we give the following results, which play significant roles in this paper.

Definition-Theorem 1.5. [AIR, Theorem 2.18, 2.28 and 2.30] Let be a finite dimen-sional algebra. Then

(i) any basic almost support -tilting pair (U, Q) is a direct summand of exactly twobasic support -tilting pairs (T, P) and (T, P). Moreover we have T > T orT < T.

Under the above setting, let X be an indecomposable -module satisfying either T =U X or P = Q X. We write (T, P) = (X,0)(T, P) if X is a direct summand ofT and (T, P) = (0,X)(T, P) if X is a direct summand of P and we say that (T

, P) isa mutation of (T, P). In particular, we say that (T, P) is a left mutation (respectively,right mutation) of (T, P) if T > T (respectively, if T < T) and write (respectively,

+

). By (i), exactly one of the left mutation or right mutation occurs.Now, assume that X is a direct summand of T and T = U X. In this case, forsimplicity, we write a left mutation T = X(T) and a right mutation T

= +X(T).Moreover the following conditions are equivalent.

(ii) (a) T > T.(b) X / FacU.

Furthermore, if T is a -tilting -module, then the following property holds.

(iii) Let Xf

U Y 0 be an exact sequence, where f is a minimal left (addU)-approximation. Then one of the following holds.(a) Y = 0 and X(T)

= U is a basic support -tilting -module.

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(b) Y = 0 and Y is a direct sum of copies of an indecomposable -module Y1,which is not in addT, and X(T)

= Y1 U is a basic -tilting -module.

Finally, we define the following useful quiver.

Definition 1.6. Let be a finite dimensional algebra. We define the support -tiltingquiver H(s-tilt) as follows.

The set of vertices is s-tilt. Draw an arrow from T to T if T is a left mutation of T.

The following theorem relates H(s-tilt) with partially orders of s-tilt.

Theorem 1.7. [AIR, Corollary 2.34] The support -tilting quiverH(s-tilt) is the Hassequiver of the partially ordered set s-tilt.

1.3. Torsion classes. The notion of torsion classes is quite important in the representa-tion theory. As we will see below, support -tilting modules have a close relationship withtorsion classes.

Definition 1.8. Let be a finite dimensional algebra and T a full subcategory in mod.

(a) We call T torsion class (respectively, torsion-free class) if it is closed under factormodules (respectively, submodules) and extensions.

(b) We call T a contravariantly finite subcategory in mod if for each X mod,

there is a map f : T X with T T such that Hom(T, T)

f Hom(T

, X) issurjective for all T T. Dually, a covariantly finite subcategory is defined. Wecall T a functorially finite subcategory if it is contravariantly and covariantly finite.

We denote by f-tors the set of functorially finite torsion classes in mod.Then, we have the following result, where we denote by P(T) the direct sum of one copy

of each of the indecomposable Ext-projective objects in T(i.e. {X mod | Ext1(X, T) =0}) up to isomorphism.

Theorem 1.9. [AIR] For a finite dimensional algebra , there are mutually inverse bi-jections

s-tilt f-tors

given by s-tilt T FacT f-tors and f-tors T P(T) s-tilt.

This explains why support -tilting modules are useful to investigate functorially finite

torsion classes. We have the following much stronger result under the assumption that|s-tilt| < .

Theorem 1.10. [IJ] For a finite dimensional algebra , the following conditions areequivalent.

(a) |s-tilt| < .(b) Any torsion class inmod is functorially finite.(c) Any torsion-free class in mod is functorially finite.

Therefore, if there are only finitely many basic support -tilting modules, then Theorem1.9 give a bijection between s-tilt and torsion classes ofmod.

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2. Support -tilting ideals and the Weyl group

The aim of this section is to give a complete description of all support -tilting modules

over preprojective algebras of Dynkin type and study their several properties.Throughout this section, unless otherwise specified, let Q be a Dynkin quiver with

Q0 = {1, . . . , n}, the preprojective algebra of Q and Ii := (1 ei) for i Q0. Wedenote by I1, . . . , I n the set of ideals of which can be written as

Ii1Ii2 Iik

for some k 0 and i1, . . . , ik Q0.

2.1. Support -tilting ideals. In this subsection, we introduce support -tilting ideals.We will show that elements of I1, . . . , I n are support -tilting ideals. This fact plays akey role in this paper.

We start with the following definition.

Definition 2.1. We call a two-sided ideal I of support -tilting if I is a left support-tilting -module and a right support -tilting -module.

The aim of this subsection is to prove the following result.

Theorem 2.2. Any T I1, . . . , I n is a basic support -tilting ideal of .

First we recall some exact sequences for later use (see [BBK]). For any i Q0, we cantake a minimal projective resolution of Si := /Ii as follows

eip2 GG

aQ1,e(a)=i

es(a)p1GG ei

p0 GG Si GG 0.

Since Imp1 = eiIi, we have the following exact sequences

(1) 0 GG eiIi GG ei

p0 GG Si GG 0.

(2) eip2GG

aQ1,e(a)=i

es(a) GG eiIi GG 0.

As a direct sum of (1) and 0 (1 ei)id (1 ei) 0, we have

(3) 0 GG Ii

00 id

GG

(p0 0)GG Si GG 0.

Moreover, we have Kerp2 = S(i), where : Q0 Q0 is a Nakayama permutation of (i.e. D(e(i)) = ei) [BBK, Proposition 4.2]. Therefore, we have the following exactsequences

(4) 0 GG S(i) GG eip2 GG

aQ1,e(a)=i

es(a).

Then we give the following lemma.

Lemma 2.3. Ii is a basic support -tilting ideal of for any i Q0.

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Proof. It is clear that Ii is basic. We will show that Ii is a right support -tilting -module; the proof for a left -module is similar. If ei is a simple module, then is a

simple algebra since is a preprojective algebra. Thus, we have Ii = 0 and it is a support-tilting module.Assume that ei is not a simple module and hence eiIi = 0. Since is selfinjective, ej

has a simple socle for any j Q0 and these simple modules are mutually non-isomorphic.Therefore, eiIi is not isomorphic to ej for any j Q0. Hence, we obtain |Ii| = ||. Thus,it is enough to show that Hom(Ii, (Ii)) = 0.

Applying the functor := D Hom(, ) to (2), we have the following exact sequence

(5) 0 GG (eiIi) GG (ei)p2 GG

aQ1,e(a)=i

(es(a))

by the definition of the AR translation. On the other hand, applying the functor to

(4), we have the following exact sequence

(6) 0 GG (S(i)) GG (ei)p2GG

aQ1,e(a)=i

(es(a)).

Since is selfinjective, we have (S(i)) = (Soc(ei)) = Top(ei) = Si. Therefore,comparing (5) with (6), we get (Ii) = (eiIi) = Si. Since Si / add(Top(Ii)), we haveHom(Ii, (Ii)) = 0.

Next we show the following property.

Lemma 2.4. LetT be a support -tilting -module. For a simple op-module S, at least

one of the statements T S = 0 and Tor1 (T, S) = 0 holds.

Proof. By [AIR, Proposition 2.5], we can take a minimal projective presentation P1 P0 T 0 such that P0 and P1 do not have a common summand. Thus, at least one ofthe statement P0 S = 0 and P1 S = 0 holds. Since we have Tor

1 (T, S)

= P1 Sand T S = P0 S, the assertion follows.

From the above lemma, we have the following corollary.

Corollary 2.5. LetT be a support -tilting-module. IfT Ii = T, then we haveT Ii =T Ii.

Proof. By the assumption T Ii = T and Lemma 1.2, we have T Si = 0. Then, byLemma 2.4, we obtain Tor1 (T, Si) = 0. By applying the functor T to the exact

sequence (3), we get the following exact sequence

0 = Tor1 (T, Si) T Ii T = T.

Hence we have T Ii = Im(T Ii T) = T Ii.

Next we give the following easy observation.

Lemma 2.6. LetI be a two-sided ideal of . For a primitive idempotent ei with i Q0,eiI is either indecomposable or zero.

Proof. Since is selfinjective, ei has a simple socle. Thus the submodule eiI of ei hasa simple socle or zero.

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Moreover, we need the following lemmas.

Lemma 2.7. LetI be a two-sided ideal of . Then there exists a surjective algebra map

End(I), (I x x I).

In particular, for an idempotent ei, i Q0, we have a surjective map

ei(1 ei) Hom((1 ei)I, eiI).

Proof. We have to show that, for any f End(I), there exists such that f = () :I I. Since is selfinjective, we have the following commutative diagram

0 GG I GG

f

h

0 GG I GG ,

where is the canonical inclusion and h : is the -module map. Put h(1) = .Then h is given by left multiplication with . Thus, we obtain f(x) = h(x) = x for anyx I and we have proved our claim. The second statement easily follows from the firstone.

Lemma 2.8. Let T be a support -tilting ideal of . If IiT = T, then we have eiT /Fac((1 ei)T). In particular, T has a left mutation

eiT

(T).

Proof. By Lemma 2.6 and the assumption, eiT = 0 is indecomposable. We assume thateiT Fac((1 ei)T). Then, there exist an integer d > 0 and a surjective map

{(1 ei)T}d eiT.

By Lemma 2.7, there exist m (1 m d) such that

dm=1

eim(1 ei) (1 ei)T = eiT.

On the other hand, since Ii = (1 ei), we get

ei(1 ei) (1 ei)T = eiIiT.

Thus we have

eiT =d

m=1

eim(1 ei)T eiIiT,

which is a contradiction. The second statement follows from Definition-Theorem 1.5 (ii).

Moreover we need to prove the following.

Lemma 2.9. LetT be a support -tilting ideal of . For the map p2 in the sequence (2),the map

ei Tp2TGG

aQ1,e(a)=i

es(a) T

is a left (add((1 ei)T))-approximation.

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Proof. We have to show that

Hom

( aQ1,e(a)=i

es(a)

T, (1 ei)T)

(p2T,(1ei)T) Hom

(e

i

T, (1 e

i)T) 0.

Let E := End(T). We have

Hom(ei T, (1 ei)T) = Hom(ei, Hom(T, (1 ei)T))= Hom(ei, (1 ei)E),

and similarly

Hom(

aQ1,e(a)=i

es(a) T, (1 ei)T) = Hom(

aQ1,e(a)=i

es(a), (1 ei)E).

Then, from their functoriality, we write the above surjection by

Hom( aQ1,e(a)=i

es(a), (1 ei)E)(p2,(1ei)E) Hom(ei, (1 ei)E) 0.

On the other hand, by Lemma 2.7, there exists a surjective algebra map

g : E,

which admits -module map. Then, we have the following commutative diagram

Hom(

aQ1,e(a)=i

es(a), (1 ei))(p2,(1ei)) GG GG

(aQ1,e(a)=i es(a), g)

Hom(ei, (1 ei))

(ei, g)

Hom(

aQ1,e(a)=i

es(a), (1 ei)E)(p2,(1ei)E) GG Hom(ei, (1 ei)E).

It is clear that Hom(

aQ1,e(a)=ies(a), g) and Hom(ei, g) are surjective since

es(a) and ei are projective. Moreover, by (4), Hom(p2, (1 ei)) is also surjective.Consequently, Hom(p2, (1 ei)E) is also surjective. Then the statement follows from theadditivity of the functors.

Now we apply the above results to the following key proposition.

Proposition 2.10. Let T I1, . . . , I n and assume that T is a basic support -tiltingideal of . Then IiT is a basic support -tilting -module.

Proof. There is nothing to show if IiT = T. Assume that IiT = T. Then, by Lemma 2.8,there exists a left mutation eiT(T).

Now let e be an idempotent of such that T is a -tilting (/e)-module. By applyingthe functor T to (2), we get exact sequence

ei Tp2T

aQ1,e(a)=i

es(a) TT eiIi T 0

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of (/e)-modules. By applying Corollary 2.5 to the support -tilting left -module T,we have eiIi T = eiIiT. Moreover, Lemma 2.6 implies that eiIiT is indecomposable.

On the other hand, from Lemma 2.9, p2 T is a left (add((1 ei)T))-approximation.If eiIiT = 0, then p2 T is clearly left minimal. Assume eiIiT = 0. Since ejT has asimple socle for any j Q0 and these simple modules are mutually non-isomorphic, Tis a radical map and hence p2 T is left minimal. Then, applying Definition-Theorem1.5 (iii), we conclude that eiT(T)

= eiIiT (1 ei)T = IiT is a basic support -tilting(/e)-module.

Now we are ready to prove Theorem 2.2.

Proof of Theorem 2.2. We use induction on the number of products of the ideal Ii, i Q0.By Lemma 2.3, Ii is a basic support -tilting ideal of for any i Q0. Assume thatIi1 Iik is a basic support -tilting ideal for i1, . . . , ik Q0 and k > 1. By Proposition2.10, Ii0(Ii1 Iik) is a basic support -tilting -module for i0 Q0. Similarly we canshow that it is a basic support -tilting left -module.

At the end of this subsection, we give the following lemma for later use.

Lemma 2.11. LetT I1, . . . , I n. If IiT = T, then there is a left mutation of T :

eiT(T)= IiT.

Proof. This follows from Lemma 2.8 and Proposition 2.10.

Now we show examples of support -tilting modules and their mutations.

Example 2.12. (a) Let be the preprojective algebra of type A2. In this case, H(s-tilt)is given as follows.

12 1

I1GG 0 1I2

22

12 21

I2bb

I1 22

0 0

2 21 I2GG 2 0

I1

bb

Here we represent modules by their radical filtrations and we write a direct sum X Yby X Y. For example, 12 1 denotes the support -tilting pair (e1 S1, 0) and 0 1 denotes(S1, P2). Note that Ii is given by left multiplications.

(b) Let be the preprojective algebra of type A3. In this case, H(s-tilt) is given asfollows

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12 YUYA MIZUNO

1 20 2 1

99

&&

12 1 23 2 1

TT

99

10 2 1

00

12 13 2 1

UU

1 22 31 23 2 1

cc

22

12 313 2 1

UU

&&

20 2 1

11

0 0 1

&&

2 23 2 1

UU

&&

1 2 32 31 23 2 1

gg

''

GG 1 32 31 23 2 1

bb

22

22 13 2

3 2 1

TT

3 0 1

cc

11

0 2 0

GG0 0 0

313 2 1

UU

2 32 13 23 2 1

bb

22

331 2

3 2 1

TT

&&

23 2 0

cc

3 0 0

hh

2 33 2 0

UU

99

32 3 23 2 1

TT

99

3

3 2 0

cc

33 2

3 2 1

UU

In these two examples, H(s-tilt) consists of a finite connected component. We willshow that this is the case for preprojective algebras of Dynkin type in the sequel. Thus,all support -tilting modules can be obtained by mutations from .

2.2. Bijection between support -tilting modules and the Weyl group. In theprevious subsection, we have shown that elements ofI1, . . . , I n are basic support -tiltingideals. In this subsection, we will show that any basic support -tilting -module is givenas an element of I1, . . . , I n. Moreover, we show that there exists a bijection betweenthe Weyl group and basic support -tilting -modules, which implies that there are only

finitely many basic support -tilting -modules.First we recall the following definition.

Definition 2.13. Let Q be a finite connected acyclic quiver with vertices Q0 = {1, . . . , n}.The Coxeter group WQ associated to Q is defined by the generators s1, . . . , sn and relations.

s2i = 1, sisj = sjsi if there is no arrow between i and j in Q, sisjsi = sjsisj if there is precisely one arrow between i and j in Q.

Each element w WQ can be written in the form w = si1 sik . If k is minimal amongall such expressions for w, then k is called the length of w and we denote by l(w) = k. Inthis case, we call si1 sik a reduced expression of w.

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Note that WQ is the Weyl group if Q is Dynkin. Then, we give the following importantresult.

Theorem 2.14. There exists a bijection WQ I1, . . . , I n. It is given by w Iw =Ii1Ii2 Iik for any reduced expression w = si1 sik .

The statement is given in [BIRS, III.1.9] for non-Dynkin quivers. The same statementalso holds for Dynkin quivers. For the convenience of the reader, we give a complete proofof Theorem 2.14 for Dynkin quivers.

For a proof, we provide the following set-up.

Notation 2.15. Let Q be an extended Dynkin quiver obtained from Q by adding a vertex0 (i.e. Q0 = {0} Q0) and the associated arrows. We denote by the completion of theassociated preprojective algebra and

Ii :=

(1 ei)

for i

Q. For each w WQ, let

Iw := Ii1 Iik , where w = si1 sik is a reduced expression (this is well-defined [BIRS]).Note that we have = /e0, Ii = Ii/e0.

Proof of Theorem 2.14. One can check that the map

WQ I1, . . . , I n, w = si1 sik Iw := Ii1Ii2 Iik

is well-defined and surjective by [BIRS, Theorem III.1.9], where the assumption that Q isnon-Dynkin is not used. We only have to show the injectivity.

Since i1, , ik Q0, we have sij = s0 for any 1 j k. Thus we have

Iw = Ii1 Iik = (

Ii1/e0) (

Iik/e0) = (

Ii1

Iik)/e0 =

Iw/e0.

Therefore, if Iw = Iw for w, w WQ, then we have Iw/e0 = Iw/e0. Since s0 doesnot appear in reduced expressions of w and w, we have e0 Iw and e0 Iw. Hence, we

obtain Iw = Iw and conclude w = w by [BIRS, Theorem III.1.9]. Thus, for any w WQ, we have support -tilting module Iw. Then we give an explicit

description of mutations using this property.Let (Iw, Pw) be a basic support -tilting pair, where Pw is a basic projective -module

which is determined from Iw (subsection 1.2). Because ei has simple socle S(i), where : Q0 Q0 is a Nakayama permutation of , if eiIw = 0, then e(i) addPw and hence

Pw =

iQ0,eiIw=0

e(i). Then, for i Q0, we write a mutation

i(Iw, Pw) := (eiIw,0)(Iw, Pw) if eiIw = 0(0,e(i))(Iw, Pw) if eiIw = 0.Then we have the following result.

Theorem 2.16. For any w WQ and i Q0, a mutation of basic support -tilting-module Iw is given as follows:

i(Iw, Pw) = (Isiw, Psiw).

In particular, WQ acts transitively and freely on s-tilt by si(Iw, Pw) := i(Iw, Pw).

Proof. By Theorem 2.2 and 2.14, Iw and Isiw are basic support -tilting modules, whichare not isomorphic.

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14 YUYA MIZUNO

Let w = si1 sik be a reduced expression. If l(siw) > l(w), then siw = sisi1 sik isa reduced expression. Hence, we get (1 ei)Isiw = (1 ei)IiIw = (1 ei)Iw. Therefore

we have(Isiw, Psiw)

= (eiIiIw (1 ei)Iw, Pw) if eiIiIw = 0

((1 ei)Iw, Pw e(i)) if eiIiIw = 0.

Thus (Iw, Pw) and (Isiw, Psiw) have a common almost complete support -tilting pair((1 ei)Iw, Pw) as a direct summand.

On the other hand, if l(siw) < l(w), then u := siw satisfies l(u) < l(siu). Similarly, itshows that (1 ei)Isiw = (1 ei)Iw and (Iw, Pw) = (Isiu, Psiu) and (Isiw, Psiw) = (Iu, Pu)have a common almost complete support -tilting pair as a direct summand. Therefore,by Definition-Theorem 1.5 (i), the first statement follows. The second one follows fromTheorem 2.14.

Using the above result, we give the following crucial lemma.

Lemma 2.17. The support -tilting quiverH(s-tilt) has a finite connected component.Proof. Let C be the connected component of H(s-tilt) containing . By Theorem 2.2,Iw is a basic support -tilting -module for any w WQ. We will show that {Iw | w WQ}gives the set of vertices in C.

We show that every Iw belongs to C by induction on the length of w WQ. Assumethat {Iw | w WQ with l(w) k} belong to C. Take w WQ with l(w) = k + 1 anddecompose w = siw

, where l(w) = k. Then, we have Iw = IiIw = Iw by Theorem 2.14and we get eiIw

(Iw) = IiIw by Lemma 2.11. Thus, Iw belongs to C from Theorem 1.7

and hence our claim follows inductively.Moreover we show that any vertex in C has a form Iw for some w WQ. This follows

the fact that the neighbours of Iw are given as Isiw for i Q by Theorem 2.16. Therefore

we obtain the assertion.

Now we recall the following useful result.

Proposition 2.18. [AIR, Corollary 2.38] If H(s-tilt) has a finite connected componentC, then C = H(s-tilt).

As a conclusion, we can obtain the following statement.

Theorem 2.19. Any basic support -tilting -module is isomorphic to an element ofI1, . . . , I n. In particular, it is isomorphic to a support -tilting ideal of .

Proof. By Lemma 2.17, H(s-tilt) has a finite connected component, and Corollary 2.18implies that it coincides with H(s-tilt). Thus, any support -tilting -module is givenby Iw for some w WQ. In particular, it is a support -tilting ideal from Theorem 2.2.

Furthermore, we give the following lemma.

Lemma 2.20. If right ideals T and U are isomorphic as -modules, then T = U.

Proof. Assume that f : T U is an isomorphism of -modules. Since is selfinjective,we have the following commutative diagram

0 GG T GG

f

h

0 GG U GG ,

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where is the canonical inclusion and h : is a -module map. Put h(1) = . Bythe commutative diagram, we have f(T) = h(T) and hence we get U = h(T) = T T.

By the similar argument for f1

, we can show that T U. Thus we conclude thatT = U.

Now we are ready to prove one of the main result of this paper.

Theorem 2.21. There exist bijections between the isomorphism classes of basic support-tilting -modules, basic support -tilting op-modules and the elements of WQ.

Proof. We will give a bijection between s-tilt and WQ. A bijection s-tiltop and WQ

can be given similarly.By Theorem 2.2 and 2.14, we have a map WQ w Iw s-tilt. This map is

surjective since any support -tilting -module is isomorphic to Iw for some w WQ byTheorem 2.19. Moreover it is injective by Theorem 2.14 and Lemma 2.20. Thus we get

the conclusion.

Remark 2.22. It is known that the order of WQ is finite and the explicit number isdetermined by the type of underlying graph of Q as follows [H, Section 2.11].

Q An Dn E6 E7 E8(n + 1)! 2n1n! 51840 2903040 696729600

Finally, we give an application related to annihilators of s-tilt using the above result.Note that tilting modules are nothing but faithful support -tilting modules [AIR, Propo-sition 2.2]. Hence, it is important to investigate annihilators of support -tilting modules.We can completely determine them for preprojective algebras of Dynkin type. Here, wedenote by ann X := { | X = 0} the annihilator of X mod.

Corollary 2.23. We have ann Iw = Iw1w0, where w0 is the longest element in WQ.Thus, we have a bijection

s-tilt s-tilt, T annT.

For a proof, we recall the following useful result.

Proposition 2.24. [ORT, Proposition 6.4] Let w0 be the longest element in WQ. Thenwe have the following isomorphisms

(a) DIw = /Iw1w0 inmodop.

(b) DIw = /Iw0w1 inmod.

Then we give a proof of Corollary 2.23.

Proof of Corollary 2.23. It is clear that the right annihilator of Iw coincides with the leftannihilator of DIw. On the other hand, we have isomorphism DIw = /Iw1w0 of

op-modules by Proposition 2.24. Thus, we get the first statement. The second one followsfrom Theorem 2.14.

2.3. Partial orders of support -tilting modules and the Weyl group. In thissubsection, we study a relationship between a partial order of support -tilting modulesand that ofWQ. In particular, we show that the bijection WQ s-tilt given in Theorem2.21 induces an isomorphism as partially ordered sets.

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16 YUYA MIZUNO

Definition 2.25. Let u, w WQ. We write u L w if there exist sik , . . . , si1 such thatw = sik . . . si1u and l(sij . . . si1u) = l(u) + j for 0 j k. Clearly L gives a partial

order on WQ, and we call L the left order (it is also called weak order). We denote byH(WQ, L) the Hasse quiver of left order on WQ.

The following assertion is immediate from the definition of the left order.

Lemma 2.26. An arrow in H(WQ, L) is given by

w siw (if l(w) > l(siw)),siw w (if l(w) < l(siw))

for any w WQ and i Q0.

Example 2.27. (a) Let Q be a quiver of type A2. Then H(WQ, L) is given as follows.

132

}}

312oo

123 321

}}

213

231oo

(b) Let Q be a quiver of type A3. Then H(WQ, L) is given as follows.

2341

vv

2314

}}

3241

vv

3214

vv

2134

3124

}}

2431

vv

3421

}}

2413

vv

1234 1324oo 2143

}}

3412

vv

4231

}}

4321oo

3142

vv

1243

1342

4213

vv

4312

}}

4123

vv

1423

4132

vv

1432

Next we give the following observation.

Lemma 2.28. Letw WQ and i Q0.

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(i) If l(w) < l(siw), then IiIw = Isiw Iw and we have a left mutation i (Iw, Pw).

(ii) If l(w) > l(siw), then IiIw = Iw Isiw and we have a right mutation +i (Iw, Pw).

Proof. Let w = si1 sik be a reduced expression. Ifl(siw) > l(w), then siw = sisi1 sikis a reduced expression. Thus, we have IiIw = Isiw Iw by Theorem 2.14 and, by Lemma2.11, we get a left mutation i (Iw, Pw).

On the other hand, if l(siw) < l(w), then u := siw satisfies l(u) < l(siu). Then,similarly, we have IiIu = Isiu = Iw Iu = Isiw. In particular, we have IiIw = I

2i Iu =

IiIu = Iw and left mutation i (Iu, Pu) = (Iw, Pw), or equivalently

+i (Iw, Pw) = (Iu, Pu) =

(Isiw, Psiw).

Summarizing previous results, we give the following equivalent conditions that a left (orright) mutation occurs.

Corollary 2.29. Letw WQ and i Q0.

(a) The following conditions are equivalent :(i) l(w) < l(siw).

(ii) IiIw = Iw.(iii) We have a left mutation i (Iw, Pw).

(b) The following conditions are equivalent :(i) l(w) > l(siw).

(ii) IiIw = Iw.(iii) We have a right mutation +i (Iw, Pw).

Proof. It is enough to show (a).By Lemma 2.11 and 2.28, (i) implies (ii) and (ii) implies (iii). Assume that we have a

left mutation i (Iw, Pw). If l(w) > l(siw), then we get a right mutation +i (Iw, Pw) by

Lemma 2.28, which contradicts Definition-Theorem 1.5. Thus (iii) implies (i).

Finally we give the following results.

Theorem 2.30. The bijection in WQ s-tilt in Theorem 2.21 gives an isomorphismof partially ordered sets (WQ, L) and (s-tilt, )

op.

Proof. It is enough to show that H(WQ, L) coincides with H(s-tilt)op since the Hasse

quivers determine the partial orders.By Theorem 1.7 and Corollary 2.29, arrows starting from Iw are given by Iw IiIw in

H(s-tilt), where i Q0 satisfies l(w) < l(siw).On the other hand, Lemma 2.26 implies that arrows to w are given by siw w

in H(WQ, L), where si satisfies l(w) < l(siw). Therefore H(s-tilt)op coincides with

H(WQ, L) and the statement follows.

We remark that the Bruhat order on WQ coincides with the reverse inclusion relationon I1, In [ORT, Lemma 6.5].

3. g-matrices and cones

In this section, we study g-vectors [DK] (which is also called index [AR2, P]) and g-matrices of support -tilting modules. We give a complete description of g-matrices interms of reflection transformations. Moreover, we study cones defined by g-matrices, whichgive geometric realizations of support -tilting modules, and we show that they coincidewith chambers of the associated root system.

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18 YUYA MIZUNO

Throughout this section, unless otherwise specified, let Q be a Dynkin quiver withvertices Q0 = {1, . . . , n}, be the preprojective algebra of Q and Ii = (1 ei) for

i Q0.First, we define g-vectors in our setting. We refer to [AIR, section 5] for a backgroundof this notion (see also section 4).

Definition 3.1. By Krull-Schmidt theorem, we identify the set of isomorphism classes ofprojective -modules with submonoid Zn0 of the free abelian group Z

n.For a -module X, take a minimal projective presentation

P1(X) GG P0(X) GG X GG 0

and let g(X) = (g1(X), , gn(X))t := P0(X)P1(X) Z

n, where t denotes the transposematrix. Then, for w WQ and i Q0, we define a g-vector by

Z

n

g

i

(w) = g(eiIw) if eiIw = 0

(0, . . . , 0, (i)1, 0, , 0)t if eiIw = 0,

where : Q0 Q0 is a Nakayama permutation of (i.e. D(e(i)) = ei)). It is shown in

[AIR, Theorem 5.1] that g1(w), , gn(w) form a basis ofZn. Then we define a g-matrixof support -tilting -module Iw, or simply of w, by

g(w) := (g1(w), , gn(w)) GLn(Z)

and its cone byC(w) := {a1g

1(w) + + angn(w) | ai R>0}.

Next, we define g-vectors for extended Dynkin cases. We follow Notation 2.15. ByTheorem 2.14, there exists a bijection

WQ

I0, I1, . . . , In, w Iw = Ii1 Ii2 Iik ,where w = si1 sik is a reduced expression of w. Note that

Iw is a tilting -modules ofprojective dimension at most one [IR] (hence eiIw = 0). For w WQ and i Q0, take aminimal projective presentation of the -module eiIw

P1(eiIw) GG P0(eiIw) GG eiIw GG 0.Then we define a g-vector by

gi(w) := (g0(eiIw), , gn(eiIw))t := P0(eiIw) P1(eiIw) Zn+1,and we define a g-matrix by

g(w) := (g0(w), , gn(w)) GLn+1(Z).

We use these notations in this section and, for simplicity, denote by () := .Then, we give the following results.

Proposition 3.2. Let w WQ and i Q0. Take a minimal projective presentation of

the -module eiIw(7) 0 GG P1

f1 GG P0f0 GG eiIw GG 0.

Then, applying the functor to (7), we have the following exact sequence

0 GG 1(ei/eiIw) GG P1f1 GG P0

f0 GG eiIw GG 0 ,

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where 1 := Hom(D, ). Moreover, if eiIw = 0, then the sequence

P1f1

GG P0f0

GG eiIw GG 0is a minimal projective presentation of the -module eiIw.

Proof. Since ei = e0 and = /e0, we have eiIw = eiIw/eiIwe0 = eiIw/eie0 =eiIw. Then, by applying the functor to (7), we have the following exact sequence

0 GG Tor1 (eiIw, ) g GG P1 f1 GG P0 f0GG eiIw GG 0 .

On the other hand, we have the following exact sequence

0 GG ei

Iw GG ei

GG ei

/ei

Iw GG 0.

Then, we obtain -module isomorphisms

Tor1 (eiIw, ) = D Ext1(eiIw, D) ([CE])

= D Ext2(ei/eiIw, D)= Hom(D, ei/eiIw) (2-CY property)= Hom(D, ei/eiIw) (ei/eiIw = ei/eiIw)= 1(ei/eiIw).

Now assume that eiIw = 0. Then, clearly 1(ei/eiIw) is not projective and hence g

is a radical map. Moreover, it is clear that f0 is a projective cover by the assumption off0. Thus we obtain the conclusion.

Remark 3.3. It is known that, for every non-projetive indecomposable module X overa preprojective algebra of Dynkin type, its third syzygy 3X is isomorphic to 1(X)[AR1, BBK]. This property appears in the above proposition.

Using the above proposition, we give the following key result.

Proposition 3.4. For any i Q0 and w WQ, the g-vector gi(w) equals the image of

gi(w) under the projectionZn+1 Zn with respect to the i-th entries (i = 1, , n).

Proof. Take a minimal projective presentation of the -module eiIw(8) 0 GG P1

f1 GG P0f0 GG ei

Iw GG 0.

We decompose P0 = P

0 (e0) and P1 = P1 (e0)m, where (e0) (respectively,(e0)m) is a maximal direct summand ofP0 (respectively, P1) which belongs to add(e0).

For gi(w) = (g0(eiIw), g1(eiIw), , gn(eiIw))t, it is clear that (g1(eiIw), . . . , gn(eiIw))t isdetermined by P0 and P

1. We will show that it coincides with g

i(w).By Proposition 3.2, applying to (8), we have the following exact sequence

(9) 0 GG 1(ei/eiIw) GG P1f1 GG P0

f0 GG eiIw GG 0.

(i) Assume that P0 = 0. It implies that eiIw = 0 and, by Proposition 3.2, the se-quence (9) gives a minimal projective presentation of eiIw. Thus g

i(w) coincides with

(g1(ei

Iw), . . . , gn(ei

Iw))

t.

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20 YUYA MIZUNO

(ii) Assume that P0 = 0. It implies that eiIw = 0 and gi(w) = (0, . . . , 0,

(i)

1, 0, , 0)t

by the definition. On the other hand, by (9), we have P1= 1(ei) = e(i) and

hence P1= e(i). Thus gi(w) = (g0(eiIw), 0, . . . , 0, (i)1, 0, , 0)t and we obtain the

conclusion.

Next we introduce the following notations [Bou, BB].

Definition 3.5. Let Q be a finite connected acyclic quiver with vertices Q0 = {1, . . . , n}.For i, j Q0, let mij := {a Q1| s(a) = i, e(a) = j} + {a Q1| s(a) = j, e(a) = i}. Lete1, . . . , en be the canonical basis ofZ

n.The geometric representation : WQ GLn(Z) of WQ is well-defined by

(si)(ej ) = i(ej ) := ej + (mij 2ij )ei,

where ij denotes the Kronecker delta. On the other hand, the contragradient : WQ

GLn(Z) of the geometric representation is defined by

(si)(ej) = i (ej) =

ej i = j

ej +

aQ1,s(a)=i

ee(a) i = j.

For an arbitrary expression w = si1 . . . sik of w, we define (w) = w := ik . . . i1 and(w) = w :=

ik

. . . i1 .

We refer to [Bou] for the following properties.

Proposition 3.6. (a) The maps : WQ GLn(Z) and : WQ GLn(Z) are

injective.

(b) We have (i)2

= id = (

i )2

for any i Q0.Then, we give the following important result, where we denote the set of g-matrices of

support -tilting -modules by g-mat().

Proposition 3.7. For any w WQ, we have g(w) = (w). In particular, the map

WQ g-mat(), w g(w) is injective.

Proof. By Proposition 3.4, the g-matrix g(w) = (g1(w), , gn(w)) is given by 1-st to n-throws and columns of g(w) = (g0(w), g1(w), , gn(w)).

On the other hand, the g-vector gi(w) Zn+1 gives the class ofeiIw in the Grothendieckgroup ofmod

for any i

Q0 since

Iw has the projective dimension at most one. Hence, by

[IR, Theorem 6.6], we have g(w) = (w), where is the contragradient W

Q

GLn+1(Z)

of the geometric representation.For w = si1 . . . sik , the 0-th column of

ij

is (1, 0, . . . , 0)t by the assumption of sij = s0for any 1 j k. Because 1-st to n-th rows and columns of ij is equal to

ij

by the

definition, we conclude that g(w) = ik . . . i1

= (w). The second statement followsfrom Proposition 3.6.

Finally we consider the following settings.

Definition 3.8. An element of := {w(ei)}iQ0,wWQ is called a root. For a, b Rn,

we write a natural pairing a, b := a bt. To each root x , we define a hyperplane

Hx := {y Rn | y, x = 0}.
http://-/?-http://-/?-

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A chamber of is a connected component ofRn \

x Hx. We denote by C0 the chamber{a1e1 + + anen | ai R>0}. Note that chambers are given by wWQ

w(C0) and there

is a bijection between WQ and chambers via w w(C0) [Bou].

Then we obtain the following conclusion.

Theorem 3.9. (a) The set ofg-matrices of support -tilting-modules coincides withthe subgroup 1, . . . ,

n of GL(Z

n) generated by i for all i Q0.(b) For w WQ, we have

C(w) = w(C0),

where C(w) denotes the cone of Iw. In particular, cones of basic support -tilting-modules give chambers of the associated root system .

Proof. From Proposition 3.7, (a) follows. Moreover, we have

C(w) = {a1g1(w) + + ang

n(w) | ai R>0}

= {a1w(e1) + + an

w(en) | ai R>0}

= w(C0).

Thus, (b) follows.

Example 3.10. (a) Let be the preprojective algebra of type A2. In this case, theg-matrices of Example 2.12 (a) are given as follows, where 1 =

1 0

1 1

and 2 =

1 10 1

.

1 10 1

1GG 0 11 1

2

55

1 00 1

2aa

1 33

0 11 0

1 01 1 2

GG 1 11 0

1

YY

Moreover, their chambers are given as follows.

11 10

10oo GG 1

0

01

yy

11

22

(b) Let be the preprojective algebra of type A3. In this case, the g-matrices ofExample 2.12 (b) are given as follows.

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22 YUYA MIZUNO

0 1 00 0 1

1 1 1@@

''

1 1 00 0 10 1 1

UU

99

0 1 10 0 11 1 0

22

1 1 10 0 10 1 0

TT

1 0 00 1 10 0 1

dd

00

1 1 10 1 10 1 0

UU

&&

0 1 00 1 11 1 1

22

0 0 10 1 11 0 0

((

1 1 01 1 10 1 1

TT

''

1 0 00 1 00 0 1

ii

%%

GG 1 1 00 1 00 1 1

dd

00

1 0 01 1 10 0 1

UU 0 0 1

1 1 11 0 0

bb

22

0 1 10 1 01 1 0

GG0 0 10 1 01 0 0

0 1 11 1 1

1 1 0

TT

1 0 01 1 00 0 1

dd

00

0 1 01 1 0

1 1 1

UU

&&

1 1 11 1 00 1 0

bb0 0 11 1 0

1 0 0

ee

1 1 11 0 00 1 0

TT

@@

1 1 0

1 0 00 1 1

UU

99

0 1 1

1 0 01 1 0

bb

0 1 01 0 0

1 1 1

TT

Remark 3.11. In the above examples, the rows of each matrix are sign-coherent, thatis, all elements are non-positive or non-negtative. This follows from the fact that, for aminimal projective presentation of a support -tilting module T

P1 GG P0 GG T GG 0,

we have addP0 addP1 = 0 [AIR, Proposition 2.5]. Note that a similar observation wasmade by [DK, section 2.4] for cluster-tilting objects in 2-CY triangulated categories.

4. Further connections

In this section, we extend our bijections by combining with other works, and explainsome relationships with other results.

We have the following bijections (We refer to [BY] for the definitions (g), (h), (i) and(j)).

Theorem 4.1. LetQ be a Dynkin quiver with vertices Q0 = {1, . . . , n} and the prepro-jective algebra of Q. There are bijections between the following objects.

(a) The elements of the Weyl group WQ.(b) The set I1, . . . , I n.

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(c) The set of isomorphism classes of basic support -tilting -modules.(d) The set of isomorphism classes of basic support -tilting op-modules.

(e) The set of torsion classes in mod.(f) The set of torsion-free classes in mod.(g) The set of isomorphism classes of basic two-term silting complexes in Kb(proj).(h) The set of intermediate bounded co-t-structures in Kb(proj) with respect to the

standard co-t-structure.(i) The set of intermediate bounded t-structures in Db(mod) with length heart with

respect to the standard t-structure.(j) The set of isomorphism classes of two-term simple-minded collections inDb(mod).

(k) The set of quotient closed subcategories in modKQ.(l) The set of subclosed subcategories inmodKQ.

We have given bijections between (a), (b), (c) and (d). Bijections between (g), (h), (i)and (j) are the restriction of [KY] and it is given in [BY, Corollary 4.3] (it is stated for

Jacobian algebras, but it holds for any finite dimensional algebra as they point out). Notethat compatibilities of mutations and partial orders of these objects are shown in [KY].

We will give bijections between (a), (e) and (f) by showing the following statement,which provides complete descriptions of torsion classes and torsion-free classes in mod.

Proposition 4.2. (i) For any torsion class T in mod, there exists w WQ suchthat T = FacIw.

(ii) For any torsion-free class F in mod, there exists w WQ such that F =Sub/Iw.

(iii) There exist bijections betweenWQ, torsion classes and torsion-free classes inmod.

Proof. Since there exists only finitely many support -tilting modules by Theorem 2.21,Theorem 1.10 shows that torsion classes and torsion-free classes are functorially finite.

Thus, Theorem 1.9 gives a bijection between basic support -tilting -modules and torsionclasses ofmod.

On the other hand, since F is functorially finite, there exists X mod such thatF = SubX by [S]. Then DSubX = FacDX is a functorially finite torsion class in modop.Similarly, via the bijection of Theorem 2.21, there exists w WQ such that DX = Iw modop. By Proposition 2.24, we have X = DIw = /Iw0w1, where w0 is the longestelement in WQ. From the above arguments, (iii) is clear.

Remark 4.3. It is shown that objects FacIw and Sub/Iw have several nice properties.For example, FacIw and Sub/Iw are Frobenius and, moreover, stable 2-CY categorieswhich have cluster-tilting objects. They also play important roles in the study of clusteralgebra structures for a coordinate ring of the unipotent cell associated with w (refer to

[BIRS, GLS3]).It is known that a left order of WQ is a lattice (i.e. any two elements x, y WQ have a

greatest lower bound, called meet x

y and a least upper bound, called join x

y). Fromthe viewpoint of Theorem 4.2, it is natural to consider what is a join and meet in torsionclasses. We give an answer to the question as follows.

Proposition 4.4. Let T and T be torsion classes of mod. Then we have T

T =T

T and T

T = Tors{T, T}, where Tors{T, T} is the smallest torsion class inmod containing T and T.

Proof. It is clear that T

T and Tors{T, T} are torsion classes. Since any torsion classin mod is functorially finite by Theorem 1.10 and 2.21, the statement follows.
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24 YUYA MIZUNO

Next we will see bijections between (a), (k) and (l), which are given by [ORT]. Webriefly see their results and explain a relationship with our results.

Let X be a -module. We denote by XKQ the KQ-module by the restriction, that is,we forget the action of the arrows a Q. Then we define ()KQ : mod modKQ,X (X)KQ := addXKQ . Take Iw for w WQ. In [ORT], the authors show that (Iw)KQ (respec-tively, (/Iw)KQ) is a quotient closed subcategory (respectively, subclosed subcategory)ofmodKQ, and any quotient closed subcategory (respectively, subclosed subcategory) isgiven in this form.

Now, we can conclude that the functor ()KQ gives a bijection between (e) and (k)(similarly (f) and (l)). Indeed, any torsion class is given by FacIw for some w WQ fromProposition 4.2 and we have (FacIw)KQ = (Iw)KQ by a result of [ORT].

Moreover, it is natural to consider when a quotient closed subcategory (Iw)KQ is a tor-sion class. The answer is given by [ORT, Proposition 10.4] and they show that elementsof WQ satisfying a certain condition defined by c-sortable [Re] are in bijection with tor-

sion classes ofmodKQ by this correspondence. We emphasize that there exist bijectionsbetween torsion classes ofmodKQ and several important objects such as clusters in thecluster algebra given by Q (we refer to [IT, Theorem 1.1] for more details). From thisobservation, we conclude that there exists a surjection from objects of Theorem 4.1 toobjects of [IT, Theorem 1.1].

Finally, we provide a bijection between (c) and (g), which is given by [AIR, Theorem3.2], and explain a definition of g-vectors. Let Kb(proj) be the homotopy category ofbounded complexes of proj. For a -module X, take a minimal projective presentationof X

P1(X) GG P0(X) GG X GG 0.

We denote by PX := (

1

P1(X)

0

P0(X)) Kb

(proj). Then, for a support -tiltingpair (X, P) for , the map (X, P) PX P[1] Kb(proj) gives a two-term silting

complex in Kb(proj) and it provides a bijection between (c) and (g). Note that the g-vector of (X, P) is given as a class of the corresponding silting complex PX P[1] in theGrothendieck group K0(K

b(proj)) [AIR, section 5]. Furthermore, we remark that, since is selfinjective, it is shown that a -stable support -tilting -module X (i.e X = (X) for := D Hom(, )) gives a two-term tilting complex by this correspondence [M]. Thus,we can also conclude that there exist only finitely many basic two-term tilting complexesin Kb(proj).

References

[AIR] T. Adachi, O. Iyama, I. Reiten, -tilting theory, arXiv: 1210.1036.[AIRT] C. Amiot, O. Iyama, I. Reiten, G. Todorov, Preprojective algebras and c-sortable words, Proc.

Lond. Math. Soc. (3) 104 (2012), no. 3, 513539.[AR1] M. Auslander, I. Reiten, DTr-periodic modules and functors, Representation theory of algebras

(Cocoyoc, 1994), 3950, CMS Conf. Proc., 18, Amer. Math. Soc., Providence, RI, 1996.[AR2] M. Auslander, I. Reiten, Modules determined by their composition factors, Ill. J. of Math 29 (1985),

280301.[ARS] M. Auslander, I. Reiten, S. O. Smal, Representation Theory of Artin Algebras, Cambridge Studies

in Advanced Mathematics, 36. Cambridge University Press, Cambridge, 1995.[BGL] D. Baer, W. Geigle, H. Lenzing, The preprojective algebra of a tame hereditary Artin algebra, Comm.

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[BB] A. Bjorner, F. Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231,Springer, New York, 2005.

[Bou] N. Bourbaki, Lie groups and Lie algebras. Chapters 46, Elements of Mathematics (Berlin), Springer-

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no. 5, 10271037.[DK] R. Dehy, B. Keller, On the combinatorics of rigid objects in 2-Calabi-Yau categories, Int. Math. Res.

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[GP] I. M. Gelfand, V. A. Ponomarev, Model algebras and representations of graphs, Funktsional. Anal. iPrilozhen. 13 (1979), no. 3, 112.

[H] J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics,29. Cambridge University Press, Cambridge, 1990.

[IT] C. Ingalls, H. Thomas, Noncrossing partitions and representations of quivers, Compos. Math. 145(2009), no. 6, 15331562.

[IJ] O. Iyama, G. Jasso, in preparation.[IR] O. Iyama, I. Reiten, Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras, Amer.

J. Math. 130 (2008), no. 4, 10871149.[KY] S. Koenig, D. Yang, Silting objects, simple-minded collections, t-structures and co-t-structures forfinite-dimensional algebras, arXiv:1203.5657.

[L1] G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4(1991), no. 2, 365421.

[L2] G. Lusztig, Semicanonical bases arising from enveloping algebras, Adv. Math. 151 (2000), no. 2,129139.

[M] Y, Mizuno, -stable -tilting modules, arXiv:1210.8322.[ORT] S. Oppermann, I. Reiten, H. Thomas, Quotient closed subcategories of quiver representations,

arXiv:1205.3268.[P] Y. Palu, Cluster characters for 2-Calabi-Yau triangulated categories, Ann. Inst. Fourier (Grenoble) 58

(2008), no. 6, 22212248.[Re] N. Reading, Clusters, Coxeter-sortable elements and noncrossing partitions, Trans. Amer. Math. Soc.,

359 (2007), no. 12, 59315958.[Ri] C. M. Ringel, The preprojective algebra of a quiver, Algebras and modules, II (Geiranger, 1996),

467480, CMS Conf. Proc., 24, Amer. Math. Soc., Providence, RI, 1998.[S] S. O. Smal, Torsion theory and tilting modules, Bull. London Math. Soc. 16(1984), 518522.

Graduate School of Mathematics, Nagoya University, Frocho, Chikusaku, Nagoya, 464-

8602, Japan

E-mail address: [email protected]
http://arxiv.org/abs/1302.6045http://arxiv.org/abs/1203.5657http://arxiv.org/abs/1210.8322http://arxiv.org/abs/1205.3268http://arxiv.org/abs/1205.3268http://arxiv.org/abs/1210.8322http://arxiv.org/abs/1203.5657http://arxiv.org/abs/1302.6045

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