MATRIX FACTORIZATIONS AND REPRESENTATIONS OF QUIVERS II: TYPE ADE CASE. Contents

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1 YITP IMS-52 MATIX FACTOIZATIONS AND EPESENTATIONS OF QUIVES II: TYPE ADE CASE HIOSHIGE KAJIUA, KYOJI SAITO, AND ATSUSHI TAKAHASHI Abstract We study a triangulated category of aded matrix factorizations for a polynomial of type ADE We show that it is equivalent to the derived category of finitely generated modules over the path algebra of the corresponding Dynkin quiver Also, we discuss a special stability condition for the triangulated category in the sense of T Bridgeland, which is naturally defined by the ading Contents Introduction 2 Triangulated categories of matrix factorizations 3 2 The triangulated category HMF A (f of matrix factorizations 3 22 The triangulated category HMF (f of aded matrix factorizations 6 3 HMF (f for type ADE and representations of Dynkin quivers 0 3 Statement of the main theorem (Theorem The proof of Theorem A stability condition on HMF (f 22 5 Tables of data for matrix factorizations of type ADE 24 eferences 35 Introduction The universal deformation and the simultaneous resolution of a simple singularity are described by the corresponding simple Lie algebra (Brieskorn [Bs] Inspired by that theory, the second named author associated in [Sa2],[Sa4] a generalization of root systems, consisting of vanishing cycles of the singularity, to any regular weight systems [Sa], and asked to construct a suitable Lie theory in order to reconstruct the primitive forms for the singularities In fact, the simple singularities correspond exactly to the weight systems having only positive exponents, and, in this case, this approach gives the classical finite root systems as in [Bs] Date: November 7th, 2005

2 2 HIOSHIGE KAJIUA, KYOJI SAITO, AND ATSUSHI TAKAHASHI As the next case, the approach is worked out for simple elliptic singularities corresponding to weight systems having only non-negative exponents, from where the theory of elliptic Lie algebras is emerging [Sa2] However, the root system in this approach in general is hard to manipulate because of the transcendental nature of vanishing cycles Hence, he asked ([Sa4], Problem in p24 in English version an algebraic and/or a combinatorial construction of the root system starting from a regular weight system In [T2], based on the mirror symmetry for the Landau-Ginzburg orbifolds and also based on the duality theory of the weight systems [Sa3],[T], the third named author proposed a new approach to the root systems, answering to the above problem He introduced a triangulated category D b Z (A f of aded matrix factorizations for a weighted homogeneous polynomial f attached to a regular weight system and showed that the category D b Z (A f for a polynomial of type A l is equivalent to the bounded derived category of modules over the path algebra of the Dynkin quiver of type A l He conjectured ([T2], Conjecture 3 further that the same type of equivalences hold for all simple polynomials of type ADE The main goal of the present paper is to answer affirmatively to the conjecture One side of this conjecture: the properties of the category of modules over a path algebra of a Dynkin quiver are already well-understood by the Gabriel s theorem [Ga], which states that the number of the indecomposable objects in the category for a Dynkin quiver coincides with the number of the positive roots of the root system corresponding to the Dynkin diaam The other side of the conjecture: the triangulated categories of (unaded matrix factorizations were introduced and developed by Eisenbud [E] and Knörrer [K] in the study of the maximal Cohen-Macaulay modules ecently, the categories of matrix factorizations are rediscovered in string theory as the categories of topological D-branes of type B in Landau- Ginzburg models (see [KL],[KL2] The category D b Z (A f of aded matrix factorizations is then motivated by the work on the categories of topological D-branes of type B in Landau- Ginzburg orbifolds (f, Z/hZ by Hori-Walcher [HW], where the orbifolding corresponds to introducing the Q-ading In fact, in [T2], the triangulated category D b Z (A f is constructed from a special A -category with Q-ading via the twisted complexes in the sense of Bondal- Kapranov [BK] Independently, D Orlov defines a triangulated category, called the category of aded D-branes of type B, which is in fact equivalent to D b Z (A f (see the end of subsection 22 Though some notions, for instance the central charge of the stability condition (see section 4, can be understood more naturally in D b Z (A f, the Orlov s construction of categories requires less terminologies and is easier to understand in a traditional way in algebraic geometry Therefore, in this paper we shall use the Orlov s construction with a slight modification of the scaling of deees and denote the modified category by HMF (f

3 MATIX FACTOIZATIONS AND EPESENTATIONS OF QUIVES II: TYPE ADE CASE 3 Let us explain details of the contents of the present paper In section 2, we recall the construction of triangulated categories of matrix factorizations Since we compare the category HMF (f with the unaded version HMF O(f in the proof of our main theorem (Theorem 3, we first introduce the unaded version HMF O (f corresponding to that given in [O] in subsection 2, and then we define the aded version HMF (f based on [O2] in subsection 22, where we also explain the relation of the category HMF (f with the category DZ b (A f introduced in [T2] Section 3 is the main part of the present paper In subsection 3, we state the main theorem (Theorem 3: for a polynomial f of type ADE, HMF (f is equivalent as a triangulated category to the bounded derived category of modules over the path algebra of the Dynkin quiver of type of f Subsection 32 is devoted to the proof of Theorem 3 The proof is based on various explicit data on the matrix factorizations; the complete list of the matrix factorizations (Table, their adings (Table 2 and the complete list of the morphisms in HMF (f (Table 3 The tables are arranged in the final section (section 5 In section 4, we construct a stability condition, the notion of which is introduced by Bridgeland [Bd], for the triangulated category HMF (f One can see that, as in the A l case [T2], the phase of objects (see Theorem 36 or Table 2 and the central charge Z (Definition 4 can be naturally given by the ading of matrix factorizations in Table 2 (cf [W] They in fact define a stability condition on HMF (f (Theorem 42, from which an abelian category is obtained as a full subcategory of HMF (f In Proposition 43, we show that this abelian category is equivalent to an abelian category of modules over the path algebra C principal, where principal is the Dynkin quiver with the orientation being taken to be the principal orientation introduced in [Sa5] Acknowledgement : We are ateful to M Kashiwara and K Watanabe for valuable discussions This work was partly supported by Grant-in Aid for Scientific esearch ant numbers , and of the Ministry of Education, Science and Culture in Japan H K is supported by JSPS esearch Fellowships for Young Scientists 2 Triangulated categories of matrix factorizations In this section, we set up several definitions which are used in the present paper The goal of this section is the introduction of the categories HMF A (f and HMF A (f attached to a weighted homogeneous polynomial f A following [O],[O2] with slight modifications 2 The triangulated category HMF A (f of matrix factorizations Let A be either the polynomial ring := C[x, y, z], the convergent power series ring O := C{x, y, z} or the formal power series ring Ô := C[[x, y, z]] in three variables x, y and z

4 4 HIOSHIGE KAJIUA, KYOJI SAITO, AND ATSUSHI TAKAHASHI Definition 2 (Matrix factorization For a polynomial f A, a matrix factorization M of f is defined by ( p 0 M := P 0 where P 0, P are right free A-modules of finite rank, and p 0 : P 0 P, p : P P 0 are A-homomorphisms such that p p 0 = f P0 and p 0 p = f P The set of all matrix factorizations of f is denoted by MF A (f p Since p 0 p and p p 0 are f times the identities, where f is nonzero element of A, the rank of P 0 coincides with that of P We call the rank the size of the matrix factorization M p 0 Definition 22 (Homomorphism Given two matrix factorizations M := ( P 0 P p P, and M := ( P 0 p 0 P, a homomorphism Φ : M M is a pair of A-homomorphisms Φ = (φ 0, φ p such that the following diaam commutes: φ 0 : P 0 P 0, φ : P P, P 0 p 0 P p P 0 P 0 φ 0 p 0 P φ p P 0 φ 0 The set of all homomorphisms from M to M, denoted by Hom MFA (f (M, M, is naturally an A-module and is finitely generated, since the sizes of the matrix factorizations are finite For three matrix factorizations M, M, M and homomorphisms Φ : M M and Φ : M M, the composition Φ Φ is defined by Φ Φ = (φ 0φ 0, φ φ This composition is associative: Φ (Φ Φ = (Φ Φ Φ for any three homomorphisms Definition 23 (HMF A (f An additive category HMF A (f is defined by the following data The set of objects is given by the set of all matrix factorizations: Ob(HMF A (f := MF A (f For any two objects M, M MF A (f, the set of morphisms is given by the quotient module: Hom HMFA (f(m, M = Hom MFA (f (M, M /, where two elements Φ, Φ in Hom MFA (f (M, M are equivalent (homotopic Φ Φ if there exists a homotopy (h 0, h, ie, a pair (h 0, h : (P 0 P, P P 0 of A-homomorphisms

5 MATIX FACTOIZATIONS AND EPESENTATIONS OF QUIVES II: TYPE ADE CASE 5 such that Φ Φ = (p h 0 + h p 0, p 0h + h 0 p The composition of morphisms on Hom HMFA (f is induced from that on Hom MFA (f since Φ Φ and Ψ Ψ imply ΨΦ Ψ Φ p 0 Note that the matrix factorization M = ( P 0 P MF A (f of size one with p (p 0, p = (, f or (p 0, p = (f, defines the zero object in HMF A (f, that is: one has Hom HMFA (f(m, M = Hom HMFA (f(m, M = 0 for any matrix factorization M MF A (f or, equivalently, M Hom MFA (f (M, M is homotopic to zero Lemma 24 For any two matrix factorizations ( M, M HMF A (f, the space of morphisms Hom HMFA (f(m, M f is a finitely generated A/, f, f -module x y z Proof Since Hom MFA (f (M, M is a finitely generated A-module and the equivalence relation is given by quotienting out by an A-submodule, Hom HMFA ( (f(m, M is also a f finitely generated A-module On the other hand, the Jacobi ideal, f, f annihilates x y z Hom HMFA (f(m, M f : that is, (φ x 0, φ f (φ y 0, φ f (φ z 0, φ 0 for any morphism Φ = (φ 0, φ This can be shown for instance by differentiating p p 0 = f P0 and p 0 p = f P by Then we have two identities p p p x x 0 + p 0 = f x x P 0 and p 0 p p x + p 0 = f x x P Multiplying φ 0 and φ by these two identities, respectively, leads to f (φ p x 0, φ 0, where (φ 0, φ x 0 p is the corresponding homotopy In a similar way one can x obtain f (φ y 0, φ f (φ z 0, φ 0 Definition 25 (Shift functor The shift functor T : HMF A (f HMF A (f is defined as p 0 follows The action of T on M = ( P 0 P p ( T P 0 p 0 p HMF A (f is given by ( p P := P p 0 For any M, M HMF A (f, the action of T on Φ = (φ 0, φ Hom HMFA (f(m, M is given by T (φ 0, φ := (φ, φ 0 P 0 Note that the square T 2 of the shift functor is isomorphic to the identity functor on HMF A (f

6 6 HIOSHIGE KAJIUA, KYOJI SAITO, AND ATSUSHI TAKAHASHI Definition 26 (Mapping cone For an element Φ = (φ 0, φ Hom MFA (f (M, M, the mapping cone C(Φ MF A (f is defined by ( c 0 C(Φ := C 0, where c C ( ( C 0 := P P 0, C := P 0 P p 0 p 0 0, c 0 =, c φ p = 0 φ 0 p The following is stated in [O] Proposition 33 Proposition 27 The additive category HMF A (f endowed with the shift functor T and the distinguished triangles forms a triangulated category, where a distinguished triangle is a sequence of morphisms which is isomorphic to the sequence M Φ M C(Φ T (M for some M, M MF A (f and Φ Hom MFA (f (M, M Proof The proof is the same as the proof of the analogous result for a usual homotopic category (see eg [GM], [KS] 22 The triangulated category HMF (f of aded matrix factorizations In this subsection, we study aded matrix factorizations for a weighted homogeneous polynomial f and construct the corresponding triangulated category, denoted by HMF (f A quadruple W := (a, b, c; h of positive integers with gcd(a, b, c = is called a weight system For a weight system W, we define the Euler vector field E = E W by E := a h x x + b h y y + c h z z For a given weight system W, becomes a aded ring by putting deg(x = 2a 2b, deg(y = h h and deg(z = 2c Let = h d 2 h Z 0 d be the aded piece decomposition, where d := {f 2Ef = df} A weight system W is called regular ([Sa] if the following equivalent conditions are satisfied: (a χ W (T := T h (T h T a (T h T b (T h T c has no poles except at T = 0 (T a (T b (T c (b A generic element of the eigenspace 2 = {f ( Ef = f} has an isolated critical f point at the origin, ie, the Jacobi ring /, f, f is finite dimensional over C x y z Such an element f of 2 as in (b shall be called a polynomial of type W In the present paper, by a aded module, we mean a aded right module with deees only in 2 h Z Namely, a aded -module P decomposes into the direct sum: P = d 2 h Z P d (2

7 MATIX FACTOIZATIONS AND EPESENTATIONS OF QUIVES II: TYPE ADE CASE 7 For two aded -modules P and P, a aded -homomorphism φ of deee s 2 Z is an h -homomorphism φ : P P such that φ( P d P d+s for any d The category of aded -modules has a deee shifting automorphism τ defined by (τ( P d = P d+ 2 h For any two aded -modules P, P and a aded -homomorphism φ : P P, we denote the induced aded -homomorphism by τ(φ : τ( P τ( P On the other hand, an - homomorphism φ : P P of deee 2m induces a deee zero -homomorphism from P to h τ m ( P, which we denote again by φ : P τ m ( P Definition 28 (Graded matrix factorization For a polynomial f of type W, a aded matrix factorization M of f is defined by ( M := p 0 P 0 P p where P 0, P are free aded right -modules of finite rank, p 0 : P0 P is a aded - homomorphism of deee zero, p : P P 0 is a aded -homomorphism of deee two such that p p 0 = f ep0 and p 0 p = f ep The set of all aded matrix factorizations of f is denoted by MF (f Definition 29 (Homomorphism Given two aded matrix factorizations M, M MF (f, a homomorphism Φ = (φ 0, φ : M M is a homomorphism in the sense of Definition 22 such that φ 0 and φ are aded -homomorphisms of deee zero The vector space of all aded -homomorphisms from M to M is denoted by Hom MF (f ( M, M For three aded matrix factorizations M, M, M MF (f and morphisms Φ : M M, Φ : M M, the composition is again a aded -homomorphism: Φ Φ Hom MF (f ( M, M Definition 20 (HMF (f An additive category HMF (f of aded matrix factorizations is defined by the following data The set of objects is given by the set of all aded matrix factorizations:, Ob(HMF (f := MF (f For any two objects M, M MF (f, the set of morphisms is given by Hom HMF (f ( M, M := Hom MF (f ( M, M /, where two elements Φ, Φ in Hom MF (f ( M, M are equivalent Φ Φ if there exists a homotopy, ie, a pair (h 0, h : ( P 0 P, P P 0 of aded -homomorphisms such that This τ is what is often denoted (for instance [Y],[O2] by (, ie, τ( P = P (

8 8 HIOSHIGE KAJIUA, KYOJI SAITO, AND ATSUSHI TAKAHASHI h 0 is of deee minus two, h is of deee zero and Φ Φ = (τ h (p h 0 +h p 0, p 0h +τ h (h 0 p The composition of morphisms is induced from that on Hom MF (f ( M, M A aded matrix factorization is the zero object in HMF direct sum of the aded matrix factorizations of the forms ( τ n ( (f if and only if it is a τ n ( f MF (f and ( τ n ( f τ n +h ( MF (f for some n, n Z Lemma 2 The category HMF (f is Krull-Schmidt, that is, (a for any two objects M, M HMF (f, Hom HMF (f ( M, M is finite dimensional; (b for any object M HMF (f and any idempotent e Hom HMF (f ( M, M, there exists a matrix factorization M HMF (f and a pair of morphisms Φ Hom HMF (f ( M, M, Φ Hom HMF (f ( M, M such that e = Φ Φ and ΦΦ = Id fm Proof (a Due to Lemma 24, n Z Hom HMF (f ( M, τ n ( M is a finitely generated aded ( ( f /, f, f f -module Since the Jacobi ring /, f, f is finite dimensional, the space x y z x y z Hom HMF (f ( M, M is finite dimensional over C (b Let + be the maximal ideal of of all positive deee elements Note that any aded matrix factorization is isomorphic in HMF (f to a aded matrix factorization whose entries belong to τ n ( + for some n Z Thus, we may assume that M := p 0 ( P0 P p HMF (f is such a aded matrix factorization Suppose that M has an idempotent e Hom HMF (f ( M, M, e 2 = e This implies that there exists ê Hom MF (f ( M, M such that ê 2 ê = (τ h (p h 0 + h p 0, p 0 h + τ h (h 0 p (22 for some homotopy (h 0, h on Hom HMF (f ( M, M However, since each entry of p 0 and p belongs to τ n ( +, each entry in the right hand side also belongs to τ n ( + Let π : Hom MF (f ( M, M Hom MF (f ( M, M be the canonical projection given by restricting each entry on / + = C Then, eq(22 in fact implies that π(ê 2 π(ê = 0 Thus, for π(ê =: (ê 0, ê, defining a matrix factorization M HMF (f by ( M ê p 0 ê 0 := ê 0 P0 ê P, ê 0 p ê one obtains a pair of morphisms Φ Hom HMF (f ( M, M and Φ Hom HMF (f ( M, M such that e = Φ Φ and ΦΦ = Id fm One can see that τ induces an automorphism on HMF (f, which we shall denote by the same notation τ : HMF (f HMF (f Explicitly, the action of τ on M =

9 MATIX FACTOIZATIONS AND EPESENTATIONS OF QUIVES II: TYPE ADE CASE 9 p 0 ( P0 P p HMF (f is given by ( τ p 0 P 0 P p ( := τ( P 0 τ(p 0 τ(p τ( P The action of τ on morphisms are naturally induced from that on aded -homomorphisms between two aded right -modules 25 2 Also, we have the shift functor T on HMF (f, the aded version of that in Definition Definition 22 (Shift functor on HMF is defined as follows The action of T on M HMF ( ( T p 0 P 0 P p (f The shift functor T : HMF (f HMF (f is given by = P p τ h (p 0 τ h ( P 0 For any M, M HMF (f, the action of T on Φ = (φ 0, φ Hom HMF (f ( M, M is given by T (φ 0, φ = (φ, τ h (φ 0 (f We remark that the square T 2 of the shift functor is not isomorphic to the identity functor on HMF (f but T 2 = τ h Definition 23 (Mapping cone For an element Φ = (φ 0, φ Hom MF (f ( M, M, the mapping cone C(Φ MF (f is defined by ( c 0 C(Φ := C 0, where c C ( ( C 0 := P P 0, C := τ h ( P 0 P p 0 τ h (p 0 0, c 0 =, c φ p = 0 τ h (φ 0 p This mapping cone is well-defined In fact, one can see that the deee of c 0 and c are zero and two, since the aded -homomorphisms p : P P 0 of deee two and p 0 : P 0 P of deee zero induce aded -homomorphisms p : P τ h ( P 0 of deee zero and τ h (p 0 : τ h ( P 0 P of deee two, respectively The following is stated in [O2] Proposition 34 2 The shift functor T is often denoted by []

10 0 HIOSHIGE KAJIUA, KYOJI SAITO, AND ATSUSHI TAKAHASHI Theorem 24 The additive category HMF (f endowed with the shift functor T and the distinguished triangles forms a triangulated category, where a distinguished triangle is defined by a sequence isomorphic to the sequence M Φ M C(Φ T ( M for some M, M MF (f and Φ Hom MF (f ( M, M Proof As in the case for HMF A (f, the proof is the same as the proof of the analogous result for a usual homotopic category (see eg [GM], [KS] Let M p 0 = ( P0 P p HMF (f be a aded matrix factorization of size r One can choose homogeneous free basis (b,, b r ; b,, b r such that P 0 = b b r and P = b b r Then, the aded matrix factorization M is expressed as a 2r by 2r matrix satisfying Q = ( ψ ϕ, ϕ, ψ Mat r ( (23 Q 2 = f 2r, SQ + QS + 2EQ = Q, (24 where S is the diagonal matrix of the form S := diag(s,, s r ; s,, s r such that s i = deg(b i and s i = deg( b i for i =, 2,, r We call this S a ading matrix of Q This procedure M (Q, S gives the equivalence between the triangulated category HMF (f and the triangulated category Db Z (A f in [T2] This implies that HMF (f is an enhanced triangulated category in the sense of Bondal-Kapranov [BK] We shall represent the matrix factorization M p 0 = ( P0 P p by (Q, S 3 HMF (f for type ADE and representations of Dynkin quivers In this section, we formulate the main theorem (Theorem 3 of the present paper in subsection 3 The proof of the theorem is given in subsection 32 3 Statement of the main theorem (Theorem 3 The main theorem states an equivalence between the triangulated category HMF (f for a polynomial f of type ADE with the derived category of modules over a path algebra

11 MATIX FACTOIZATIONS AND EPESENTATIONS OF QUIVES II: TYPE ADE CASE of a Dynkin quiver In order to formulate the results, we recall (i the weighted homogeneous polynomials of type ADE and (ii the notion of the path algebras of the Dynkin quivers (i ADE polynomials For a regular weight system W, we have the following facts [Sa] (a There exist integers m,, m l, called the exponents of W, such that χ W (T = T m + + T m l, where the smallest exponent is given by ɛ := a + b + c h (b The regular weight systems with ɛ > 0 are listed as follows A l : (, b, l + b; l +, b l, D l : (l 2, 2, l ; 2(l, (3 E 6 : (4, 3, 6; 2, E 7 : (6, 4, 9; 8, E 8 : (0, 6, 5; 30 Here, the naming in the left hand side is given according to the identifications of the exponents of the weight systems with those of the simple Lie algebras As a consequence, one obtains ɛ = for all regular weight system with ɛ > 0 For the polynomials of type ADE, without loss of generality we may choose the followings: x l+ + yz, h = l +, A l (l, x 2 y + y l + z 2, h = 2(l, D l (l 4, f(x, y, z = x 3 + y 4 + z 2, h = 2, E 6, x 3 + xy 3 + z 2, h = 8, E 7, x 3 + y 5 + z 2, h = 30, E 8 We denote by X(f the type of f (ii Path algebras (a The path algebra C of a quiver is defined as follows (see [Ga], [] and [Ha] Chapter, 5 A quiver is a pair ( 0, of the set 0 of vertices and the set of arrows (oriented edges Any arrow in has a starting point and end point in 0 A path of length r from a vertex v to a vertex v in a quiver is of the form (v α,, α r v with arrows α i satisfying the starting point of α is v, the end point of α i is equal to the starting point of α i+ for all i r, and the end point of α r is v In addition, we also define a path of length zero (v v for any vertex v in The path algebra C of a quiver is then the C-vector space with basis the set of all paths in The product structure is defined by the composition of paths, where the product of two non-composable paths is set to be zero The category of finitely generated right modules over the path algebra C is denoted by mod - C It is an abelian category, and its derived category is denoted by D b (mod - C

12 2 HIOSHIGE KAJIUA, KYOJI SAITO, AND ATSUSHI TAKAHASHI If 0 and are finite sets and does not have any oriented cycle, then D b (mod - C is a Krull-Schmidt category (b A Dynkin quiver X of type X of ADE is a Dynkin diaam X of type X listed in Figure together with an orientation for each edge of the Dynkin diaam It is known [Ga] that the number of all the isomorphism classes of the indecomposable objects of the abelian category mod - C of a quiver is finite if and only if the quiver is a Dynkin quiver (of type ADE 2 b l l : X = A l, l 2 l 3 l 2 l : X = D l, : X = E 6, : X = E 7, : X = E 8 FIGUE ADE Dynkin diaam For a Dynkin diaam X, we shall denote by Π X the set of vertices For later convenience, the elements of Π X are labeled by the integers {,, l} as in the above figures, and we shall sometimes confuse vertices in Π X with labels in {,, l} The following is the main theorem of the present paper Theorem 3 Let f(x, y, z C[x, y, z] be a polynomial of type X(f, and let X(f be a Dynkin quiver of type X(f with a fixed orientation Then, we have the following equivalence of the triangulated categories HMF (f Db (mod - C X(f

13 MATIX FACTOIZATIONS AND EPESENTATIONS OF QUIVES II: TYPE ADE CASE 3 32 The proof of Theorem 3 The construction of the proof of Theorem 3 is as follows Step We describe the Auslander-eiten (A-quiver for the triangulated category HMFÔ(f of matrix factorizations due to [E], [A2] and [A] and give the matrix factorizations explicitly Step 2 We determine the structure of the triangulated category HMF (f of aded matrix factorizations (Theorem 36 Step 3 By comparing the A-quiver of the category HMFÔ(f with the category HMF (f we find the exceptional collections corresponding to X(f in HMF (f and complete the proof of the main theorem (Theorem 3 Step The Auslander-eiten quiver for HMFÔ(f We recall the known results on the equivalence of the McKay quiver for Kleinean oup and the A-quiver for the simple singularities [A2], [A] and [E] For a Krull-Schmidt category C over C, an object X Ob(C is called indecomposable if any idempotent e Hom C (X, X is zero or the identity Id X For two objects X, Y Ob(C, denote by rad C (X, Y the subspace of Hom C (X, Y of non-invertible morphisms from X to Y We denote by rad 2 C(X, Y rad C (X, Y the space of morphisms each of which is described as a composition Φ Φ with Φ rad C (X, Z, Φ rad C (Z, Y for some object Z Ob(C For two indecomposable objects X, Y Ob(C, an element in rad C (X, Y \rad 2 C(X, Y is called an irreducible morphism The space Irr C (X, Y := rad C (X, Y /rad 2 C(X, Y in fact forms a subvector space of Hom C (X, Y We call by the A-quiver Γ(C of a Krull-Schmidt category C the quiver Γ(C := (Γ 0, Γ whose vertex set Γ 0 consists of the isomorphism classes [X] of the indecomposable objects X Ob(C and whose arrow set Γ consists of dim C (Irr C (X, Y arrows from [X] Γ 0 to [Y ] Γ 0 for any [X], [Y ] Γ 0 (see [],[Ha],[Y] On the other hand, for a Dynkin diaam X of type X listed in Figure, we define a quiver consisting of the vertex set Π X and arrows in both directions k of X between vertices k, k Π X The resulting quiver is denoted by X k for each edge Note that the category HMFÔ(f is Krull-Schmidt (see [Y], Proposition 8 Theorem 32 Let f be a polynomial f of type ADE, which we regard as an element of Ô (i ([A2],[A],[E] The A-quiver of the category HMFÔ(f is isomorphic to the quiver X of type X(f corresponding to f : Γ(HMFÔ(f X(f (ii According to (i, fix an identification Π X(f Γ 0 (HMFÔ(f, k [M k ] A representative M k of the isomorphism classes [M k ] of the indecomposable matrix factorizations of

14 4 HIOSHIGE KAJIUA, KYOJI SAITO, AND ATSUSHI TAKAHASHI minimum size is given explicitly in Table The size of M k is 2ν k, where ν k is the coefficient of the highest root for k Π X(f Proof (i This statement follows from the combination of results of [A2] and [E], where the Auslander-eiten quivers of the categories of the maximal Cohen-Macaulay modules over Ô/(f for type ADE are determined in [A2], and the equivalence of the category of Maximal Cohen-Macaulay modules with the category HMFÔ(f of the matrix factorizations is given in [E] (ii For each type X(f, since we have (Π X(f non-isomorphic matrix factorizations (Table, these actually complete all the vertices of the A-quiver Γ(HMFÔ(f emark 33 In [Y], matrix factorizations for a polynomial of type ADE in two variables x, y are listed up completely On the other hand, for type A l and D l in both two and three variables, all the matrix factorizations and the A-quivers are presented in [Sc], where the relation of the results in two variables and those in three variables is given This gives a method of finding the matrix factorizations of a polynomial of type E l, l = 6, 7, 8, in three variables from the ones in two variables case [Y] For a recent paper in physics, see also [KL3] Hereafter we fix an identification of Γ(HMFÔ(f with X(f by k [M k ] Step 2 Indecomposable objects in HMF (f ecall that one has the inclusion O Ô We prepare some definitions for any fixed weighted homogeneous polynomial f Definition 34 (Forgetful functor from HMF (f to HMF Ô (f For a fixed weighted homogeneous polynomial f, there exists a functor F : HMF (f HMF Ô (f given by F ( M := M Ô for M HMF (f and the naturally induced homomorphism F : Hom HMF HMF (f ( M, M Hom HMFÔ(f(F ( M, F ( M for any two objects M, M (f We call this F the forgetful functor It is known that F : HMF (f HMF Ô (f brings an indecomposable object to an indecomposable object ([Y], Lemma 52 Let us introduce the notion of distance between two indecomposable objects in HMFÔ(f and in HMF (f as follows Definition 35 (Distance For any two indecomposable objects M, M HMFÔ(f, define the distance d(m, M Z 0 from M to M by the minimal length of the paths from [M] to [M ] in the A-quiver Γ(HMFÔ(f of HMFÔ(f In particular, we have d(m, M = 0 if and only if M M in HMFÔ(f

15 MATIX FACTOIZATIONS AND EPESENTATIONS OF QUIVES II: TYPE ADE CASE 5 For any two indecomposable objects M, M HMF (f, the distance d( M, M Z 0 from M to M is defined by d( M, M := d(f ( M, F ( M By definition, an irreducible morphism exists in Hom HMFÔ(f(M, M if and only if d(m, M = Let us return to the case that f is of type ADE In this case, the distance is in fact symmetric: d(m k, M = d(m, M k for any [M k ], [M ] Π, since, due to Theorem 32, there exists an arrow from [M k ] to [M ] if and only if there exists an arrow from [M ] to [M k ] For two indecomposable objects M k, M HMF (f such that F ( M k = M k, F ( M = M, we denote d( M k, M = d(m k, M := d(k, k Since we fix a polynomial f of type ADE and then the corresponding type X(f of the Dynkin quiver X(f, hereafter we drop the subscript X(f In order to state the following Theorem 36, it is convenient to introduce the ordered decomposition Π = {Π, Π 2 } of Π, called a principal decomposition of Π [Sa5], as follows We first define the base vertex [M o ] Π For a diaam of D l or E l, l = 6, 7, 8, we choose [M o ] as the vertex on which three edges join Explicitly, it is [M k=l 2 ] for type D l, [M k=2 ] for type E 6, [M k=4 ] for type E 7 and [M k=5 ] for type E 8 (Figure For type A l, we set [M o ] := [M k=b ] (see Figure depending on the index b, b l (see eq(3 Then, we define the decomposition of Π by Π := {k Π d(m o, M k 2Z 0 + }, Π 2 := {k Π d(m o, M k 2Z 0 } ecall that we express a aded matrix factorization M HMF (f by the pair M = (Q, S, where Q denotes a matrix factorization in eq(23 and S is the ading matrix defined in eq(24 Theorem 36 Let f be a polynomial of type ADE The triangulated category HMF (f is described as follows (i (Objects: is given by The set of isomorphism classes of all indecomposable objects of HMF (f [ M k n := (Q k, S k n], k Π, n Z Here, Q k is the matrix factorizations of size 2ν k given in Table, the ading matrix S k n for k Π σ, σ =, 2, and n Z is given by: S k n := diag ( q k, q k,, q k ν k, q k ν k ; q k, q k,, q k ν k, q k ν k + φ k n 4νk, (32

16 6 HIOSHIGE KAJIUA, KYOJI SAITO, AND ATSUSHI TAKAHASHI where the data of the first term, called the traceless part: q k j 2 h Z σ h and qk j 2 h Z σ h for j ν k are given in Table 2, and the coefficient of the second term, called the phase, is given by φ k n := φ( M k n = 2n+σ h (ii (Morphisms: (ii-a An irreducible morphism exists in Hom HMF (f ( M n, k only if d(k, k = and φ n = φk n + h (ii-b dim C (Hom HMF (f ( M n, k M n = for φ n φk n = d(k, h M n if and (iii (The Serre duality: The automorphism S := T τ : HMF (f HMF (f satisfies the following properties n, n Z : (iii-a Hom HMF (f ( M k n, S( M k n C for any indecomposable object M k n HMF (f (iii-b This induces the following nondegenerate bilinear map for any k, k Π and Hom HMF (f ( M n, k M n Hom HMF (f ( M n, S( M n k C Proof (i For each M k, by direct calculations based on the explicit form of M k in Table, we can attach ading matrices S satisfying eq(24 This, together with the fact that F : HMF (f HMF Ô (f brings an indecomposable object to an indecomposable object, implies that the union l k= F (M k gives the set of all indecomposable objects in HMF (f and then F : HMF (f HMF Ô (f is a surjection (see also [A3] and [Y], Theorem 54 Actually, S is unique up to an addition of a constant multiple of the identity Therefore, we decompose S into the traceless part and the phase part as in eq(32 Due to the restriction of deees (2, one has ±qj k + φ k n 2 Z and h ± qk j + φ k n 2 Z for any k Π h and j ν k (see below eq(24, and by solving these conditions, we obtain Statement (i, ie, Table 2 In particular, one has F (M k = { M k n n Z} and τ( M k n = M k n+ (ii-a For any two indecomposable objects M k n, F : Hom HMF (f ( M k n, M n HMF (f, Statement (i implies that M n Hom HMFÔ(f(M k, M is injective and then F : n ZHom HMF (f ( M n, k τ n ( M n Hom HMFÔ(f(M k, M is an isomorphism of vector spaces This induces the following isomorphisms: F : F : F : n ZHom HMF (f ( M n, k τ n ( M n Hom HMFÔ(f(M k, M, n Zrad HMF (f ( M n, k τ n ( M n rad HMFÔ(f(M k, M, n Zrad 2 HMF (f( M n, k τ n ( M n rad 2 HMFÔ(f(M k, M, and hence, the isomorphism F : n ZIrr HMF (f ( M k n, τ n ( M n Irr HMFÔ(f(M k, M

17 MATIX FACTOIZATIONS AND EPESENTATIONS OF QUIVES II: TYPE ADE CASE 7 For k, k Π, define a multi-set C(k, k of non-negative integers by C(k, k := {h(φ(τ n ( M n φ( M n k Hom HMF (f ( M n, k τ n ( M n 0, n Z}, where the integer h(φ(τ n ( M n φ( M n k = h(φ n φk n + 2n appears with multiplicity dim C (Hom HMF (f ( M n, k τ n ( M n The multi-set C(k, in fact depends only on k and k, and is independent of n and n For k, k Π such that d(k, k =, by calculating Hom HMFÔ(f(M k, M using the explicit forms of the matrix factorizations M k, M in Table, one can see that C(k, k consists of positive odd integers including of multiplicity one This implies that, for two indecomposable objects M n, k M n HMF (f, a morphism in Hom HMF (f ( M n, k M n is an irreducible morphism if and only if h(φ n φk n = (Statement (ii-a (A morphism in Hom HMF (f ( M n, k M n which corresponds to c C(k, with c 3 can be obtained by composing irreducible morphisms (ii-b By direct calculations of Hom HMF (f ( M k n, M k n and Hom HMF (f ( M k n, S( M k n using the explicit forms of the matrix factorizations in Table again, one gets: Lemma 37 For an indecomposable object M n k HMF (f, one has -a Hom HMF (f ( M k n, M k n C, -b Hom HMF (f ( M k n, S( M k n C 2 For an indecomposable object M n k HMF (f, a morphism Ψ Hom HMF (f ( M n, k S( M n k and an indecomposable object M n HMF (f such that d(k, =, 2-a ΨΦ 0 holds for an irreducible morphism Φ Hom HMF (f ( M n, M k n, 2-b S(ΦΨ 0 holds for an irreducible morphism Φ Hom HMF (f ( M k n, M n Note that Lemma 37-a also follows from the A-quiver Γ(HMFÔ(f of HMFÔ(f in Theorem 32 and Statement (ii-a Due to Lemma 37-b, for any indecomposable object M n k HMF (f, one can consider the cone C(Ψ HMF (f of a nonzero morphism Ψ Hom HMF (f (S ( M n, k M n, k that is, the cone of a lift of the morphism Ψ Hom HMF (f (S ( M n, k M n k to a homomorphism in Hom MF (f (S ( M n, k M n k We denote it simply by C(Ψ, since two different lifts of the morphism Ψ Hom HMF (f (S ( M n, k M n k lead to two cones which are isomorphic to each other in HMF (f ecall that an Auslander-eiten (A-triangle (see [Ha],[Y], and also an Auslander- eiten sequence or equivalently an almost split sequence [A] in a Krull-Schmidt triangulated category C is a distinguished triangle satisfying the following conditions: X u Y v Z w T (X (33

18 8 HIOSHIGE KAJIUA, KYOJI SAITO, AND ATSUSHI TAKAHASHI (A X, Z are indecomposable objects in Ob(C (A2 w 0 (A3 If Φ : W Z is not a split epimorphism, then there exists Φ : W Y such that vφ = Φ Then, it is known (see [Ha], Proposition in 43 that u and v are irreducible morphisms Lemma 38 For an indecomposable object M n k HMF (f and the cone C(Ψ HMF (f of a nonzero morphism Ψ Hom HMF (f (S ( M n, k M n, k the distinguished triangle is an A-triangle M k n C(Ψ τ( M k n T (Ψ T ( M k n (34 Proof Since by definition the condition (A and (A2 are already satisfied, it is enough to show that the condition (A3 holds Consider the functor Hom HMF (f ( M n, on the distinguished triangle (34 for an indecomposable object M n HMF (f One obtains the long exact sequence as follows: Hom HMF (f ( M n, S ( M k n Ψ Hom HMF (f ( M n, M k n Hom HMF (f ( M n, C(Ψ Hom HMF (f ( M n, τ( M k n T (Ψ Hom HMF (f ( M n, T ( M k n (35 A nonzero morphism in Hom HMF (f ( M n, τ( M n k is a split epimorphism if and only if M n = τ( M n, k in which case Hom HMF (f ( M n, τ( M n k C is spanned by the identity fm k n Hom HMF, M For M τ( M n, k due to Lemma 372-a, the map T (Ψ in the long (f ( M n n n exact sequence (35 turns out to be the zero map, which implies that we have the surjection Hom HMF (f ( M n, C(Ψ Hom HMF (f ( M n, τ( M k n and then the condition (A3 The existence of the A-triangle together with the corresponding long exact sequence (35 leads to the two key lemmas as follows Lemma 39 Let M n k HMF (f be an indecomposable object The cone C(Ψ of a nonzero morphism Ψ Hom HMF (f (S ( M n, k M n k is isomorphic to the direct sum of indecomposable objects M k n M k m nm for some m Z >0 such that {k,, k m } = {k Π d(k, k = } and φ( M k i n i = φ( M n k + for any i =,, m h Proof For the A-triangle in eq(34, the morphisms M k n C(Ψ and C(Ψ τ( M k n are irreducible and hence one has φ( M k i n i = φ( M k n + h and d(, k i =, for any i =,, m, with the direct sum decomposition of indecomposable objects C(Ψ m i= M k i n i above Also, the fact that the distinguished triangle (34 is an A-triangle further guarantees

19 MATIX FACTOIZATIONS AND EPESENTATIONS OF QUIVES II: TYPE ADE CASE 9 that, for an indecomposable object M n HMF (f, there exists an irreducible morphism in Hom HMF (f ( M n, k M n if and only if (, n = (k i, n i for some i =,, m (see Lemma in p40 of [Ha] Thus, the rest of the proof is to show k i k j if i j, for which it is enough to check this lemma at the level of the ading matrices (see eq(5 below Table 2 Lemma 30 For an indecomposable object M n k HMF (f, let C(Ψ be the cone of a nonzero morphism Ψ Hom HMF (f (S ( M n, k M n, k and let C(Ψ m M i= k i n i, m Z >0, be the direct sum decomposition of indecomposable objects as above Then, for an indecomposable object M n HMF (f and the given multi-set C(, k, one obtains m C(k, k i = {c c C(k, k, c 0} {c + c C(k, k, c h 2 if k = k S }, i= where, for k Π, k S n Z Π denotes the vertex such that [F (S( M k n] = [M ks ] Π for any Proof Consider the long exact sequence (35 for the A-triangle (34 As discussed in the proof of Lemma 38, for M n τ( M n, k the map T (Ψ in eq(35 defines the zero map due to Lemma 372-a Similarly, for M n S ( M n, k the map Ψ in eq(35 defines the zero map due to Lemma 372-b On the other hand, if M n = τ( M n, k the map T (Ψ in eq(35 defines an isomorphism and hence Hom HMF (f ( M n, C(Ψ Hom HMF (f ( M n, τ( M n k turns out to be a zero map Similarly, if M n = S ( M n, k the map Ψ in eq(35 defines an isomorphism and hence Hom HMF (f ( M n, M n k Hom HMF (f ( M n, C(Ψ turns out to be a zero map Combining these facts, one has the exact sequences as follows: 0 Hom( M k n, τ( M n and M k n S ( M 0 Hom( M n, k M n Hom HMF (f ( M n, k C(Ψ Hom( M n, k τ( M n 0 if M k n n, M n Hom HMF (f ( M n, k C(Ψ 0 if M n k = τ( M n, and 0 Hom HMF (f ( M n, k C(Ψ Hom( M n, k τ( M n 0 if M k n = S ( M n in this case Hom( M n, k τ( M n = 0 From these results follows the statement of this lemma, though Calculate Hom HMF (f (M k, τ n (M k, n Z, where we set k = for type A l, D l and E 8, k = 5 or 6 for E 6, and k = 7 for E 7, by using the explicit forms of matrix factorizations M k in Table Then, one obtains C(k, k Using Lemma 39 and Lemma 30 repeatedly, one can actually obtain C(k, k for all k, k Π (Table 3 In particular, by a use of Table 3, one can get Statement (ii-b and Corollary 3 The following equivalent statements hold: for two indecomposable objects M n, k M n HMF (f,

20 20 HIOSHIGE KAJIUA, KYOJI SAITO, AND ATSUSHI TAKAHASHI (a If φ(s( M n φ( M n k < d(k, h S, then Hom HMF (f ( M n, M n k = 0 (b If φ( M n k φ( M n < h d(, k, then Hom HMF (f ( M n, k S( M n = 0 (iii-a This is already proven in Lemma 37-b (iii-b Suppose that there exists a nonzero morphism Φ Hom HMF (f ( M n, M n k for two indecomposable objects M n, k M n HMF (f and let us show that there exists a nonzero morphism Φ S Hom HMF (f ( M n, k S( M n such that ΦS Φ is nonzero in Hom HMF (f ( M n, S( M n If M n k = S( M n, then one may take ΦS = S( M fk, so assume that M k n n S( M n, ie, 0 φ k n φ n < 2 As in the proof of Lemma 39, from the long exact sequence (35 of h the A-triangle (34 one obtains the exact sequence 0 Hom HMF (f ( M n, M k n Hom HMF (f ( M n, C(Ψ for the cone C(Ψ of a nonzero morphism Ψ : S ( M n k M n k Namely, there exists an indecomposable object M n HMF (f such that d(k, =, φ n φk n = and h Φ Φ is not zero in Hom HMF (f ( M n, M n with an irreducible morphism Φ Hom HMF (f ( M n, k M n epeating this procedure together with Corollary 3 (a leads that the path corresponding to the morphism arrives at S( M n Namely, for a given morphism Φ in Hom HMF (f ( M n, M n, k there exists a nonzero morphism Φ S Hom HMF (f ( M n, k S( M n such that ΦS Φ is nonzero in Hom HMF (f ( M n, S( M n Conversely, suppose that there exists a morphism Φ S Hom HMF (f ( M n, k S( M n for two indecomposable objects M n, k M n HMF (f One can show that there exists a nonzero morphism Φ Hom HMF (f ( M n, M n k such that Φ S Φ is nonzero in Hom HMF (f ( M n, S( M n by considering the functor Hom HMF (f (, S( M n on the A-triangle (34 together with Corollary 3 (b, instead of Hom HMF, with Corollary 3 (a Thus, one gets Statement (iii-b of Theorem 36 (f ( M n Step 3 Exceptional collections in HMF (f In this step, for a fixed polynomial f of type ADE and a Dynkin quiver X(f, we find exceptional collections in HMF (f and then show the equivalence of the category (f with the derived category of modules over the path algebra of the Dynkin quiver HMF X(f (which we again denote simply by Lemma 32 For any Dynkin quiver, there exists n := (n,, n l Z l such that one has an isomorphism of C-algebras: C C Γ(n := k,k Π Hom HMF (f ( M k n k, M n k,

21 MATIX FACTOIZATIONS AND EPESENTATIONS OF QUIVES II: TYPE ADE CASE 2 with the natural correspondence between the path from k Π to k Π and the space Hom( M n k k, M n for any k, k Π Such a tuple of integers is unique up to the Z shift (n + n,, n l + n for some n Z Proof Let us fix n Z of M n For each k Π, d(, k =, take M n k k HMF (f such that φ k n k φ n = (resp if there exists an arrow in from Π to k Π (resp h h from k Π to Π epeating this process leads to the tuple n = (n,, n l such that C Γ(n C On the other hand, Theorem 36 (ii-b implies that there exists a morphism for any given path and there are no relations among them Thus, one gets this lemma Corollary 33 Let S l be the permutation oup of Π = {,, l} Given a Dynkin quiver and n Z l such that C Γ(n C, one can take an element σ S l such that φ σ( n σ(k φ σ(k n σ(k if k > k Thus, for {E,, E l }, E k σ(k := M n σ(k, Hom HMF (f (E k, E 0 only if k > k Lemma 34 Given a Dynkin quiver and n Z l Hom HMF (f (τ n ( M k n k, M n k = 0 for any k, Π if n such that C Γ(n C, one has Proof Let us first concentrate on irreducible morphisms For any k, k Π such that d(k, k =, one obtains φ n k φk n k +n = ± 2n since φ h n k φk n k = ± On the other hand, Theorem 36 (ii-a implies that there exists an irreducible morphism in Hom HMF h M (f ( M k n k +n, only if d(k, k = and n = 0 or n =, since ± 2n can be only if n = 0 or n = h h Since, by definition, all morphisms except the identities can be described by the composition of the irreducible morphisms, one gets this lemma By Lemma 34 and the Serre duality (Theorem 36 (iii, we obtain the following lemma Lemma 35 Given a Dynkin quiver and n Z l such that C Γ(n C, {E,, E l } given in Corollary 33 is a strongly exceptional collection, that is, Hom HMF (f (E i, E j = 0 for i > j, Hom HMF (f (E i, T k (E j = 0 for k 0 and any i, j n k Proof Due to Corollary 33, it is sufficient to show that Hom HMF (f (E i, T n (E j = 0 for n On the other hand, by the Serre duality, dim C (Hom HMF (f (E i, T n (E j = dim C (Hom HMF (f (T n τ(e j, E i holds If n =, Hom HMF (f (T n τ(e j, E i = Hom HMF (f (τ(e j, E i = 0 follows from Lemma 34 If n 2, first we have φ(t n τ(e j = φ(e j + (n + 2 and then h ( φ(e i φ(t n τ(e j = (φ(e i φ(e j n + 2 h

22 22 HIOSHIGE KAJIUA, KYOJI SAITO, AND ATSUSHI TAKAHASHI Here, it is clear that φ(e i φ(e j l h holds for {E,, E l } Thus, we have the following inequality: φ(e i φ(t n τ(e j l h ( n + 2 h < 0 if n 2 On the other hand, since all morphisms except the identities can be obtained by the composition of the irreducible morphisms, Theorem 36 (ii-a implies that Hom HMF (f (E j, E i 0 only if φ(e i φ(e j h d(ej, E i 0 This implies Hom HMF (f (T n τ(e j, E i = 0 Corollary 36 D b (mod - C is a full triangulated subcategory of HMF (f Proof Use the fact that {E,, E l } is a strongly exceptional collection and C l Hom HMF (f (E i, E j i,j= Since HMF (f is an enhanced triangulated category, we can apply the theorem by Bondal- Kapranov ([BK] Theorem which implies that D b (mod - C is full as a triangulated subcategory ecall that the number of indecomposable objects of HMF (f up to the shift functor T is l h, which coincides with the number of the positive roots for the root system of type 2 ADE This number is the number of indecomposable objects of D b (mod - C up to the shift functor T by Gabriel s theorem [Ga] Therefore, one obtains D b (mod - C HMF (f 4 A stability condition on HMF (f Let K 0 (HMF (f be the Grothendieck oup of the category HMF (f [Gr] For a triangulated category C, let F be a free abelian oup generated by the isomorphism classes of objects [X] for X Ob(C Let F 0 be the suboup generated by [X] [Y ] + [Z] for all distinguished triangles X Y Z T (M in C The Grothendieck oup K 0 (C of the triangulated category C is, by definition, the factor oup F/F 0 Definition 4 For a aded matrix factorization M = (Q, S HMF (f, we associate a complex number as follows: Z( M := Tr(e π S The map Z extends to a oup homomorphism Z : K 0 (HMF (f C

23 MATIX FACTOIZATIONS AND EPESENTATIONS OF QUIVES II: TYPE ADE CASE 23 Theorem 42 Let f be a polynomial of type ADE Let P(φ be the full additive subcategory of HMF (f additively generated by the indecomposable objects of phase φ Then P(φ and Z define a stability condition on HMF (f in the sense of Bridgeland [Bd] Proof By definition, what we should check is that P(φ and Z satisfy the following properties: (i if M P(φ, then Z( M = m( M exp(π φ for some m( M >0, (ii for all φ, P(φ + = T (P(φ, (iii if φ > φ 2 and M i P(φ i, i =, 2, then Hom HMF (f ( M, M 2 = 0, (iv for each nonzero object M HMF (f, there is a finite sequence of real numbers φ > φ 2 > > φ n and a collection of distinguished triangles 0 M0 M M 2 Mn Mn M Ñ Ñ 2 Ñ n with Ñj P(φ j for all j =,, n Here, for any indecomposable object M k n HMF (f, Z( M k n is of the form: Z( M k n = m( M k ne π φ k n, m( M k n >0 In fact, by a straightforward calculation, one obtains ν k ( Z( M n k = e π (qj k+φk n + e π ( qj k+φk n + e π ( q j k+φk n + e π ( q j k+φk n j= = 2e π φ k n ν k j= ( cos(πq k j + cos(π q k j = m( M k ne π φ k n, m( M k n := 2 ν k j= ( cos(πq k j + cos(π q k j This shows Statement (i in Theorem 42 It is clear that Statement (ii,(iii and (iv follow from Theorem 36 Proposition 53 in [Bd] states that a stability condition (Z, P on a triangulated category defines an abelian category In our case, for the triangulate category HMF (f, we obtain an abelian category P((0, ] which consists of objects M HMF (f having the decomposition Ñ,, Ñn given by Theorem 42 (iv such that φ > > φ n > 0

24 24 HIOSHIGE KAJIUA, KYOJI SAITO, AND ATSUSHI TAKAHASHI Proposition 43 Given a polynomial f of type ADE, the following equivalence of abelian categories holds: P((0, ] mod - C principal, where principal is the Dynkin quiver of the corresponding ADE Dynkin diaam with the principal orientation [Sa5], ie, all arrows start from Π and end on Π 2 Proof Let us take the following collection { M 0,, M k 0 } The corresponding C-algebra is denoted by C Γ(n = 0 (see Lemma 32 First, we show that C Γ(n = 0 C principal, (4 for which it is enough to say that M k 0 P((0, ] is projective for each k Π For any Ñ P(φ(Ñ such that 0 < φ(ñ, the Serre duality implies Hom HMF (f ( M k 0, T (Ñ Hom HMF (f (τ(ñ, M k 0 Here, one has Hom HMF (f (τ(ñ, M k 0 0 only if φ(ñ+ 2 h φ( M k 0 However, such Ñ does not exist since φ( M k 0 = σ h for k Π σ, σ =, 2 Thus, one has Hom HMF (f ( M k 0, T (Ñ = 0 and hence eq(4 On the other hand, the collection M 0,, M k 0 is a full sub-abelian category of P((0, ] and is equivalent to the abelian category mod - C principal Also, the Gabriel s theorem [Ga] asserts that the number of the indecomposable objects in mod - C principal is equal to the number of the positive roots of, which coincides with the number of the indecomposable objects in P((0, ] Thus, one obtains the equivalence mod - C principal P((0, ] emark 44 (Principal orientation In the triangulated category HMF (f, the principal oriented quiver principal is realized by a strongly exceptional collection { M n,, M n l l } with n = = n l = n Z for some n Z as above It is interesting that a strongly exceptional collection of this type has minimum range of the phase: 2n+ h φ( M k n k 2n+2 h for any k Π 5 Tables of data for matrix factorizations of type ADE Table We list up the Auslander-eiten (A-quiver of the category HMFÔ(f for a polynomial f of type ADE We label each isomorphism class of the indecomposable objects (matrix factorizations in MFÔ(f by upperscript {,, k,, l} such as [M k ] We also list up a representative ( M k of [M k ] by giving the pair (ϕ k, ψ k for each matrix factorization Q k := (see eq(23 In addition, for type D l and E l (l = 6, 7, 8 cases, we attach ψ k ϕ k

25 MATIX FACTOIZATIONS AND EPESENTATIONS OF QUIVES II: TYPE ADE CASE 25 the indices ( qk q k ν k defined in Theorem 36 (eq(32 to each of those pairs as ϕ k 0 q k B A qν k k, ψ k 0 q k B A qν k k Those indices are for later convenience and irrelevant for the statement of Theorem 32 They will be listed up again in the next table (Table 2 Type A l The A-quiver for the polynomial f = x l+ + yz is of the form: [M ] [M 2 ] [M l ] [M l ], where, for each k Π, (ϕ k, ψ k is given by ( ( y x ϕ k l+ k z x :=, ψ k l+ k := x k z x k y Type D l The A-quiver for the polynomial f = x 2 y + y l + z 2 is given by: [M ] [M 2 ] [M l ] [M l 3 ] [M l 2 ] [M l ], where, for each k Π, (ϕ k, ψ k is given by ( z x ϕ 2 ( l 3 = ψ ( l 3 = + y l 2, y z z 0 xy y k 2 ϕ k ş ť l 2 k = ψş k ť l 2 k = 0 z y l k 2 x l k l k x y k 2 z 0, k: even (2 k l 2 y l k 2 xy 0 z z y k 2 xy 0 k+ ϕ k ş ť l 2 k = ψş k l ť l 2 k = y 2 z 0 x l k l k x 0 z y k 3, k: odd (3 k l 2 2 k l 0 xy y 2 z The forms of (ϕ l, ψ l and (ϕ l, ψ l depend on whether l is even or odd