DERIVED PICARD GROUPS OF FINITE DIMENSIONAL HEREDITARY ALGEBRAS

JUN-ICHI MIYACHI AND AMNON YEKUTIELI

Abstract. Let A be a finite dimensional algebra over a field k. The derived Picard group DPic_{k}(A) is the group of triangle auto-equivalences of D^{b}(mod A) induced by two-sided tilting complexes.

We study the group DPic_{k}(A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPic_{k}(A), as well as explicit calculations for many cases, including all finite and tame representation types.

Our method is to construct a representation of DPic_{k}(A) on a certain infinite quiver Γ⃗^{irr}. This representation is faithful when the quiver Δ⃗ of A is a tree, and then DPic_{k}(A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPic_{k}(A).

When A is hereditary, DPic_{k}(A) coincides with the full group of k-linear triangle auto-equivalences of D^{b}(mod A). Hence we can calculate the group of such auto-equivalences for any triangulated category D equivalent to D^{b}(mod A). These include the derived categories of piecewise hereditary algebras, and of certain noncommutative spaces introduced by Kontsevich-Rosen-berg.

0. Introduction and Statement of Results

Let k be a field and A an associative unital k-algebra. We write Mod A for the category of left A-modules, and D^{b}(Mod A) for the bounded derived category. Let A^{∘} be the opposite algebra and A^{e} := A ⊗_{k} A^{∘} the enveloping algebra, so that Mod A^{e} is the category of k-central A-A-bimodules.

A two-sided tilting complex a complex T ∈ D^{b}(Mod A^{e}) for which there exists another complex T^{∨} ∈ D^{b}(Mod A^{e}) satisfying T^{∨} ⊗_{A}^{L} T ≅ T ⊗_{A}^{L} T^{∨} ≅ A. This notion is due to Rickard [Rd]. The derived Picard group of A (relative to k) is

DPic_{k}(A) := \frac{\{two-sided tilting complexes T ∈ D^{b}(Mod A^{e})\}}{isomorphism}

with identity element A, product (T_{1}, T_{2}) ↦ T_{1} ⊗_{A}^{L} T_{2} and inverse T ↦ T^{∨} := R Hom_{A}(T, A). See [Ye] for more details.

Since every invertible bimodule is a two-sided tilting complex, DPic_{k}(A) contains the (noncommutative) Picard group Pic_{k}(A) as a subgroup. It also contains a central subgroup ⟨σ⟩ ≅ ℤ, where σ is the class of the two-sided tilting complex A[1]. In [Ye] we showed that when A is either local or commutative one has DPic_{k}(A) = Pic_{k}(A) × ⟨σ⟩. This was discovered independently by Rouquier-Zimmermann [Zi],

Date: 19 July 2000. 1991 Mathematics Subject Classification. Primary: 18E30, 16G20; Secondary: 16G70, 16G60,

16E20, 14C22.

Keywords: derived category; Picard group; finite dimensional algebra; quiver. The second author was partially supported by the US-Israel Binational Science Foundation.

[RZ]. On the other hand in the smallest example of a k-algebra A that is neither commutative nor local, namely the 2 × 2 upper triangular matrix algebra, this equality fails. These observations suggest that the group structure of DPic_{k}(A) should carry some information about the geometry of the noncommutative ring A.

This prediction is further motivated by another result in [Ye], which says that DPic_{k}(A) classifies the dualizing complexes over A. The geometric significance of dualizing complexes is well known (cf. [RD] and [YZ]).

From a broader perspective, DPic_{k}(A) is related to the geometry of noncommutative schemes on the one hand, and to mirror symmetry and deformations of (commutative) smooth projective varieties on the other hand. See [BO], [Ko], [KR] and [Or].

A good starting point for the study of the group DPic_{k}(A) is to consider finite dimensional k-algebras. The geometric object associated to a finite dimensional kalgebra A is its quiver Δ⃗, as defined by Gabriel (cf. [GR] or [ARS]). It is worthwhile to note that from the point of view of noncommutative localization theory (cf. [MR] Section 4.3) Δ⃗ is the link graph of A. More on this in Remark 1.2.

Some calculations of the groups DPic_{k}(A) for finite dimensional algebras have already been done. Let us mention the work of Rouquier-Zimmermann [RZ] on Brauer tree algebras, and the work of Lenzing-Meltzer [LM] on canonical algebras.

In this paper we present a systematic study the group DPic_{k}(A) when A is a finite dimensional hereditary algebra over an algebraically closed field k. We obtain general results on the structure of DPic_{k}(A), as well as explicit calculations. These results carry over to piecewise hereditary algebras, as well as to certain noncommutative schemes. The rest of the Introduction is devoted to stating our main results.

The group Aut_{k}(A) = Aut_{Alg k}(A) of k-algebra automorphisms is a linear algebraic group over k, via the inclusion into Aut_{Mod k}(A) = GL(A). This induces a structure of linear algebraic group on the quotient Out_{k}(A) of outer automorphisms. We denote by Out_{k}^{0}(A) the identity component of Out_{k}(A).

Recall that A is a basic k-algebra if A/𝔯 ≅ k × ⋯ × k, where 𝔯 is the Jacobson radical. For a basic algebra one has Out_{k}(A) = Pic_{k}(A). A hereditary basic algebra A is isomorphic to the path algebra kΔ⃗ of its quiver. An algebra A is indecomposable iff the quiver Δ⃗ is connected.

For Morita equivalent k-algebras A and B one has DPic_{k}(A) ≅ DPic_{k}(B), and the quivers of A and B are isomorphic. According to a result of Brauer (see [Po] Section 2) one has Out_{k}^{0}(A) ≅ Out_{k}^{0}(B). If A ≅ ∏_{i=1}^{n} A_{i} then DPic_{k}(A) ≅ G ⋉ ∏_{i=1}^{n} DPic_{k}(A_{i}), where G ⊂ S_{n} is a permutation group (cf. [Ye] Lemma 2.6). Also Δ⃗(A) ≅ ∐ Δ⃗(A_{i}) and Out_{k}^{0}(A) ≅ ∏ Out_{k}^{0}(A_{i}). Since the main result Theorem 0.1 is stated in terms of Δ⃗ and Out_{k}^{0}(A), we allow ourselves to assume throughout that A is a basic indecomposable algebra.

Given a quiver Q⃗ we denote by Q⃗_{0} its vertex set. For a pair of vertices x, y ∈ Q⃗_{0} we write d(x, y) for the arrow-multiplicity, i.e. the number of arrows α : x → y. Let Aut(Q⃗_{0}) be the permutation group of Q⃗_{0}, and let Aut(Q⃗_{0}; d) be the subgroup of arrow-multiplicity preserving permutations, namely

Aut(Q⃗_{0}; d) = {π ∈ Aut(Q⃗_{0}) | d(π(x), π(y)) = d(x, y) for all x, y ∈ Q⃗_{0}}.

Write Aut(Q⃗) for the automorphism group of the quiver Q⃗. Then Aut(Q⃗_{0}; d) is the image of the canonical homomorphism Aut(Q⃗) → Aut(Q⃗_{0}). The surjection Aut(Q⃗) ↠ Aut(Q⃗_{0}; d) is split, and it is bijective iff Q⃗ has no multiple arrows.

Of particular importance to us is a certain countable quiver Γ⃗^{irr}. This is a full subquiver of the Auslander-Reiten quiver Γ⃗(D^{b}(mod A)) of D^{b}(mod A) as defined by Happel [Ha]. Here mod A is the category of finitely generated A-modules. If A has finite representation type (i.e. Δ⃗ is a Dynkin quiver) then Γ⃗^{irr} ≅ ℤ⃗Δ⃗, where ℤ⃗Δ⃗ is the quiver introduced by Riedtmann [Rn]. Otherwise Γ⃗^{irr} ≅ ℤ×ℤ⃗Δ⃗. See Definitions 2.2 and 2.3 for the definition of the quivers Γ⃗^{irr} and ℤ⃗Δ⃗, and see Figures 3 and 4 for illustrations. The group DPic_{k}(A) acts on Γ⃗_{0}^{irr} by arrow-multiplicity preserving permutations, giving rise to a group homomorphism q : DPic_{k}(A) → Aut(Γ⃗_{0}^{irr}; d).

Define the bimodule A^{∗} := Hom_{k}(A, k). Then A^{∗} is a two-sided tilting complex, the functor M ↦ A^{∗} ⊗_{A}^{L} M ≅ R Hom_{A}(M, A^{∗}) is the Serre functor of D^{b}(mod A) in the sense of [BK], and M ↦ A^{∗}[−1] ⊗_{A}^{L} M is the translation functor in the sense of [Ha] Section I.4. We write τ ∈ DPic_{k}(A) for the element represented by A^{∗}[−1]. Then τ is translation of the quiver Γ⃗^{irr}. Let us denote by Aut(Γ⃗_{0}^{irr}; d)^{⟨τ,σ⟩} the subgroup of Aut(Γ⃗_{0}^{irr}; d) consisting of permutations that commute with τ and σ.

Here is the main result of the paper.

Theorem 0.1. Let A be an indecomposable basic hereditary finite dimensional algebra over an algebraically closed field k, with quiver Δ⃗.

1. There is an exact sequence of groups

1 → Out_{k}^{0}(A) → DPic_{k}(A) ⟶^{q} Aut(Γ⃗_{0}^{irr}; d)^{⟨τ,σ⟩} → 1.

This sequence splits.

2. If A has finite representation type then there is an isomorphism of groups

DPic_{k}(A) ≅ Aut(ℤ⃗Δ⃗)^{⟨τ⟩}.

3. If A has infinite representation type then there is an isomorphism of groups

DPic_{k}(A) ≅ (Aut((ℤ⃗Δ⃗)_{0}; d)^{⟨τ⟩} ⋉ Out_{k}^{0}(A)) × ℤ.

The factor ℤ of DPic_{k}(A) in part 3 is generated by σ. If Δ⃗ has no multiple arrows then so does ℤ⃗Δ⃗, and hence Aut((ℤ⃗Δ⃗)_{0}; d) = Aut(ℤ⃗Δ⃗). The proof of Theorem 0.1 is in Section 3 where it is stated again as Theorem 3.8.

Recall that a finite dimensional k-algebra B is called piecewise hereditary of type Δ⃗ if D^{b}(mod B) ≈ D^{b}(mod A) where A = kΔ⃗ for some finite quiver Δ⃗ without oriented cycles. By [Rd] Corollary 3.5 one knows that DPic_{k}(B) ≅ DPic_{k}(A). The next corollary follows.

Corollary 0.2. Suppose B is a piecewise hereditary k-algebra of type Δ⃗. Then DPic_{k}(B) is described by Theorem 0.1 with A = kΔ⃗.

In Section 4 we work out explicit descriptions of the groups Pic_{k}(A) and DPic_{k}(A) for the Dynkin and affine quivers, as well as for some wild quivers with multiple arrows. As an example we present below the explicit description of DPic_{k}(A) for a Dynkin quiver of type A_{n} (which corresponds to upper triangular n × n matrices). The corollary is extracted from Theorem 4.1.

Corollary 0.3. Suppose Δ⃗ is a Dynkin quiver of type A_{n} and A = kΔ⃗. Then DPic_{k}(A) is an abelian group generated by τ and σ, with one relation

τ^{n+1} = σ^{−2}.

The relation τ^{n+1} = σ^{−2} was already discovered by E. Kreines (cf. [Ye] Appendix). This relation has been known also to Kontsevich, and in his terminology D^{b}(mod A) is “fractionally Calabi-Yau of dimension \frac{n−1}{n+1}” (see [Ko]; note that the Serre functor is τσ).

Suppose D is a k-linear triangulated category that’s equivalent to a small category. Denote by Out_{k}^{tr}(D) the group of k-linear triangle auto-equivalences of D modulo functorial isomorphisms. For a finite dimensional algebra A one has DPic_{k}(A) ⊂ Out_{k}^{tr}(D^{b}(mod A)), with equality when A is hereditary (cf. Corollary 1.9).

In [KR] Kontsevich-Rosenberg introduce the noncommutative projective space NP_{k}^{n}, n ≥ 1. They state that D^{b}(Coh NP_{k}^{n}) is equivalent to D^{b}(mod kΩ⃗_{n+1}), where Ω⃗_{n+1} is the quiver in Figure 6, and Coh NP_{k}^{n} is the category of coherent sheaves. By Beilinson’s results in [Be], there is an equivalence D^{b}(Coh P_{k}^{1}) ≈ D^{b}(mod kΩ⃗_{2}). Combining Theorem 4.3 and Corollary 1.9 we get the next corollary.

Corollary 0.4. Let X be either NP_{k}^{n} (n ≥ 1) or P_{k}^{n} (n = 1). Then

Out_{k}^{tr}(D^{b}(Coh X)) ≅ ℤ × (ℤ ⋉ PGL_{n+1}(k)).

In Section 5 we look at a tree Δ with n vertices. Every orientation ω of Δ gives a quiver Δ⃗_{ω}. The equivalences between the various categories D^{b}(mod kΔ⃗_{ω}) form the derived Picard groupoid DPic_{k}(Δ). The subgroupoid generated by the two-sided tilting complexes of [APR] is called the reflection groupoid Ref(Δ). We show that there is a surjection Ref(Δ) ↠ W (Δ), where W (Δ) ⊂ GL_{n}(ℤ) is the Weyl group as in [BGP]. We also prove that for any orientation ω, Ref(Δ)(ω, ω) = ⟨τ_{ω}⟩ where τ_{ω} ∈ DPic_{k}(A_{ω}) is the translation.

Acknowledgments. We wish to thank A. Bondal, I. Reiten and M. Van den Bergh for very helpful conversations and correspondences. Thanks to the referee for suggestions and improvements to the paper. Some of the work on the paper was done during visits to MIT and the University of Washington, and we thank the departments of mathematics at these universities for their hospitality. The second author was supported by the Weizmann Institute of Science throughout most of this research.

1. Conventions and Preliminary Results

In this section we fix notations and conventions to be used throughout the paper. This is needed since there are conflicting conventions in the literature regarding quivers and path algebras. We also prove two preliminary results.

Throughout the paper k denotes a fixed algebraically closed field. Our notation for a quiver is Q⃗ = (Q⃗_{0}, Q⃗_{1}); Q⃗_{0} is the set of vertices, and Q⃗_{1} is the set of arrows. For x, y ∈ Q⃗_{0}, d(x, y) denotes the number of arrows x → y.

In this section the letter A denotes a k-linear category that’s equivalent to a small full subcategory of itself (this assumption avoids some set theoretical problems). Let us write Aut_{k}(A) for the class of k-linear auto-equivalences of A. Then the set

(1.1) Out_{k}(A) = \frac{Aut_{k}(A)}{functorial isomorphism}

is a group.

Suppose A is a k-linear additive Krull-Schmidt category (i.e. dim_{k} Hom_{A}(M, N) < ∞ and all idempotents split). We define the quiver Γ⃗(A) of A as follows: Γ⃗_{0}(A) is the set of isomorphism classes of indecomposable objects of A. For two vertices x, y there are d(x, y) arrows α : x → y, where we choose representatives M_{x} ∈ x, M_{y} ∈ y, Irr(M_{x}, M_{y}) = rad(M_{x}, M_{y})/ rad^{2}(M_{x}, M_{y}) is the space of irreducible morphisms and d(x, y) := dim_{k} Irr(M_{x}, M_{y}). See [Rl] Section 2.2 for full details.

If A is a k-linear category (possibly without direct sums) we can embed it in the additive category A × ℕ, where a morphism (x, m) → (y, n) is an n × m matrix with entries in A(x, y) = Hom_{A}(x, y). Of course if A is additive then A ≈ A × ℕ. If A × ℕ is Krull-Schmidt then we shall write Γ⃗(A) for the quiver Γ⃗(A × ℕ).

Let Q⃗ be a quiver. Assume that for every vertex x ∈ Q⃗_{0} the number of arrows starting or ending at x is finite, and for every two vertices x, y ∈ Q⃗_{0} there is only a finite number of oriented paths from x to y. Let k⟨Q⃗⟩ be the path category, whose set of objects is Q⃗_{0}, the morphisms are generated by the identities and the arrows, and the only relations arise from incomposability of paths. Observe that this differs from the definition in [Rl], where the path category corresponds to k⟨Q⃗⟩ × ℕ in our notation. The morphism spaces of k⟨Q⃗⟩ are ℤ-graded, where the arrows have degree 1. If I ⊂ k⟨Q⃗⟩ is any ideal contained in rad_{k⟨Q⃗⟩}^{2} = ⨁_{n≥2} k⟨Q⃗⟩_{n}, and

k⟨Q⃗, I⟩ := k⟨Q⃗⟩/I is the quotient category, then the additive category k⟨Q⃗, I⟩ × ℕ is Krull-Schmidt, and the quiver of k⟨Q⃗, I⟩ is Γ⃗(k⟨Q⃗, I⟩) = Q⃗.

Let A be a finite dimensional k-algebra. In representation theory there are three equivalent ways to define the quiver Δ⃗ = Δ⃗(A) of A. The set Δ⃗_{0} enumerates either a complete set of primitive orthogonal idempotents {e_{x}}_{x∈Δ⃗_{0}}, as in [ARS] Section III.1; or it enumerates the simple A-modules {S_{x}}_{x∈Δ⃗_{0}}, as in [Rl] Section 2.1; or it enumerates the indecomposable projective A-modules {P_{x}}_{x∈Δ⃗_{0}}, as in [Rl] Section 2.4. The arrow multiplicity is in all cases

d(x, y) = dim_{k} e_{x}(𝔯/𝔯^{2})e_{y} = dim_{k} Ext_{A}^{1}(S_{y}, S_{x}) = dim_{k} Irr_{proj A}(P_{x}, P_{y}).

Here 𝔯 is the Jacobson radical and proj A is the category of finitely generated projective modules, which is Krull-Schmidt. Observe that the third definition is just Δ⃗(A) = Γ⃗(proj A).

Remark 1.2. The set Δ⃗_{0} also enumerates the prime spectrum of A, Spec A ≅ {𝔭_{x}}_{x∈Δ⃗_{0}}. One can show that 𝔯/𝔯^{2} ≅ ⨁_{x,y∈Δ⃗_{0}}(𝔭_{x} ∩ 𝔭_{y})/𝔭_{x}𝔭_{y} as A-A-bimodules. This implies that d(x, y) > 0 iff there is a second layer link 𝔭_{x} ⇝ 𝔭_{y} (cf. [MR] Section 4.3.7). Thus if we ignore multiple arrows, the quiver Δ⃗ is precisely the link graph of A.

Recall that a translation τ is an injective function from a subset of Q⃗_{0}, called the set of non-projective vertices, to Q⃗_{0}, such that d(τ(y), x) = d(x, y). Q⃗ is a stable translation quiver if it comes with a translation τ such that all vertices are non-projective. A polarization μ is an injective function defined on the set of arrows β : x → y ending in non-projective vertices, with μ(β) : τ(y) → x. Cf. [Rl] Section 2.2.

Notation 1.3. Suppose the quiver Q⃗ has a translation τ and a polarization μ. Given a non-projective vertex y ∈ Q⃗_{0} let x_{1}, …, x_{m} be some labeling, without repetition, of the set of vertices {x | there is an arrow x → y} . Correspondingly label the arrows β_{i,j} : x_{i} → y and α_{i,j} : τ(y) → x_{i}, where i = 1, …, m; j = 1, …, d_{i} = d(x_{i}, y); and α_{i,j} = μ(β_{i,j}). The mesh ending at y is the subquiver with vertices {τ(y), x_{i}, y} and arrows {α_{i,j}, β_{i,j}}.

Figure 1. The mesh ending at the vertex y when d_{i} = 1.

If Q⃗ has no multiple arrows then d_{i} = 1 and the picture of the mesh ending at y is shown in Figure 1.

The mesh relation at y is defined to be

∑_{i=1}^{m} ∑_{j=1}^{d_{i}} β_{i,j}α_{i,j} ∈ Hom_{k⟨Q⃗⟩}(τ(y), y). (1.4)

It is a homogeneous morphism of degree 2.

Definition 1.5. Let I_{m} be the mesh ideal in the category k⟨Q⃗⟩, i.e. the two sided ideal generated by the mesh relations (1.4) where y runs over all non-projective vertices. The quotient category

k⟨Q⃗, I_{m}⟩ := k⟨Q⃗⟩/I_{m}

is called the mesh category.

Observe that in [Rn], [Rl] and [Ha] the notation for k⟨Q⃗, I_{m}⟩ is k(Q⃗).

Now let Δ⃗ be a finite quiver without oriented cycles and A = kΔ⃗ the path algebra. Our convention for the multiplication in A is as follows. If x ⟶^{α} y and y ⟶^{β} z are paths in Δ⃗, and if x ⟶^{γ} z is the concatenated path, then γ = αβ in A. We note that the composition rule in the path category k⟨Δ⃗⟩ is opposite to that in A, so that ⨁_{x,y} Hom_{k⟨Δ⃗⟩}(x, y) = A^{∘}.

For every x ∈ Δ⃗_{0} let e_{x} ∈ A be the corresponding idempotent, and let P_{x} = Ae_{x} be the indecomposable projective A-module. So {P_{x}}_{x∈Δ⃗_{0}} is a set of representatives of the isomorphism classes of indecomposable projective A-modules. Define P ⊂ mod A to be the full subcategory on the objects {P_{x}}_{x∈Δ⃗_{0}}. Then P × ℕ ≈ proj A and Δ⃗ ≅ Γ⃗(P) ≅ Γ⃗(proj A).

There is an equivalence of categories k⟨Δ⃗⟩ ⟶^{≈} P that sends x ↦ P_{x}, and an arrow α : x → y goes to the right multiplication P_{x} = Ae_{x} ⟶^{α} P_{y} = Ae_{y}. We will identify P and k⟨Δ⃗⟩ in this way.

Recall that the automorphism group Aut_{k}(A) is a linear algebraic group. Let H be the closed subgroup

H := {F ∈ Aut_{k}(A) | F (e_{x}) = e_{x} for all x ∈ Δ⃗_{0}}.

Lemma 1.6. H is connected.

Proof. For each pair x, y ∈ Δ⃗_{0} the k-vector space P(x, y) := Hom_{P}(x, y) ≅ e_{x}Ae_{y} is graded. Let P(x, y)_{i} be the homogeneous component of degree i, and

Y := ∏_{x,y∈Δ⃗_{0}}(Aut_{k}(P(x, y)_{1}) × Hom_{k}(P(x, y)_{1}, P(x, y)_{≥2})).

This is a connected algebraic variety. Since A is generated as k-algebra by the idempotents and the arrows, and the only relations in A are the monomial relations arising from incomposability of paths, it follows that any element F′ ∈ Y extends uniquely to a k-algebra automorphism F of A that fixes the idempotents. Conversely any automorphism F ∈ H restricts to an element F′ of Y. This bijection Y → H is an isomorphism of varieties. Hence H is connected.

The next result is partially proved in [GS] Theorem 4.8 (they assume k has characteristic 0).

Proposition 1.7. Let A be a basic hereditary finite dimensional algebra over an algebraically closed field k, with quiver Δ⃗.

1. There is a split exact sequence of groups

1 → Out_{k}^{0}(A) → Pic_{k}(A) → Aut(Δ⃗_{0}; d) → 1.

2. The group Out_{k}^{0}(A) is trivial when Δ⃗ is a tree.

Proof. 1. Since A is basic we have Out_{k}(A) = Pic_{k}(A). By Morita theory we have Pic_{k}(A) ≅ Out_{k}(Mod A). Any auto-equivalence of the category P extends to an auto-equivalence of Mod A (using projective resolutions), and this induces an isomorphism of groups Out_{k}(P) →^{≃} Out_{k}(Mod A).

The class of auto-equivalences Aut_{k}(P) is actually a group here. In fact Aut_{k}(P) can be identified with the subgroup of Aut_{k}(A) consisting of automorphisms that permute the set of idempotents {e_{x}} ⊂ A.

Define a homomorphism of groups q : Out_{k}(A) → Aut(Δ⃗_{0}; d) by q(F)(x) = y if F P_{x} ≅ P_{y}. Thus we get a commutative diagram

[figure: H; Aut_{k}(P); f; Aut_{k}(A); g; Out_{k}(P); ≅; Out_{k}(A); q; Aut(Δ⃗_{0}; d).]

For an element F ∈ Aut_{k}(P) we have F |_{{e_{x}}} = qf(F), and hence Ker(q) = f(H) = g(H). According to Lemma 1.6, H is connected. Because g is a morphism of varieties we see that Ker(q) is connected. But the index of Ker(q) is finite, so we get Ker(q) = Out_{k}^{0}(A).

In order to split q we choose any splitting of Aut(Δ⃗) ↠ Aut(Δ⃗_{0}; d) and compose it with the homomorphism Aut(Δ⃗) → Aut_{k}(P).

2. When Δ⃗ is a tree the group H is a torus: H ≅ ∏_{x,y∈Δ⃗_{0}} Aut_{k}(P(x, y)_{1}). In fact H consists entirely of inner automorphisms that are conjugations by elements of the form ∑ λ_{x}e_{x} with λ_{x} ∈ k^{×}. Thus g(H) = 1.

The next theorem seems to be known to some experts, but we could not locate any reference in the literature. Since it is needed in the paper we have included a short proof. For a left coherent ring A (e.g. a hereditary ring) we denote by mod A the category of coherent A-modules. In the theorem k could be any field.

Theorem 1.8. Suppose A is a hereditary k-algebra. Then any k-linear triangle auto-equivalence of D^{b}(mod A) is standard.

Proof. Let F be a k-linear triangle auto-equivalence of D^{b}(mod A). By [Rd] Corollary 3.5 there exits a two-sided tilting complex T with T ≅ F A in D^{b}(mod A). Replacing F with (T^{∨} ⊗_{A}^{L} −)F we may assume that F A ≅ A. Hence F (mod A) ⊂ mod A, and F |_{mod A} is an equivalence. Classical Morita theory says that F |_{mod A} ≅ (P ⊗_{A} −) for some invertible bimodule P. So replacing F by (P^{∨} ⊗_{A} −)F we can assume that there is an isomorphism ϕ^{0} : F |_{mod A} ≅ 1_{mod A}.

Now for every object M ∈ D^{b}(mod A) we can choose an isomorphism M ≅ ⨁_{i} M_{i}[−i] with M_{i} ∈ mod A (cf. [Ha] Lemma I.5.2). Define ϕ_{M} : F M →^{≃} M to be the composition

F M ≅ ⨁(F M_{i})[−i] \underrightarrow{∑ ϕ_{M_{i}}^{0} [−i]} ⨁_{i} M_{i}[−i] ≅ M.

According to the proof of [BO] Proposition A.3, for any morphism α : M → N one has ϕ_{N}F (α) = αϕ_{M}, so ϕ : F → 1_{D^{b}(mod A)} is an isomorphism of functors.

Corollary 1.9. Suppose A is a hereditary k-algebra. Then

DPic_{k}(A) ≅ Out_{k}^{tr}(D^{b}(mod A)).

Proof. The group homomorphism DPic_{k}(A) → Out_{k}^{tr}(D^{b}(mod A)) is injective, say by [Ye] Proposition 2.2, and it is surjective by the theorem.

2. An Equivalence of Categories

In this section we prove the technical result Theorem 2.6. It holds for any finite dimensional hereditary k-algebra A. In the special case of finite representation type, Theorem 2.6 is just [Ha] Proposition I.5.6. Our result is the derived category counterpart of [Rl] Lemma 2.3.3. For notation see Section 1 above.

We use a few facts about Auslander-Reiten triangles in D^{b}(mod A). These facts are well known to experts in representation theory, but for the benefit of other readers we have collected them in Theorems 2.1 and 2.4.

Let D be a k-linear triangulated category, which is Krull-Schmidt (as additive category). As in any Krull-Schmidt category, sink and source morphisms can be defined in D; cf. [Rl] Section 2.2. In [Ha] Section I.4, Happel defines Auslander-Reiten triangles in D, generalizing the Auslander-Reiten (or almost split) sequences in an abelian Krull-Schmidt category. A triangle M′ ⟶^{g} M ⟶^{f} M′′ → M′[1] in D is an Auslander-Reiten triangle if g is a source morphism, or equivalently if f is a sink morphism. As before, we denote by M_{x} ∈ D an indecomposable object in the isomorphism class x ∈ Γ⃗(D).

Now let Δ⃗ be a finite quiver without oriented cycles, and A = kΔ⃗ the path algebra. For M ∈ mod k let M^{∗} := Hom_{k}(M, k). Define auto-equivalences σ and τ of D^{b}(mod A) by σM := M[1] and τM := R Hom_{A}(M, A)^{∗}[−1] ≅ A^{∗}[−1] ⊗_{A}^{L} M.

Theorem 2.1 (Happel, Ringel). Let A = kΔ⃗. Then the following hold. 1. As an additive k-linear category, D^{b}(mod A) is a Krull-Schmidt category.

2. The quiver Γ⃗ := Γ⃗(D^{b}(mod A)) is a stable translation quiver, and the translation τ satisfies M_{τ(x)} ≅ τM_{x}.

3. The Auslander-Reiten triangles in D^{b}(mod A) (up to isomorphism) correspond bijectively to the meshes in Γ⃗. In the notation 1.3 with Q⃗ = Γ⃗ these triangles are

M_{τ(y)} ⟶^{(g_{i,j})} ⨁_{i=1}^{m} ⨁_{j=1}^{d_{i}} M_{x_{i}} ⟶^{(f_{i,j})^{t}} M_{y} → M_{τ(y)}[1].

4. A morphism (g_{i,j}) : M_{τ(y)} → ⨁_{i=1}^{m} ⨁_{j=1}^{d_{i}} M_{x_{i}} is a source morphism iff for all i, {g_{i,j}}_{j=1}^{d_{i}} is a basis of Irr_{D^{b}(mod A)}(M_{τ(y)}, M_{x_{i}}). Likewise a morphism (f_{i,j})^{t} : ⨁_{i=1}^{m} ⨁_{j=1}^{d_{i}} M_{x_{i}} → M_{y} is a sink morphism iff for all i, {f_{i,j}}_{j=1}^{d_{i}} is basis of Irr_{D^{b}(mod A)}(M_{x_{i}}, M_{y}).

Proof. 1. This is implicit in [Ha] Sections I.4 and I.5. In particular [Ha] Lemma I.5.2 shows that for any indecomposable object M ∈ D^{b}(mod A) the ring End_{D^{b}(mod A)}(M) is local.

2. See [Ha] Corollary I.4.9.

3. According to [Ha] Theorem I.4.6 and Lemma I.4.8, for each y ∈ Γ⃗_{0} there exists such an Auslander-Reiten triangle. By [Ha] Proposition I.4.3 these are all the Auslander-Reiten triangles, up to isomorphism.

4. Since source and sink morphism depend only on the structure of k-linear additive category on D^{b}(mod A) (cf. [Ha] Section I.4.5) we may use [Rl] Lemma 2.2.3.

The Auslander-Reiten quiver Γ⃗(D^{b}(mod A)) contains the quiver Δ⃗, as the full subquiver with vertices corresponding to the indecomposable projective A-modules, under the inclusion mod A ⊂ D^{b}(mod A).

Definition 2.2. We call a connected component of Γ⃗(D^{b}(mod A)) irregular if it is isomorphic to the connected component containing Δ⃗, and we denote by Γ⃗^{irr} the disjoint union of all irregular components of Γ⃗(D^{b}(mod A)).

The name “irregular” is inspired by [ARS] Section VIII.4, where regular components of Γ⃗(mod A) are discussed. The quiver Γ⃗^{irr} will be of special interest to us. It’s structure is explained in Theorem 2.4 below. But first we need to recall the following definition due to Riedtmann [Rn],

Definition 2.3. From the quiver Δ⃗ one can construct another quiver, denoted by ℤ⃗Δ⃗. The vertex set of ℤ⃗Δ⃗ is ℤ × Δ⃗_{0}, and for every arrow x ⟶^{α} y in Δ⃗ there are

arrows (n, x) ⟶^{(n,α)} (n, y) and (n, y) ⟶^{(n,α^{∗})} (n + 1, x) in ℤ⃗Δ⃗.

The function τ(n, x) = (n − 1, x) makes ℤ⃗Δ⃗ into a stable translation quiver. Observe that τ is an automorphism of the quiver ℤ⃗Δ⃗, not just of the vertex set (ℤ⃗Δ⃗)_{0}. ℤ⃗Δ⃗ is equipped with a polarization μ, given by μ(n + 1, α) = (n, α^{∗}) and μ(n, α^{∗}) = (n, α). See Figures 3 and 4 in Section 5 for examples. We identify Δ⃗ with the subquiver {0} × Δ⃗ ⊂ ℤ⃗Δ⃗.

Next let us define a quiver ℤ × (ℤ⃗Δ⃗) := ∐_{m∈ℤ} ℤ⃗Δ⃗; the connected components are {m} × (ℤ⃗Δ⃗), m ∈ ℤ. Define an automorphism σ of ℤ × (ℤ⃗Δ⃗) by the action σ(m) = m + 1 on the first factor. There is a translation τ and a polarization μ of ℤ × (ℤ⃗Δ⃗) that extend those of ℤ⃗Δ⃗ ≅ {0} × (ℤ⃗Δ⃗) and commute with σ.

The auto-equivalences σ and τ of D^{b}(mod A) induce commuting permutations of Γ⃗_{0}, which we also denote by σ and τ respectively.

Theorem 2.4 (Happel). 1. If A has finite representation type then there is a unique isomorphism of quivers

ρ : ℤ⃗Δ⃗ →^{≃} Γ⃗^{irr}

which is the identity on Δ⃗ and commutes with τ on vertices. Furthermore Γ⃗^{irr} = Γ⃗(D^{b}(mod A)).

2. If A has infinite representation type then there exists an isomorphism of quivers

ρ : ℤ × (ℤ⃗Δ⃗) →^{≃} Γ⃗^{irr}

which is the identity on Δ⃗ and commutes with τ and σ on vertices. If Δ⃗ is a tree then the isomorphism ρ is unique.

Proof. This is essentially [Ha] Proposition I.5.5 and Corollary I.5.6.

Fix once and for all for every vertex x ∈ Γ⃗_{0}^{irr} an indecomposable object M_{x} ∈ D^{b}(mod A) which represents x, and such that M_{x} = P_{x} for x ∈ Δ⃗_{0}. Define B ⊂ D^{b}(mod A) to be the full subcategory with objects {M_{x} | x ∈ (ℤ⃗Δ⃗)_{0}}.

The additive category B × ℕ is also Krull-Schmidt, so for M_{x}, M_{y} ∈ B the two k-modules Irr_{B×ℕ}(M_{x}, M_{y}) and Irr_{D^{b}(mod A)}(M_{x}, M_{y}) could conceivably differ (cf.

[Rl] Section 2.2). But this is not the case as we see in the lemma below.

Lemma 2.5. Suppose I ⊂ ℤ is a segment (i.e. I = {i ∈ ℤ | a ≤ i ≤ b} with a, b ∈ ℤ ∪ {±∞}). Let B(I) ⊂ D^{b}(mod A) be the full subcategory on the objects M_{x}, x ∈ I × Δ⃗_{0} ⊂ Γ⃗(D^{b}(mod A))_{0}. Then for any M_{x}, M_{y} ∈ B(I) one has

Irr_{B(I)×ℕ}(M_{x}, M_{y}) ≅ Irr_{D^{b}(mod A)}(M_{x}, M_{y}).

Proof. Consider a sink morphism in D^{b}(mod A) ending in M_{(n,y)}, (n, y) ∈ (ℤ⃗Δ⃗)_{0}. By Theorem 2.1(3) and Theorem 2.4, it is of the form (f_{i,j})^{t} : ⨁_{i=1}^{m} ⨁_{j=1}^{d_{i}} M_{(n−ϵ_{i},x_{i})} → M_{(n,y)} with ϵ_{i} ∈ {0, 1} (cf. Notation 1.3). From the definition of a sink morphism we see that this is also a sink morphism in the category B × ℕ.

According to [Rl] Lemma 2.2.3 (dual form), both k-modules Irr_{B×ℕ}(M_{(n−ϵ_{i},x_{i})}, M_{(n,y)}) and Irr_{D^{b}(mod A)}(M_{(n−ϵ_{i},x_{i})}, M_{(n,y)}) have the morphisms f_{i,1}, …, f_{i,d_{i}} as basis. And there are no irreducible morphisms N → M_{(n,y)} for indecomposable objects N not isomorphic to one of the M_{(n−ϵ_{i},x_{i})}, in either category. Thus the lemma is proved for B(I) = B.

Let x, y ∈ Δ⃗_{0} and l, n ∈ ℤ. If Hom(M_{(l,x)}, M_{(n,y)}) ≠ 0 then necessarily l ≤ n. This is clear for l = 0, since M_{(0,x)} is a projective module, and an easy calculation shows that for n < 0,

H^{0}(M_{(n,y)}) ≅ H^{0}(A^{∗}[−1] ⊗_{A}^{L} ⋯ ⊗_{A}^{L} A^{∗}[−1] ⊗_{A}^{L} M_{(0,y)}) = 0.

In general we can translate by τ^{−l}.

Now take an arbitrary segment I. The paragraph above implies that for n, l ∈ I and i ≥ 0, rad_{B(I)×ℕ}^{i}(M_{(l,x)}, M_{(n,y)}) = rad_{B×ℕ}^{i}(M_{(l,x)}, M_{(n,y)}). Hence Irr_{B(I)×ℕ}(M_{(l,x)}, M_{(n,y)}) = Irr_{B×ℕ}(M_{(l,x)}, M_{(n,y)}).

Henceforth we shall simply write Irr(M_{x}, M_{y}) when x, y ∈ (ℤ⃗Δ⃗)_{0}. The lemma implies that the quiver of the category B(I) is the full subquiver I⃗Δ⃗ ⊂ ℤ⃗Δ⃗.

Note that for I = {0} we get B(I) = P. Since P is canonically equivalent to k⟨Δ⃗⟩, there is a full faithful k-linear functor G_{0} : k⟨Δ⃗⟩ → B such that G_{0}x = M_{x} = P_{x} for every vertex x ∈ Δ⃗_{0}, and {G_{0}(α_{j})}_{j=1}^{d(x,y)} is a basis of Irr(M_{x}, M_{y}) for every pair of vertices x, y, where α_{1}, ⋯, α_{d(x,y)} are the arrows α_{j} : x → y.

Theorem 2.6. Let Δ⃗ be a finite quiver without oriented cycles, A = kΔ⃗ its path algebra, k⟨ℤ⃗Δ⃗, I_{m}⟩ the mesh category (Definitions 2.3 and 1.5) and B ⊂ D^{b}(mod A) the full subcategory on the objects {M_{x}}_{x∈(ℤ⃗Δ⃗)_{0}}. Then there is a k-linear functor

G : k⟨ℤ⃗Δ⃗, I_{m}⟩ → B

such that (i) Gx = M_{x} for each vertex x ∈ (ℤ⃗Δ⃗)_{0}. (ii) G|_{k⟨Δ⃗⟩} = G_{0}. (iii) G is full and faithful. Moreover, the functor G is unique up to isomorphism.

In other words, there is a unique equivalence G extending G_{0}.

Proof. Let Q⃗^{+} ⊂ ℤ⃗Δ⃗ be the full subquiver with vertex set {(n, y) | n ≥ 0}. Given a vertex (n, y) in Q⃗^{+}, denote by p(n, y) the number of its predecessors, i.e. the number of vertices (m, x) such that there is a path (m, x) → ⋯ → (n, y) in Q⃗^{+}. For any p ≥ 0 let Q⃗_{p}^{+} be the full subquiver with vertex set {(n, y) | n ≥ 0, p(n, y) ≤ p}. Q⃗_{p}^{+} is a translation quiver with polarization, and k⟨Q⃗_{p}^{+}, I_{m}⟩ ⊂ k⟨ℤ⃗Δ⃗, I_{m}⟩ is a full subcategory.

By recursion on p, we will define a functor G : k⟨Q⃗_{p}^{+}, I_{m}⟩ → B satisfying conditions (i), (ii) and (iv) Let x, y be a pair of vertices and let α_{1}, ⋯, α_{d(x,y)} be the arrows α_{j} : x → y.

Then {G(α_{j})}_{j=1}^{d(x,y)} is a basis of Irr(M_{x}, M_{y}).

Take p ≥ 0. It suffices to define G(α) for an arrow α in Q⃗_{p}^{+}. These arrows fall into three cases, according to their end vertex (n, y):

(a) p(n, y) < p, in which case any arrow α ending in (n, y) is in Q⃗_{p−1}^{+}, and G(α) is already defined.

(b) p(n, y) = p and n = 0. Any arrow α ending in (n, y) is in Δ⃗, so we define G(α) := G_{0}(α). By Lemma 2.5 condition (iv) holds.

(c) p(n, y) = p and n ≥ 1. In this case (n, y) is a non-projective vertex in Q⃗_{p}^{+}, and we consider the mesh ending at (n, y). The vertices with arrows to (n, y) are (n − ϵ_{i}, x_{i}), where i = 1, …, m; x_{i} ∈ Δ⃗_{0} and ϵ_{i} = 0, 1 (cf. Notation 1.3). Since p(n − 1, y) < p(n − ϵ_{i}, x_{i}) < p the arrows α_{i,j} are all in the quiver Q⃗_{p−1}^{+}, and hence G(α_{i,j}) are defined.

According to condition (iv), Lemma 2.5 and Theorem 2.1(4) it follows that there exists an Auslander-Reiten triangle

M_{(n−1,y)} ⟶^{(G(α_{i,j}))} ⨁_{i=1}^{m} ⨁_{j=1}^{d_{i}} M_{(n−ϵ_{i},x_{i})} ⟶^{(f_{i,j})^{t}} M_{(n,y)} → M_{(n−1,y)}[1] (2.7)

in D^{b}(mod A). Define

G(β_{i,j}) := f_{i,j} : M_{(n−ϵ_{i},x_{i})} → M_{(n,y)}.

Note that the mesh relation ∑ β_{i,j}α_{i,j} in k⟨Q⃗_{p}^{+}⟩ is sent by G to ∑ G(β_{i,j})G(α_{i,j}) = 0, so we indeed have a functor G : k⟨Q⃗_{p}^{+}, I_{m}⟩ → B. Also, by Theorem 2.1(4), for any i the set {G(β_{i,j})}_{j=1}^{d_{i}} is a basis of Irr(M_{(n−ϵ_{i},x_{i})}, M_{(n,y)}).

Thus we obtain a functor G : k⟨Q⃗^{+}, I_{m}⟩ → B.

By symmetry we construct a functor G : k⟨Q⃗^{−}, I_{m}⟩ → B for negative vertices (i.e. n ≤ 0), extending G_{0}. Putting the two together we obtain a functor G : k⟨ℤ⃗Δ⃗, I_{m}⟩ → B satisfying conditions (i), (ii) and (iv).

Let us prove G is fully faithful. For any n ∈ ℤ there is a full subquiver ℤ⃗_{≥n}Δ⃗ ⊂ ℤ⃗Δ⃗, on the vertex set {(i, x) | i ≥ n}. Correspondingly there are full subcategories k⟨ℤ⃗_{≥n}Δ⃗, I_{m}⟩ ⊂ k⟨ℤ⃗Δ⃗, I_{m}⟩ and B(ℤ_{≥n}) ⊂ B. It suffices to prove that G : k⟨ℤ⃗_{≥n}Δ⃗, I_{m}⟩ → B(ℤ_{≥n}) is fully faithful. By Lemma 2.5 the quiver of B(ℤ_{≥n}) is ℤ⃗_{≥n}Δ⃗, which is pre-projective. So we can use the last two paragraphs in the proof of [Rl] Lemma 2.3.3 almost verbatim.

Finally we shall prove that G is unique up to isomorphism. Suppose G′ : k⟨ℤ⃗Δ⃗, I_{m}⟩ → B is another k-linear functor satisfying conditions (i)-(iii). We will show there is an isomorphism ϕ : G →^{≃} G′ that is the identity on k⟨Δ⃗⟩.

By recursion on p we shall exhibit an isomorphism ϕ : G|_{k⟨Q⃗_{p}^{+},I_{m}⟩} →^{≃} G′|_{k⟨Q⃗_{p}^{+},I_{m}⟩}. It suffices to consider case (c) above, so let (n, y) be such a vertex. Then, because G′(α_{i,j}) = ϕ_{(n−ϵ_{i},x_{i})}G(α_{i,j})ϕ_{(n−1,y)}^{−1}, we have

∑_{i,j} G′(β_{i,j})ϕ_{(n−ϵ_{i},x_{i})}G(α_{i,j}) = G′(∑_{i,j} β_{i,j}α_{i,j})ϕ_{(n−1,y)} = 0.

Applying Hom(−, M_{(n,y)}) to the triangle (2.7) we obtain a morphism a ∈ End(M_{(n,y)}) such that G′(β_{i,j})ϕ_{(n−ϵ_{i},x_{i})} = aG(β_{i,j}). Because G′ is faithful we see that a ≠ 0, and since End(M_{(n,y)}) ≅ k it follows that a is invertible. Set ϕ_{(n,y)} := a ∈ Aut(M_{(n,y)}). This yields the desired isomorphism ϕ : G|_{k⟨Q⃗_{p}^{+},I_{m}⟩} →^{≃} G′|_{k⟨Q⃗_{p}^{+},I_{m}⟩}.

By symmetry the isomorphism ϕ extends to Q⃗^{−}.

The uniqueness of G gives the next corollary.

Corollary 2.8. Let F be a k-linear auto-equivalence of k⟨ℤ⃗Δ⃗, I_{m}⟩ fixing all objects, and such that F |_{k⟨Δ⃗⟩} ≅ 1_{k⟨Δ⃗⟩}. Then F ≅ 1_{k⟨ℤ⃗Δ⃗,I_{m}⟩}.

Remark 2.9. Beware that if A has infinite representation type then k⟨Γ⃗^{irr}, I_{m}⟩ is not equivalent to the full subcategory of D^{b}(mod A) on the objects {M_{x}}_{x∈Γ⃗_{0}^{irr}}.

This is because there are nonzero morphisms from the projective modules (vertices in the component ℤ⃗Δ⃗) to the injective modules (vertices in {1} × ℤ⃗Δ⃗).

3. The Representation of DPic_{k}(A) on the Quiver Γ⃗^{irr}

This section contains the proof of the main result of the paper, Theorem 0.1 (restated here as Theorem 3.8). It is deduced from the more technical Theorem 3.7. Throughout k is an algebraically closed field, Δ⃗ is a connected finite quiver without oriented cycles, and A = kΔ⃗ is the path algebra. We use the notation of previous sections.

Recall that A^{∗} = Hom_{k}(A, k) is a tilting complex. We shall denote by τ the class of A^{∗}[−1] in DPic_{k}(A), and by σ the class of A[1]. We identify an element T ∈ DPic_{k}(A) and the induced auto-equivalence F = T ⊗_{A}^{L} − of D^{b}(mod A).

Lemma 3.1. τ and σ are in the center of DPic_{k}(A).

Proof. The fact that σ is in the center of DPic_{k}(A) is trivial. As for τ, this follows immediately from [Rd] Proposition 5.2 (or by [BO] Proposition 1.3, since A^{∗} ⊗_{A}^{L} − is the Serre functor of D^{b}(mod A)).

In Definition 2.2 we introduced the quiver Γ⃗^{irr}. Recall that for a vertex x ∈ Γ⃗_{0}^{irr}, M_{x} ∈ D^{b}(mod A) is the representative indecomposable object.

Lemma 3.2. There is a group homomorphism

q : DPic_{k}(A) → Aut(Γ⃗_{0}^{irr}; d)^{⟨τ,σ⟩}

such that q(F)(x) = y iff F M_{x} ≅ M_{y}.

Proof. Given an auto-equivalence F of D^{b}(mod A), the formula q(F)(x) = y iff F M_{x} ≅ M_{y} defines a permutation q(F) of Γ⃗_{0}(D^{b}(mod A)) that preserves arrowmultiplicities. Hence it restricts to a permutation of Γ⃗_{0}^{irr}. By Lemma 3.1, q(F) commutes with τ and σ.

The group Out_{k}^{0}(A) was defined to be the identity component of Out_{k}(A).

Lemma 3.3. Ker(q) = Out_{k}^{0}(A).

Proof. Let T ∈ DPic_{k}(A). By Theorem 2.4 we know that Γ⃗_{0}^{irr} = ⋃_{i,j∈ℤ} τ^{i}σ^{j}(Δ⃗_{0}). Hence by Lemma 3.1, T ∈ Ker(q) iff T acts trivially on the set Δ⃗_{0}. In particular we see that Ker(q) ⊂ Pic_{k}(A). Now use Proposition 1.7.

Lemma 3.4. Suppose A has finite representation type. Then σ is in the center of the group Aut(Γ⃗^{irr})^{⟨τ⟩}.

Proof. According to [Rn] Section 2, the group Aut(ℤ⃗Δ⃗)^{⟨τ⟩} is abelian in all cases except D_{4}. But a direct calculation in this case (cf. Theorem 4.1) gives σ = τ^{−3}.

Before we can talk about the mesh category k⟨Γ⃗^{irr}, I_{m}⟩ of the quiver Γ⃗^{irr}, we have to fix a polarization μ on it. If the quiver Δ⃗ has no multiple arrows then so does Γ⃗^{irr} (by Theorem 2.4), and hence there is a unique polarization on it. If Δ⃗ isn’t a tree let us choose an isomorphism ρ : ℤ × (ℤ⃗Δ⃗) →^{≃} Γ⃗^{irr} as in that theorem. This determines a polarization μ on Γ⃗^{irr}. We also get a lifting of the permutation σ to an auto-equivalence of k⟨Γ⃗^{irr}, I_{m}⟩.

Lemma 3.5. There are group homomorphisms

p : Out_{k}(k⟨Γ⃗^{irr}, I_{m}⟩) → Aut(Γ⃗_{0}^{irr}; d)^{⟨τ⟩}

and

r : Aut(Γ⃗_{0}^{irr}; d)^{⟨τ⟩} → Out_{k}(k⟨Γ⃗^{irr}, I_{m}⟩)

satisfying p(F)(x) = F x for an auto-equivalence F and a vertex x; pr = 1; and both p and r commute with σ.

Proof. Since Γ⃗^{irr} ≅ Γ⃗(k⟨Γ⃗^{irr}, I_{m}⟩) we get a permutation p(F) ∈ Aut(Γ⃗_{0}^{irr}; d). Let’s prove that p(F) commutes with τ in Aut(Γ⃗_{0}^{irr}). Consider a vertex y ∈ Γ⃗_{0}^{irr}. In the Notation 1.3, there are vertices x_{i} and irreducible morphisms {F (α_{i,j})}_{j=1}^{d_{i}} and {F (β_{i,j})}_{j=1}^{d_{i}} that form bases of Irr_{k⟨Γ⃗^{irr},I_{m}⟩}(F τy, F x_{i}) and Irr_{k⟨Γ⃗^{irr},I_{m}⟩}(F x_{i}, F y) respectively. Since we have

∑ F (β_{i,j})F (α_{i,j}) = 0 ∈ rad_{k⟨Γ⃗^{irr},I_{m}⟩}^{2}(F τy, F y)/ rad_{k⟨Γ⃗^{irr},I_{m}⟩}^{3}(F τy, F y)

this must be a multiple of a mesh relation. Hence F τy = τF y.

Finally to define r we have to split Aut(Γ⃗^{irr}) ↠ Aut(Γ⃗_{0}^{irr}; d) consistently with μ. It suffices to order the set of arrows {α : x → y} for every pair of vertices x, y ∈ Γ⃗_{0}^{irr} consistently with μ. We only have to worry about this when A has infinite representation type. For any x, y ∈ Δ⃗_{0} choose some ordering of the set {α : x → y}. Using μ and σ this ordering can be transported to all of ℤ × (ℤ⃗Δ⃗). By the isomorphism ρ of Theorem 2.4 the ordering is copied to Γ⃗^{irr}.

Lemma 3.6. There exists a group homomorphism

q̃: DPic_{k}(A) → Out_{k}(k⟨Γ⃗^{irr}, I_{m}⟩)

such that pq̃ = q.

Proof. Choose an equivalence G : k⟨ℤ⃗Δ⃗, I_{m}⟩ → B as in Theorem 2.6. If A has infinite representation type then the isomorphism ρ we have chosen (as in Theorem 2.4) tells us how to extend G to an equivalence G : k⟨Γ⃗^{irr}, I_{m}⟩ → ∐_{l∈ℤ} B[l] that commutes with σ (cf. Remark 2.9).

Let F be a triangle auto-equivalence of D^{b}(mod A). Then F induces a permutation π = q(F) of the set Γ⃗_{0}^{irr} that commutes with σ. For every vertex x ∈ Γ⃗_{0}^{irr} choose an isomorphism

ϕ_{x} : F M_{x} →^{≃} M_{π(x)}

in D^{b}(mod A). Given an arrow α : x → y in Γ⃗^{irr}, define the morphism q̃_{{ϕ_{x}}}(F)(α) : π(x) → π(y) by the condition that the diagram

F M_{x}  ⟶^{F G(α)} F M_{y}
ϕ_{x}↓  ^{ϕ_{y}} ↓
M_{π(x)} ⟶^{Gq̃_{{ϕx}}(F)(α)} M_{π(y)}

commutes. Then q̃_{{ϕ_{x}}}(F) ∈ Aut_{k}(k⟨Γ⃗^{irr}, I_{m}⟩).

If {ϕ′_{x}} is another choice of isomorphisms ϕ′_{x} : F M_{x} →^{≃} M_{π(x)} then {ϕ′_{x}ϕ_{x}^{−1}} is an isomorphism of functors q̃_{{ϕ_{x}}}(F) → q̃_{{ϕ′_{x}}}(F), so the map q̃ : DPic_{k}(A) → Out_{k}(k⟨Γ⃗^{irr}, I_{m}⟩) is independent of these choices.

It is easy to check that q̃ respects composition of equivalences.

Theorem 3.7. Let A be an indecomposable basic hereditary finite dimensional kalgebra with quiver Δ⃗. Then the homomorphism q̃ of Lemma 3.6 induces an isomorphism of groups

DPic_{k}(A) ≅ Out_{k}(k⟨Γ⃗^{irr}, I_{m}⟩)^{⟨σ⟩} ≅ \begin{cases}Out_{k}(k⟨ℤ⃗Δ⃗, I_{m}⟩) if A has finite rep- resentation type \\ Out_{k}(k⟨ℤ⃗Δ⃗, I_{m}⟩) × ⟨σ⟩ otherwise. \end{cases}

Proof. The proof has three parts.

1. We show that the homomorphism

q̃: DPic_{k}(A) → Out_{k}(k⟨Γ⃗^{irr}, I_{m}⟩)

of Lemma 3.6 is injective. Let T be a two-sided tilting complex such that q̃(T) ≅ 1_{k⟨Γ⃗^{irr},I_{m}⟩}. Then the permutation q(T) fixes the vertices of Δ⃗ ⊂ Γ⃗^{irr}. Using the fact that A ≅ ⨁_{x∈Δ⃗_{0}} M_{x} we see that T ≅ A in D^{b}(mod A). Replacing T with H^{0} T we may assume T is a single bimodule. According to [Ye] Proposition 2.2, we see that T is actually an invertible bimodule. Since k⟨Δ⃗⟩ → k⟨ℤ⃗Δ⃗, I_{m}⟩ is full we get q̃(T)|_{k⟨Δ⃗⟩} ≅ 1_{k⟨Δ⃗⟩}. Hence by Morita theory we have T ≅ A as bimodules.

2. Assume A has finite representation type, so that Γ⃗^{irr} ≅ ℤ⃗Δ⃗. We prove that

q̃: DPic_{k}(A) → Out_{k}(k⟨ℤ⃗Δ⃗, I_{m}⟩)

is surjective.

Consider a k-linear auto-equivalence F of k⟨ℤ⃗Δ⃗, I_{m}⟩. Let π := p(F) ∈ Aut((ℤ⃗Δ⃗)_{0}; d)^{⟨τ⟩} ≅ Aut(ℤ⃗Δ⃗)^{⟨τ⟩} as in the proof of Lemma 3.6. According to Lemma 3.4, π commutes with σ. Define

M := ⨁_{x∈Δ⃗_{0}} M_{π(0,x)} ∈ D^{b}(mod A).

Then for any x, y ∈ Δ⃗_{0} and integers n, i the equivalence G : k⟨ℤ⃗Δ⃗, I_{m}⟩ → B of Theorem 2.6 produces isomorphisms

Hom_{D^{b}(mod A)}(M_{π(0,x)}, M_{(n,y)}[i]) ≅ Hom_{k⟨ℤ⃗Δ⃗,I_{m}⟩}(π(0, x), σ^{i}(n, y))
≅ Hom_{k⟨ℤ⃗Δ⃗,I_{m}⟩}((0, x), σ^{i}π^{−1}(n, y))
≅ Hom_{D^{b}(mod A)}(M_{(0,x)}, M_{π^{−1}(n,y)}[i]).

Therefore

Hom_{D^{b}(mod A)}(M, M[i]) ≅ ⨁_{x,y∈Δ⃗_{0}} Hom_{D^{b}(mod A)}(M_{(0,x)}, M_{(0,y)}[i])

≅ \begin{cases}A^{∘} & if i = 0 \\ 0 & otherwise. \end{cases}

Also for any (n, y) there is some integer i and x ∈ Δ⃗_{0} such that

Hom_{D^{b}(mod A)}(M_{π(0,x)}, M_{(n,y)}[i]) ≠ 0.

Since any object N ∈ D^{b}(mod A) is a direct sum of indecomposables M_{(n,y)}, this implies that R Hom_{A}(M, N) ≠ 0 if N ≠ 0. By [Ye] Theorem 1.8 and the proof of “(ii) ⇒ (i)” of [Ye] Theorem 1.6 there exists a two-sided tilting complex T with T ≅ M in D(Mod A) (cf. [Rd] Section 3). Replacing F with q̃(T^{∨})F, where T^{∨} := R Hom_{A}(T, A), we can assume that p(F) is trivial.

Now that p(F) is trivial, F restricts to an auto-equivalence of k⟨Δ⃗⟩, and by Proposition 1.7 we have F |_{k⟨Δ⃗⟩} ≅ 1_{k⟨Δ⃗⟩}. Then Corollary 2.8 tells us F ≅ 1_{k⟨ℤ⃗Δ⃗,I_{m}⟩}.

3. Assume A has infinite representation type. Then the quiver isomorphism ρ of Theorem 2.4 induces a group isomorphism

Out_{k}(k⟨Γ⃗^{irr}, I_{m}⟩)^{⟨σ⟩} ≅ Out_{k}(k⟨ℤ⃗Δ⃗, I_{m}⟩) × ⟨σ⟩,

and ⟨σ⟩ ≅ ℤ. We prove that

q̃: DPic_{k}(A) → Out_{k}(k⟨ℤ⃗Δ⃗, I_{m}⟩) × ℤ

is surjective.

Take an auto-equivalence F of k⟨ℤ⃗Δ⃗, I_{m}⟩, and write π := p(F) ∈ Aut((ℤ⃗Δ⃗)_{0}; d)^{⟨τ⟩}. After replacing F with τ^{j}F for suitable j ∈ ℤ, we can assume that π(0, x) ∈ ℤ⃗_{≥0}Δ⃗ for all x ∈ Δ⃗_{0}. Because ℤ⃗_{≥0}Δ⃗ is the preprojective component of Γ⃗(mod A) (cf. [Rl]), we get

M := ⨁_{x∈Δ⃗_{0}} M_{π(0,x)} ∈ mod A.

As in part 2 above, End_{A}(M) = A^{∘}. Since M is a complete slice, [HR] Theorem 7.2 says that M is a tilting module. So M is a two-sided tilting complex over A. Replacing F by q̃(M^{∨})F we can assume p(F) is trivial. Let P be an invertible bimodule such that q̃(P)|_{k⟨Δ⃗⟩} ≅ F |_{k⟨Δ⃗⟩}. Replacing F with q̃(P^{∨})F we get F |_{k⟨Δ⃗⟩} ≅ 1_{k⟨Δ⃗⟩}. Then by Corollary 2.8 we get F ≅ 1_{⟨ℤ⃗Δ⃗,I_{m}⟩}.

The next theorem is Theorem 0.1 in the Introduction.

Theorem 3.8. Let A be an indecomposable basic hereditary finite dimensional algebra over an algebraically closed field k, with quiver Δ⃗.

1. There is an exact sequence of groups

1 → Out_{k}^{0}(A) → DPic_{k}(A) ⟶^{q} Aut(Γ⃗_{0}^{irr}; d)^{⟨τ,σ⟩} → 1.

This sequence splits.

2. If A has finite representation type then there is an isomorphism of groups

DPic_{k}(A) ≅ Aut(ℤ⃗Δ⃗)^{⟨τ⟩}.

3. If A has infinite representation type then there is an isomorphism of groups

DPic_{k}(A) ≅ (Aut((ℤ⃗Δ⃗)_{0}; d)^{⟨τ⟩} ⋉ Out_{k}^{0}(A)) × ℤ.

Proof. 1. By Theorem 3.7 and Lemma 3.5 the homomorphism q is surjective. Lemma 3.3 identifies Ker(q).

2. If A has finite representation type then Δ⃗ is a tree, so Out_{k}^{0}(A) = 1 by Proposition 1.7. By Theorem 2.4 and Lemma 3.4 we get

Aut(Γ⃗_{0}^{irr}; d)^{⟨τ,σ⟩} ≅ Aut(Γ⃗^{irr})^{⟨τ,σ⟩} ≅ Aut(ℤ⃗Δ⃗)^{⟨τ⟩}.

Type  Aut(Δ⃗)  Aut(ℤ⃗Δ⃗)^{⟨τ⟩}  Relation
A_{n}, n even  1  ⟨τ, σ⟩ ≅ ℤ  τ^{n+1} = σ^{−2}
A_{n}, n odd  1  ⟨τ, σ⟩ ≅ ℤ × (ℤ/2ℤ)  τ^{n+1} = σ^{−2}
D_{4}  S_{3}  Aut(Δ⃗) × ⟨τ⟩ ≅ S_{3} × ℤ  τ^{3} = σ^{−1}
D_{n}, n ≥ 5  S_{2}  Aut(Δ⃗) × ⟨τ⟩ ≅ S_{2} × ℤ τ^{n−1} = θσ^{−1}, n odd
τ^{n−1} = σ^{−1}, n even
E_{6}  S_{2}  Aut(Δ⃗) × ⟨τ⟩ ≅ S_{2} × ℤ  τ^{6} = θσ^{−1}
E_{7}  1  ⟨τ⟩ ≅ ℤ  τ^{9} = σ^{−1}
E_{8}  1  ⟨τ⟩ ≅ ℤ  τ^{15} = σ^{−1}

Table 1. The group Aut(ℤ⃗Δ⃗)^{⟨τ⟩} for a Dynkin quiver. The orientation of Δ⃗ is shown in Figure 2. In types D_{n} and E_{6}, θ is the element of order 2 in Aut(Δ⃗).

3. If A has infinite representation type then

Aut(Γ⃗_{0}^{irr}; d)^{⟨τ,σ⟩} ≅ Aut((ℤ⃗Δ⃗)_{0}; d)^{⟨τ⟩} × ⟨σ⟩

by Theorem 2.4. We know that σ is in the center of DPic_{k}(A).

We end the section with the following problem.

Problem 3.9. The Auslander-Reiten quiver Γ⃗(D^{b}(mod A)) is defined for any finite dimensional k-algebra A of finite global dimension. Can the action of DPic_{k}(A) on Γ⃗(D^{b}(mod A)) be used to determine the structure of DPic_{k}(A) for any such A?

4. Explicit Calculations

In this section we calculate the group structure of DPic_{k}(A) for the path algebra A = kΔ⃗ for several types of quivers. Throughout S_{m} denotes the permutation group of {1, …, m}.

Suppose Δ is a tree. Given an orientation ω of the edge set Δ_{1}, denote by Δ⃗_{ω} the resulting quiver, and by A_{ω} := kΔ⃗_{ω}. If ω and ω′ are two orientations of Δ then D^{b}(mod A_{ω}) ≈ D^{b}(mod A_{ω′}). This equivalence will be discussed in the next section. For now we just note that the groups DPic_{k}(A_{ω}) ≅ DPic_{k}(A_{ω′}), so we are allowed to choose any orientation of Δ when computing these groups. This observation is relevant to Theorems 4.1 and 4.2 below.

Theorem 4.1. Let Δ⃗ be a Dynkin quiver as shown in Figure 2, and let A := kΔ⃗ be the path algebra. Then Pic_{k}(A) ≅ Aut(Δ⃗) and DPic_{k}(A) ≅ Aut(ℤ⃗Δ⃗)^{⟨τ⟩}. The groups Aut(Δ⃗) and Aut(ℤ⃗Δ⃗)^{⟨τ⟩} are described in Table 1.

Proof. The isomorphisms are by Theorem 0.1 and Proposition 1.7. The data in the third column of Table 1 was calculated in [Rn] Section 4, except for the shift σ which did not appear in that paper. So we have to do a few calculations involving σ. Below are the calculations for types A_{n} and D_{4}; the rest are similar and are left to the reader as an exercise.

Figure 2. Orientations for the Dynkin graphs

Type A_{n}: Choose the orientation in Figure 2. The quiver ℤ⃗Δ⃗ looks like Figure 3. Therefore Aut(ℤ⃗Δ⃗)^{⟨τ⟩} = ⟨τ, η⟩ where η(0, 1) = (0, n) and η(0, n) = (n − 1, 1).

Now by [Ha] Section I.5.5 and [ARS] Sections VII.1 and VIII.5, the quiver Γ⃗(mod A) ⊂ ℤ⃗Δ⃗ is the full subquiver on the vertices in the triangle {(m, i) | m ≥ 0, m + i ≤ n}. The projective vertices are (0, i) and the injective vertices are (n − i, i), where i ∈ {1, …, n}. We see that σ(0, i) = (i, n + 1 − i), and the quiver Γ⃗((mod A)[1]) = σ(Γ⃗(mod A)) is the full subquiver on the vertices in the triangle {(m, i) | m ≤ n, m + i ≥ n + 1}. Hence η = τσ and Aut(ℤ⃗Δ⃗)^{⟨τ⟩} = ⟨τ, σ⟩. The relation τ^{−(n+1)} = σ^{2} is easily verified.

Type D_{4}: The quiver ℤ⃗Δ⃗ is in Figure 4, and Γ⃗(mod A) ⊂ ℤ⃗Δ⃗ is a full subquiver. From the shape of Δ⃗ we know that mod A should have 4 indecomposable projective modules, 3 having length 2 and one of them simple. From the shape of the opposite quiver Δ⃗^{∘} we also know that mod A should have 4 indecomposable injective modules, 3 of them simple and one of length 4. Counting dimensions using Auslander-Reiten sequences we conclude that Γ⃗(mod A) is the full subquiver on the vertices {0, 1, 2} × Δ⃗_{0}. The projective vertices are {(0, 1), (0, i)}, the injective vertices are {(2, 1), (2, i)}, and the simple vertices are {(0, 1), (2, i)}, where i ∈ {2, 3, 4}.

For i ∈ {1, 2, 3, 4} let P_{i}, S_{i} and I_{i}, be the projective, simple and injective modules respectively, indexed such that P_{i} ↠ S_{i} ↣ I_{i}, and with P_{i} = M_{(0,i)}. So P_{1} = S_{1} and I_{i} = S_{i} for i ∈ {2, 3, 4}. By the symmetry of the quiver it follows that there is a nonzero morphism (0, i) → (2, i) in k⟨ℤ⃗Δ⃗⟩ for i ∈ {2, 3, 4}, and hence M_{(2,i)} ≅ S_{i}

The rule for connecting Γ⃗(mod A) with Γ⃗(mod A[1]) (see [Ha] Section I.5.5) implies that M_{(3,1)} ≅ M_{(0,1)}[1] = P_{1}[1]. Therefore M_{(3,i)} ≅ P_{i′}[1] for i, i′ ∈ {2, 3, 4}. Now for each such i there is an Auslander-Reiten triangle M_{(2,i)} → M_{(3,1)} → M_{(3,i)} → M_{(2,i)}[1]. When this triangle is turned it gives an exact sequence 0 → P_{1} → P_{i′} → S_{i} → 0, and hence i′ = i. The conclusion is that σ(m, i) = (m + 3, i) for all (m, i) ∈ (ℤ⃗Δ⃗)_{0}, so σ = τ^{−3}.

Figure 3. The quiver ℤ⃗Δ⃗ for Δ⃗ of type A_{3}. The vertices in mod A are labeled.

Figure 4. The quiver ℤ⃗Δ⃗ for Δ⃗ of type D_{4}. The vertices in mod A are labeled.

Theorem 4.2. Let Δ⃗ be a quiver of type D̃_{n}, Ẽ_{6}, Ẽ_{7} or Ẽ_{8}, with the orientation shown in Figure 5. Then Pic_{k}(A) ≅ Aut(Δ⃗) and

DPic_{k}(A) ≅ ℤ × Aut(ℤ⃗Δ⃗)^{⟨τ⟩}.

The structure of the group Aut(ℤ⃗Δ⃗)^{⟨τ⟩} is given in Table 2.

Proof. The isomorphisms follow from Theorem 0.1 and Proposition 1.7. The structure of Aut(ℤ⃗Δ⃗)^{⟨τ⟩} is quite easy to check in all cases. In type D̃_{n}, n ≥ 5 odd, the automorphism η ∈ Aut(ℤ⃗Δ⃗)^{⟨τ⟩} is

η(i, j) = \begin{cases}(i, n + 2 − j) & if j = 2, n \\ (i − \frac{1−(−1)^{j}}{2} , n + 2 − j) & otherwise. \end{cases}

Theorem 4.3. For any n ≥ 2 let Ω⃗_{n} be the quiver shown in Figure 6, and let A := kΩ⃗_{n} be the path algebra. Then Pic_{k}(A) ≅ PGL_{n}(k) and

DPic_{k}(A) ≅ ℤ × (ℤ ⋉ PGL_{n}(k)).

In the semidirect product the action of a generator ρ ∈ ℤ on a matrix F ∈ PGL_{n}(k) is ρF ρ^{−1} = (F^{−1})^{t}.

Proof. As in the proof of Lemma 1.6 and Proposition 1.7, the group of autoequivalences of the path category is Aut_{k}(k⟨Ω⃗_{n}⟩) = Aut_{k}^{0}(k⟨Ω⃗_{n}⟩) ≅ GL_{n}(k). Hence Pic_{k}(A) ≅ Out_{k}(k⟨Ω⃗_{n}⟩) ≅ PGL_{n}(k).

20 JUN-ICHI MIYACHI AND AMNON YEKUTIELI

Figure 5. Orientations for the affine tree graphs

Type  Aut(Δ⃗)  Aut(ℤ⃗Δ⃗)^{⟨τ⟩}  Relations
D̃_{4}  S_{4}  Aut(Δ⃗) × ⟨τ⟩ ≅ S_{4} × ℤ
D̃_{n}, n ≥ 5 even S_{2} ⋉ S_{2}^{2} Aut(Δ⃗) × ⟨τ⟩ ≅ (S_{2} ⋉ S_{2}^{2}) × ℤ
D̃_{n}, n ≥ 5 odd  S_{2}^{2}  Aut(Δ⃗) × ⟨η⟩ ≅ S_{2}^{2} × ℤ  η^{2} = τ
Ẽ_{6}  S_{3}  Aut(Δ⃗) × ⟨τ⟩ ≅ S_{3} × ℤ
Ẽ_{7}  S_{2}  Aut(Δ⃗) × ⟨τ⟩ ≅ S_{2} × ℤ
Ẽ_{8}  1  ⟨τ⟩ ≅ ℤ

Table 2. The groups Aut(ℤ⃗Δ⃗)^{⟨τ⟩} for the affine tree quivers shown in Figure 5.

Figure 6. The quivers Ω⃗_{n} and T⃗_{p,q}.

Given F ∈ Aut_{k}(k⟨Ω⃗_{n}⟩) let [a_{i,j}] ∈ GL_{n}(k) be its matrix w.r.t. to the basis {α_{i}}, and let [b_{i,j}] := ([a_{i,j}]^{−1})^{t}. Define an auto-equivalence F̃ ∈ Aut_{k}^{0}(k⟨ℤ⃗Ω⃗_{n}⟩) with F̃(m, α_{i}) = ∑_{j} a_{i,j}(m, α_{j}) and F̃(m, α_{i}^{∗}) = ∑_{j} b_{i,j}(m, α_{j}^{∗}), m ∈ ℤ. Then F̃ preserves all mesh relations, and by a linear algebra argument we see that up to scalars at each vertex, the only elements of Aut_{k}^{0}(k⟨ℤ⃗Ω⃗_{n}⟩) are of the form F̃.

Let ρ ∈ Aut(ℤ⃗Ω⃗_{n}) be ρ(m, 1) = (m, 2) and ρ(m, 2) = (m + 1, 1), with the obvious action on arrows to make it commute with the polarization μ. Then Out_{k}(k⟨ℤ⃗Ω⃗_{n}, I_{m}⟩) is generated by PGL_{n}(k) and ρ, so Out_{k}(k⟨ℤ⃗Ω⃗_{n}, I_{m}⟩) ≅ ℤ ⋉ PGL_{n}(k). The formula for F̃ above shows that ρF ρ^{−1} = (F^{−1})^{t} for F ∈ PGL_{n}(k).

Finally use Theorem 3.7.

Remark 4.4. By [Be] and [BO] we see that for n = 2 in the theorem above, DPic_{k}(A) ≅ ℤ × ℤ × PGL_{2}(k). The apparent discrepancy is explained by the fact that ℤ ⋉ PGL_{2}(k) ≅ ℤ × PGL_{2}(k) via (m, F) ↦ (m, H^{m}F), where H = [ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ].

For integers p ≥ q ≥ 1 let T⃗_{p,q} be the quiver shown in Figure 6. Let Δ⃗ be a quiver with underlying graph Ã_{n}. Then Δ⃗ can be brought to one of the quivers T⃗_{p,q}, p + q = n + 1, by a sequence of admissible reflections s_{x}^{−} at source vertices (see Section 6). Therefore

DPic_{k}(kΔ⃗) ≅ DPic_{k}(kT⃗_{p,q}).

Theorem 4.5. Let A be the path algebra kT⃗_{p,q}.

1. If p = q = 1 then Pic_{k}(A) ≅ PGL_{2}(k) and DPic_{k}(A) ≅ ℤ × (ℤ ⋉ PGL_{2}(k)).

2. If p > q = 1 then Pic_{k}(A) ≅ [ \begin{matrix}k^{×} & k \\ 0 & 1 \end{matrix} ] and DPic_{k}(A) ≅ ℤ × (ℤ ⋉ [ \begin{matrix}k^{×} & k \\ 0 & 1 \end{matrix} ]).

3. If p = q > 1 then Pic_{k}(A) ≅ S_{2} ⋉ k^{×} and DPic_{k}(A) ≅ ℤ^{2} × (S_{2} ⋉ k^{×}).

4. If p > q > 1 then Pic_{k}(A) ≅ k^{×} and DPic_{k}(A) ≅ ℤ^{2} × k^{×}.

Proof. 1. This is because T⃗_{1,1} = Ω⃗_{2}.

2. Here the group of auto-equivalences of k⟨T⃗_{p,q}⟩ is, in the notation of the proof of Proposition 1.7, Aut_{k}(k⟨T⃗_{p,q}⟩) ≅ (k^{×})^{p+1} × k, and the group of isomorphisms is (k^{×})^{p}. Therefore Out_{k}(k⟨T⃗_{p,q}⟩) is isomorphic to k^{×} × k as varieties, and as matrix group Out_{k}(k⟨T⃗_{p,q}⟩) ≅ [ \begin{matrix}k^{×} & k \\ 0 & 1 \end{matrix} ]. The auto-equivalence associated to [ \begin{matrix}a & b \\ 0 & 1 \end{matrix} ] ∈ [ \begin{matrix}k^{×} & k \\ 0 & 1 \end{matrix} ] is α_{i} ↦ α_{i} and β_{1} ↦ aβ_{1} + bα_{p} ⋯ α_{1}.

The quiver ℤ⃗T⃗_{p,q} has no multiple arrows. Let ρ be the symmetry ρ(m, i) = (m, i − 1) for i ≥ 2, and ρ(m, 1) = (m − 1, p). Then ρ generates Aut(ℤ⃗T⃗_{p,q})^{⟨τ⟩}, and we can use Theorem 0.1. The action of ρ on Out_{k}(k⟨T⃗_{p,q}⟩) is ρ [ \begin{matrix}a & b \\ 0 & 1 \end{matrix} ] ρ^{−1} = [ \begin{matrix}a & −b \\ 0 & 1 \end{matrix} ].

3. Here Aut_{k}^{0}(k⟨T⃗_{p,q}⟩) ≅ (k^{×})^{2p}, and the subgroup of isomorphisms is (k^{×})^{2p−1}. The symmetry θ ∈ Aut(T⃗_{p,q}) of order 2 acts on k^{×} by θaθ^{−1} = a^{−1}.

Let ρ be the symmetry ρ(m, 1) = (m−1, p+q), ρ(m, i) = (m, i−1) if 2 ≤ i ≤ p+1, and ρ(m, i) = (m − 1, i − 1) if p + 2 ≤ i ≤ p + q. Then ρ and θ commute, and they generate Aut(ℤ⃗T⃗_{p,q})^{⟨τ⟩}. The action of ρ on Aut_{k}(k⟨T⃗_{p,q}⟩) is trivial.

4. Similar to case 3.

5. The Reflection Groupoid of a Graph

In this section we interpret the reflection functors of [BGP] and the tilting modules of [APR] in the setup of derived categories.

Let Δ be a tree with n vertices. Denote by Or(Δ) the set of orientations of the edge set Δ_{1}. For ω ∈ Or(Δ) let Δ⃗_{ω} be the resulting quiver, and let A_{ω} be the path algebra kΔ⃗_{ω}.

Given two orientations ω, ω′ let

DPic_{k}(ω, ω′) := \frac{\{two-sided tilting complexes T ∈ D^{b}(Mod(A_{ω′} ⊗_{k} A_{ω}^{∘}))\}}{isomorphism}.

The derived Picard groupoid of Δ is the groupoid DPic_{k}(Δ) with object set Or(Δ) and morphism sets DPic_{k}(ω, ω′). Thus when ω = ω′ we recover the derived Picard group DPic_{k}(A_{ω}).

For an orientation ω and a vertex x let P_{x,ω} ∈ mod A_{ω} be the corresponding indecomposable projective module. Denote by τ_{ω} the translation functor of D^{b}(mod A_{ω}), i.e. the functor τ_{ω} = A_{ω}^{∗}[−1] ⊗_{A_{ω}}^{L} −.

Suppose x ∈ (Δ⃗_{ω})_{0} is a source. Define s_{x}^{−}ω to be the orientation obtained from ω by reversing the arrows starting at x. Let

T_{x,ω} := τ_{ω}^{−1}P_{x,ω} ⊕ (⨁_{y≠x} P_{y,ω}) ∈ mod A_{ω}.

According to [APR] Section 3, T_{x,ω} is a tilting module, with End_{A_{ω}}(T_{x,ω})^{∘} ≅ A_{s_{x}^{−}ω}. It is called an APR tilting module. One has isomorphisms in mod A_{s_{x}^{−}ω}:

Hom_{A_{ω}}(T_{x,ω}, P_{y,ω}) ≅ P_{y,s_{x}^{−}ω} if y ≠ x,
Hom_{A_{ω}}(T_{x,ω}, τ_{ω}^{−1}P_{x,ω}) ≅ P_{x,s_{x}^{−}ω}. (5.1)

Under the anti-equivalence between mod A_{ω} and the category of finite dimensional representations of the quiver Δ⃗_{ω}, the reflection functor of [BGP] is sent to Hom_{A_{ω}}(T_{x,ω}, −) : mod A_{ω} → mod A_{s_{x}^{−}ω}.

Definition 5.2. The reflection groupoid of Δ is the subgroupoid Ref(Δ) ⊂ DPic_{k}(Δ) generated by the two-sided tilting complexes T_{x,ω} ∈ D^{b}(Mod(A_{ω} ⊗_{k} A_{s_{x}^{−}ω}^{∘})), as ω runs over Or(Δ) and x runs over the sources in Δ⃗_{ω}.

Given an orientation ω the set {[P_{x,ω}]}_{x∈Δ_{0}} is a basis of the Grothendieck group K_{0}(A_{ω}) = K_{0}(D^{b}(mod A_{ω})). Let ℤ^{Δ_{0}} be the free abelian group with basis {e_{x}}_{x∈Δ_{0}}. Then [P_{x,ω}] ↦ e_{x} determines a canonical isomorphism K_{0}(A_{ω}) →^{≃} ℤ^{Δ_{0}}. For a two-sided tilting complex T ∈ DPic_{k}(ω, ω′) let χ_{0}(T) : K_{0}(A_{ω}) →^{≃} K_{0}(A_{ω′}) be χ_{0}(T)([M]) := [T ⊗_{A_{ω}}^{L} M]. Using the projective bases we get a functor (when we consider a group as a groupoid with a single object)

χ_{0} : DPic_{k}(Δ) → Aut_{ℤ}(ℤ^{Δ_{0}}) ≅ GL_{n}(ℤ).

Recall that for a vertex x ∈ Δ_{0} one defines the reflection s_{x} ∈ Aut_{ℤ}(ℤ^{Δ_{0}}) by

\begin{aligned}s_{x}e_{x} & := −e_{x} + ∑_{\{x,y\}∈Δ_{1}} e_{y}, \\ s_{x}e_{y} & := e_{y} if y ≠ x. \end{aligned}

The Weyl group of Δ is the subgroup W (Δ) ⊂ Aut_{ℤ}(ℤ^{Δ_{0}}) generated by the reflections s_{x}.

Proposition 5.3. Let x be a source in the quiver Δ⃗_{ω}. Then

χ_{0}(T_{x,ω}) = s_{x}.

Proof. There is an Auslander-Reiten sequence

0 → P_{x,ω} → ⨁_{(x→y)∈(Δ⃗_{ω})_{1}} P_{y,ω} → τ_{ω}^{−1}P_{x,ω} → 0

in mod A_{ω}. Applying the functor T_{x,ω}^{∨} ⊗_{A_{ω}}^{L} − ≅ R Hom_{A_{ω}}(T_{x,ω}, −) to this sequence, and using formula (5.1), we get a triangle

T_{x,ω}^{∨} ⊗_{A_{ω}}^{L} P_{x,ω} → ⨁_{{x,y}∈Δ_{1}} P_{y,s_{x}^{−}ω} → P_{x,s_{x}^{−}ω} → (T_{x,ω}^{∨} ⊗_{A_{ω}}^{L} P_{x,ω})[1]

in D^{b}(mod A_{s_{x}^{−}ω}). Hence

[T_{x,ω}^{∨} ⊗_{A_{ω}}^{L} P_{x,ω}] = −[P_{x,s_{x}^{−}ω}] + ∑_{{x,y}∈Δ_{1}}[P_{y,s_{x}^{−}ω}] ∈ K_{0}(A_{s_{x}^{−}ω}).

On the other hand for y ≠ x we have [T_{x,ω}^{∨} ⊗_{A_{ω}}^{L} P_{y,ω}] = [P_{y,s_{x}^{−}ω}]. This proves that χ_{0}(T_{x,ω}^{∨}) = s_{x}; but s_{x} = s_{x}^{−1}.

An immediate consequence is:

Corollary 5.4. χ_{0}(Ref(Δ)) = W (Δ).

An ordering (x_{1}, …, x_{n}) of Δ_{0} is called source-admissible for an orientation ω if x_{i} is a source in the quiver Δ⃗_{s_{x_{i−1}}^{−} ···s_{x_{1}}^{−} ω} for all 1 ≤ i ≤ n. Any orientation has source-admissible orderings of the vertices.

Proposition 5.5. Let (x_{1}, …, x_{n}) be a source-admissible ordering of Δ_{0} for an orientation ω. Write ω_{i} := s_{x_{i}}^{−} ⋯ s_{x_{1}}^{−}ω, A_{i} := A_{ω_{i}} and T_{i} := T_{x_{i},ω_{i−1}}. Then

T_{n}^{∨} ⊗_{A_{n−1}}^{L} ⋯ ⊗_{A_{2}}^{L} T_{2}^{∨} ⊗_{A_{1}}^{L} T_{1}^{∨} ≅ A_{ω}^{∗}[−1]

in D^{b}(Mod A_{ω}^{e}).

Proof. For an orientation ω let Γ⃗_{ω}^{irr} ⊂ Γ⃗(D^{b}(mod A_{ω})) be the quiver of definition 2.2. As usual (Γ⃗_{ω}^{irr})_{0} denotes the set of vertices of Γ⃗_{ω}^{irr}. Let G(Δ) be the groupoid with object set Or(Δ), and morphism sets Iso((Γ⃗_{ω}^{irr})_{0}, (Γ⃗_{ω′}^{irr})_{0}) for ω, ω′ ∈ Or(Δ). The groupoid G(Δ) acts faithfully on the family of sets X(Δ) := {(Γ⃗_{ω}^{irr})_{0}}_{ω∈Or(Δ)}. According to Theorem 0.1 there is an injective map of groupoids q : DPic_{k}(Δ) ↣ G(Δ).

Let us first assume Δ is a Dynkin graph. Then there is a canonical isomorphism of sets X(Δ) ≅ ℤ × Δ_{0} × Or(Δ). The action of q(τ_{ω}) on X(Δ) is q(τ_{ω})(i, x, ω) = (i − 1, x, ω). By formula (5.1), the action of q(T_{x,ω}^{∨}) on X(Δ) is q(T_{x,ω}^{∨})(0, y, ω) = (0, y, s_{x}^{−}ω) if y ≠ x, and q(T_{x,ω}^{∨})(1, x, ω) = (0, x, s_{x}^{−}ω). Since q(τ_{ω}) commutes with q(T_{x,ω}^{∨}) we have

q(T_{n}^{∨} ⊗_{A_{n−1}}^{L} ⋯ ⊗_{A_{1}}^{L} T_{1}^{∨})(i, x, ω) = (i − 1, x, ω) = q(τ_{ω})(i, x, ω)

for any x ∈ Δ_{0} and i ∈ ℤ.

If Δ is not Dynkin then X(Δ) ≅ ℤ × ℤ × Δ_{0} × Or(Δ), q(τ_{ω})(j, i, x, ω) = (j, i − 1, x, ω), etc., and the proof is the same after these modifications.

Proposition 5.6. For any orientation ω,

Ref(Δ)(ω, ω) = ⟨τ_{ω}⟩.

Proof. We will only treat the Dynkin case; the general case is proved similarly with modifications like in the previous proof.

Let T ∈ Ref(Δ)(ω, ω). From the proof above we see that q(T)(0, x, ω) = (i(x), x, ω) for some i(x) ∈ ℤ. A quiver map π : Δ⃗_{ω} → ℤ⃗Δ⃗_{ω} with π(x) = (i(x), x) must have i(x) = i for all x, since Δ is a tree. Therefore q(T) = q(τ_{ω}^{−i}).

Remark 5.7. The explicit calculations in Section 4 show that the shift σ = A[1] is not in ⟨τ⟩ ⊂ DPic_{k}(A) for most algebras A. Thus Ref(Δ) ⫋ DPic_{k}(Δ) for most graphs Δ.

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J. Miyachi: Department of Mathematics, Tokyo Gakugei University, Koganei-shi, Tokyo, 184, Japan

E-mail address: miyachi@u-gakugei.ac.jp

A. Yekutieli: Department Mathematics and Computer Science, Ben Gurion University, Be’er Sheva 84105, Israel

E-mail address: amyekut@math.bgu.ac.il