Braid action on derived category of Nakayama algebras

Intan Muchtadi-Alamsyah

Abstract

We construct an action of a braid group associated to a complete graph on the derived Picard group of a certain symmetric Nakayama algebra which is also a Brauer star algebra with no exceptional vertex. We connect this action with the affine braid group action on Brauer star algebras defined by Schaps and Zakay-Illouz. We show that for Brauer star algebras with no exceptional vertex, the action is faithful.

Introduction

For a commutative ring R we shall study the group of self-equivalences of the derived category of an R-algebra A which is projective as an R-module. We denote such a group by T rP ic(A) and call it the Derived Picard group.

Rouquier and Zimmermann [16], motivated by Broué’s conjecture, studied the T rP ic(A) of a Brauer tree algebra by defining an action of the Artin braid group B(A_{n}) on it. Khovanov and Seidel [10], motivated by mirror symmetry, defined an action of B(A_{n}) on derived categories of certain algebras similar to Brauer tree algebras, which turn out to be related to certain classes of representations of simple Lie algebras, and they proved that the action is faithful.

In this paper we will define an action of B(K_{n}), the braid group whose associated Coxeter group is given by the complete graph on n vertices, on T rP ic(N_{n}^{n}), the derived Picard group of the Nakayama algebra which is a basic Brauer star algebra with no exceptional vertex.

We will also define a group homomorphism η from B(K_{n}) to B(A_{n}) and give a categorification of η. This categorification provides a link between the action of B(K_{n}) on T rP ic(N_{n}^{n}) and the faithful action of B(A_{n}) on the derived Picard group of Brauer tree algebras defined by Rouquier and Zimmermann [16] and Khovanov and Seidel [10].

2000 Mathematics Subject Classification 16D90,16G20,18E30,20F36 The author was supported by EPSRC Grant number GR/S35387/01.

In [17] Schaps and Zakay-Illouz defined an action of affine braid group B(Ã_{n−1}) on the derived Picard group of Brauer star algebras and left open the question whether this action is faithful.

We will provide a positive answer to this question for Brauer star algebras with no exceptional vertex. We will use the categorification of η stated above, another categorification of braid embeddings defined in [9], [5] and [3] and the faithfulness of the action in [10] and [16].

The paper is organized as follows. In Section 1, we review Rickard’s Morita theory for derived equivalence and we give the definition of derived Picard groups T rP ic and Nakayama algebras.

In Section 2, we recall the Nakayama algebra N_{n}^{n} which is also a basic Brauer star algebra with n edges and no exceptional vertex. We define explicit elements of T rP ic(N_{n}^{n}).

The action of the braid group B(K_{n}) on T rP ic(N_{n}^{n}) is defined is Section 3. And in Section 4 we show that this action is not faithful using a categorification of the group homomorphism η.

In the last section, we use the results of sections 2, 3 and 4 to show that the action defined in [17] is faithful for Brauer star algebras with no exceptional vertex.

Acknowledgement I would like to thank Robert Marsh and Steffen König for numerous discussions on studying the braid action on derived Picard groups. I also thank Alexander Zimmermann for encouraging me to prove the faithfulness of the action defined in [17].

1 Derived Picard groups and Nakayama algebras

1.1 Derived equivalence

Let A and B be algebras over a commutative ring R and assume that A and B are projective as modules over R. The bounded derived category D^{b}(A) is the category whose objects are complexes of finitely generated projective modules which are bounded to the right and which have nonzero homology only in finitely many degrees. Morphisms are morphisms of complexes up to homotopy. We refer to [11] for the definition of derived categories.

Denote by A − per the full subcategory of D^{b}(A) consisting of the perfect complexes, i.e. bounded complexes of finitely generated projective Amodules.

Rickard [14] and Keller [8] have given a necessary and sufficient criterion for the existence of derived equivalences between two rings A and B.

Theorem 1.1 (Rickard [14] and Keller [8]) The following statements are equivalent :

1. D^{b}(A) ≃ D^{b}(B) as triangulated categories.

2. There is a complex T in A − per such that

(a) Hom(T, T [i]) = 0 for i ≠ 0,

(b) A − per is generated by T, and

(c) End_{D^{b}(A)}(T) ≃ B.

3. There is a bounded complex X in D^{b}(A ⊗_{R} B^{op}) whose restrictions to A and to B^{op} are perfect and a bounded complex Y in D^{b}(B ⊗_{R} A^{op}) whose restrictions to B and A^{op} are perfect such that

X ⊗_{B}^{L} Y ≃ A in D^{b}(A ⊗ A^{op}) and Y ⊗_{A}^{L} X ≃ B in D^{b}(B ⊗ B^{op}).

A complex T satisfying the condition in 2 is called a tilting complex for A and the complexes X and Y in 3 are called two-sided tilting complexes inverse to each other. The image and the pre-image of a projective indecomposable B-module under derived equivalence D^{b}(A) ≃ D^{b}(B) is a partial tilting complex in A, i.e. a complex satisfying the condition in 2(a).

We define the Derived Picard group of A as

T rP ic(A) = \{ \begin{matrix}isomorphism classes of two-sided tilting complexes \\ in D^{b}(A ⊗_{R} A^{op}) \end{matrix} \}

where the product of the classes of X and Y is given by the class of X ⊗_{A} Y.

It is clear that if A and B are projective as R-modules then D^{b}(A) ≃ D^{b}(B) implies

T rP ic(A) ≃ T rP ic(B).

The Picard group P ic(A) (i.e. the group of isomorphism classes of invertible A ⊗ A^{op}-modules M where M ⊗_{A} M^{∗} ≃ A) can be embedded into T rP ic(A) by sending an invertible A ⊗ A^{op}-module M to a stalk complex concentrated in degree 0.

1.2 Nakayama algebras

The self-injective Nakayama algebra N_{m}^{n} is the path algebra over a field K of the following quiver

2 → 3
↗  ↘
1  4
↖  ↙
m ← ⋯

modulo the ideal generated by all compositions of n + 1 consecutive arrows. By [2] N_{m}^{n} is a representative of the class of standard self-injective algebras of finite representation type associated to the Dynkin diagram of type A_{n}.

The algebra N_{m}^{n} is symmetric if and only if m divides n. In this case the Nakayama algebra N_{m}^{n} is a basic Brauer star algebra with n edges and multiplicity m, a particular Brauer tree algebra where all the edges are adjacent to the exceptional vertex. A Brauer tree algebra with n edges and multiplicity m is (up to Morita equivalence) a p-block with cyclic defect groups of order p^{d} = mn + 1 where the number of its simple modules is n. (See [1] for the definition of Brauer tree algebras).

By [13, Theorem 4.2], up to derived equivalence, a Brauer tree algebra is determined by the number of edges of the Brauer tree and the multiplicity of the exceptional vertex. Hence, an arbitrary Brauer tree algebra is derived equivalent to a Brauer star algebra associated with a star having the same number of edges.

If m = n, the Nakayama algebra N_{n}^{n} is the trivial extension algebra of the path algebra of the quiver

1 → 2 → 3 → ⋯ → n − 1 → n.

It is a Brauer star algebra with n edges and multiplicity 1.

Notation : From now on, A is the Nakayama algebra N_{n}^{n} over K and B is the Brauer tree algebra associated to a line without exceptional vertex. For projective A-modules we use the letter P and for projective B-modules we use the letter Q. We use the letters F, R, and H to denote two-sided tilting complexes (elements of T rP ic) and the letters F, R, and H for the corresponding auto-equivalences. The symbol ⊗ means ⊗_{K}.

2 Two-sided tilting complexes for Nakayama algebra N_{n}^{n}

Denote by (i_{1} i_{2} ⋯ i_{l+1}) the path starting at i_{1} and ending at i_{l+1}. Note that we have i_{k+1} − i_{k} = 1 (mod n) for all 1 ≤ k ≤ l.

Let A be the Nakayama algebra N_{n}^{n}. The algebra A is the path algebra of the quiver

2 → 3
↗  ↘
1  4
↖  ↙
n ← ⋯

modulo the relations : (i i + 1 ... n 1 2 ... i i + 1) = 0 for all 1 ≤ i ≤ n.

The paths (i) of length 0 are mutually orthogonal idempotents and their sum is the unit element. The Loewy series of the projective indecomposable modules P_{i} = A(i) are as follows :

P_{i} = S_{i} ; S_{i+1} ; ⋮ ; S_{n} ; S_{1} ; S_{2} ; ⋮ ; S_{i}

The dimension of homomorphisms between projective modules is given by :

dim_{K}Hom_{A}(P_{i}, P_{j}) = \{ \begin{matrix}2 & if i = j, \\ 1 & if i ≠ j. \end{matrix}

Define _{i}P = (i)A ≅ Hom_{A}(P_{i}, A). Define the complexes

F_{i} := 0 → P_{i} ⊗ _{i}P ⟶^{α_{i}} A → 0

and

F′_{i} := 0 → A ⟶^{β_{i}} P_{i} ⊗ _{i}P → 0

where A is in degree 0 and

\begin{aligned}α_{i} & : & P_{i} & ⊗ & _{i}P & → & A \\ & & (i) & ⊗ & (i) & ↦ & (i) \end{aligned}

β_{i} : A → P_{i} ⊗ _{i}P
1 ↦ (i ⋯ n 1 ⋯ i) ⊗ (i) + (i) ⊗ (i ⋯ n 1 ⋯ i)
+ (i + 1 ⋯ n 1 ⋯ i) ⊗ (i i + 1)
+ (i + 2 ⋯ n 1 ⋯ i) ⊗ (i i + 1 i + 2)
+ ⋯ + (n 1 ⋯ i) ⊗ (i i + 1 ⋯ n)
+ (1 ⋯ i) ⊗ (i ⋯ n 1) + (2 ⋯ i) ⊗ (i ⋯ n 1 2)
+ ⋯ + (i − 1 i) ⊗ (i ⋯ n 1 ⋯ i − 1).

Theorem 2.1 The complexes F_{i} and F′_{i} are elements of T rP ic(A).

Proof We will show that F_{i} and F′_{i} are two-sided tilting complexes and inverses each other and to do that, we will show that F_{i} ⊗_{A} F′_{i} is homotopy equivalent to A. (See [15, Theorem 6]). We follow the method of [16, Theorem 4.1] and [10, Proposition 2.4]. We have

F_{i} ⊗_{A} F′_{i} = (0 → P_{i} ⊗ _{i}P ⟶^{d^{1}} A ⊕ (P_{i} ⊗ U ⊗ _{i}P) ⟶^{d^{0}} P_{i} ⊗_{A} _{i}P → 0)

where U is the 2-dimensional space _{i}P ⊗_{A} P_{i} with a basis u_{1} = (i) ⊗ (i) and u_{2} = (i ⋯ i) ⊗ (i), and the differentials are given by

d^{1} = α_{i} + τ and d^{0} = (β_{i}, −δ) where
τ(x ⊗ y) = x ⊗ u_{1} ⊗ (i ⋯ i)y + x ⊗ u_{2} ⊗ y,
δ(x ⊗ u_{1} ⊗ y) = x ⊗ y,
δ(x ⊗ u_{2} ⊗ y) = x(i ⋯ i) ⊗ y.

The map d^{0} is surjective since δ is surjective. Therefore, since P_{i} ⊗ _{i}P is projective, d^{0} is a split surjection.

Denote by B_{i} (resp. _{i}B) the basis of P_{i} (resp. of _{i}P) where

B_{i} = {(i), (i−1 i), (i−2 i−i i), ⋯, (n 1 2 ⋯ i), ⋯, (i i+1 ⋯ n 1 ⋯ i)} and

iB = {(i), (i i+1), (i i+1, i+2), ⋯, (i i+1 ⋯ n 1), ⋯, (i i+1 ⋯ n 1 ⋯ i)}.

Let x ∈ B_{i} and y ∈ _{i}B. Then,

τ(x ⊗ y) = \{ \begin{matrix}x ⊗ u_{1} ⊗ (i ⋯ i) + x ⊗ u_{2} ⊗ (i) & if & y = (i), \\ x ⊗ u_{2} ⊗ y & if & y ≠ (i). \end{matrix}

Therefore, if x ⊗ y ∈ P_{i} ⊗ _{i}P, then x ⊗ y ≠ 0 implies τ(x ⊗ y) ≠ 0. This means that τ is injective, therefore so is d^{1}. This implies d^{1} is a split injection.

Therefore, F_{i}⊗_{A}F′_{i} is homotopy equivalent to a module V which satisfies

(P_{i} ⊗ _{i}P) ⊕ (P_{i} ⊗ _{i}P) ⊕ V ≃ (P_{i} ⊗ _{i}P ⊗_{A} P_{i} ⊗ _{i}P) ⊕ A.

Since dim_{k}(P_{i} ⊗_{A} _{i}P) = dim_{k}Hom_{A}(P_{i}, P_{i}) = 2, we obtain V ≃ A as a left module and this finishes the proof. □

Denote by

F_{i} := F_{i} ⊗_{A} −.

It follows from above that :

F_{i}^{−1} = F′_{i} ⊗_{A} −.

Lemma 2.2 We have

F_{i}(P_{j}) = \{ \begin{matrix}0 & → & P_{i} & → & 0 & & & if i = j \\ 0 & → & P_{i} & → & P_{j} & → & 0 & if i ≠ j \end{matrix}

F_{i}^{−1}(P_{j}) = \{ \begin{matrix}& & 0 & → & P_{i} & → & 0 & if i = j \\ 0 & → & P_{j} & → & P_{i} & → & 0 & if i ≠ j \end{matrix}

where P_{j} is in degree 0.

Proof The images of P_{j} under F_{i} and F_{i}^{−1} are

F_{i}(P_{j}) = (0 → P_{i} ⊗ _{i}P ⊗_{A} P_{j} ⟶^{α_{i}⊗id_{P_{j}}} A ⊗_{A} P_{j} → 0) and

F_{i}^{−1}(P_{j}) = (0 → A ⊗_{A} P_{j} ⟶^{β_{i}⊗id_{P_{j}}} P_{i} ⊗ _{i}P ⊗_{A} P_{j} → 0).

When i ≠ j, _{i}P ⊗_{A} P_{j} is generated by the path (i ⋯ j), hence

α_{i} ⊗ id_{P_{j}}(x ⊗ y ⊗_{A} z) = x(i ⋯ j) for all x ⊗ y ⊗_{A} z ∈ P_{i} ⊗ _{i}P ⊗_{A} P_{j},

β_{i} ⊗ id_{P_{j}}(1 ⊗_{A} x) = x(j ⋯ i) ⊗ (i ⋯ j) for all x ∈ P_{j}.

Since dim_{k}(_{i}P ⊗_{A}P_{j}) = dim_{k}Hom_{A}(P_{i}, P_{j}) = 1 we have P_{i}⊗ _{i}P ⊗_{A}P_{j} ≃ P_{i}. The only map between P_{i} and P_{j} is

μ_{ij} : P_{i} → P_{j} where μ_{ij}(x) = x(i ⋯ j) for all x ∈ P_{i}.

Therefore, α_{i} ⊗ id_{P_{j}} ≃ μ_{ij}, β_{i} ⊗ id_{P_{j}} ≃ μ_{ji} and

F_{i}(P_{j}) = (0 → P_{i} →^{μ_{ij}} P_{j} → 0) and F_{i}^{−1}(P_{j}) = (0 → P_{j} →^{μ_{ji}} P_{i} → 0).

When i = j, P_{i} ⊗ _{i}P ⊗_{A} P_{i} ⟶^{α_{i}⊗id_{P_{i}}} A ⊗_{A} P_{i} is surjective since for all x ∈ P_{i},

(α_{i} ⊗ id_{P_{i}})(x ⊗ (i) ⊗ (i)) = x.

Now if x ∈ P_{i}, then one can check that

β_{i} ⊗ id_{P_{i}}(1 ⊗_{A} x) = x ⊗ (i ⋯ i) + x(i ⋯ i) ⊗ (i).

Hence, if x ≠ 0, then β_{i} ⊗ id_{P_{i}}(1 ⊗_{A} x) ≠ 0; therefore, A ⊗_{A} P_{i} ⟶^{β_{i}⊗id_{P_{i}}} P_{i} ⊗ _{i}P ⊗_{A} P_{i} is injective.

Hence, F_{i}(P_{i}) (resp. F_{i}^{−1}(P_{i})) has homology concentrated in degree 1 (resp. −1). As dim_{k}(_{i}P ⊗_{A}P_{i}) = 2, F_{i}(P_{i}) ≃ P_{i}[1] and F_{i}^{−1}(P_{i}) ≃ P_{i}[−1].□

Denote by Δ the subgroup of T rP ic(A) generated by F_{1}, ⋯ F_{n}.

Lemma 2.3 For F, F′ ∈ Δ, F ≃ F′ if and only if

F ⊗_{A} P ≃ F′ ⊗_{A} P

for any projective indecomposable module P.

Proof Let us assume that F ⊗_{A}P ≃ F′⊗_{A}P for any projective indecomposable A-module P. Denote by B the Brauer tree algebra associated to a line with n edges and multiplicity 1. Denote by Φ the isomorphism of groups

Φ : T rP ic(A) → T rP ic(B)

and by G and G′ the image of F and F′ under Φ (G := Φ(F) and G′ := Φ(F′)). Denote by X the associated two-sided tilting complex in D^{b}(B⊗A^{op}) giving the derived equivalence between A and B, so we have

G^{−1} ⊗_{B} X ≃ X ⊗_{A} F^{−1} and G′ ⊗_{B} X ≃ X ⊗_{A} F′.

→^{F ⊗_{A}−}

D^{b}(A)  ←^{F^{−1}⊗_{A}−}  D^{b}(A)
X⊗_{A}− ↓  ↓ _{X⊗_{A}−}
D^{b}(B)  →^{G⊗_{B}−}  D^{b}(B)

←^{G^{−1}⊗_{B}−}

We will show that G^{−1} ⊗_{B} Q ≃ G^{′−1} ⊗_{B} Q, or equivalently

Q ≃ G′ ⊗_{B} G^{−1} ⊗_{B} Q

for any projective indecomposable B-module Q. Let Q be a projective indecomposable B-module and denote by T the partial tilting complex of A where Q = Φ(T) = X ⊗_{A} T and denote by P the projective indecomposable A-module where P := F^{−1} ⊗_{A} T. Then we have

G′ ⊗_{B} G^{−1} ⊗_{B} Q ≃ G′ ⊗_{B} G^{−1} ⊗_{B} X ⊗_{A} T
≃ G′ ⊗_{B} X ⊗_{A} F^{−1} ⊗_{A} T
≃ G′ ⊗_{B} X ⊗_{A} P
≃ X ⊗_{A} F′ ⊗_{A} P
≃ X ⊗_{A} F ⊗_{A} P by assumption
≃ X ⊗_{A} F ⊗_{A} F^{−1} ⊗_{A} T
≃ X ⊗_{A} T ≃ Q.

Hence, by [16, Remark 3], using the identity component Out_{0}(B) of the outer automorphism group of B we obtain G^{−1} ≃ G^{′−1}, which implies G ≃ G′. And since Out_{0} is invariant under derived equivalence (see [6]) we obtain F ≃ F′. □

3 Braid action

Denote by B(K_{n}) the braid group whose associated Coxeter group has Coxeter graph given by the complete graph on n vertices, generated by a_{1}, ⋯, a_{n} with relations

a_{i}a_{j}a_{i} = a_{j}a_{i}a_{j}

for all 1 ≤ i, j ≤ n, i ≠ j. We will define an action of B(K_{n}) on T rP ic(A).

Theorem 3.1 There is a group homomorphism

φ : B(K_{n}) → T rP ic(A)

where φ(a_{i}) = F_{i}.

Proof By Lemma 2.3, it is enough to show that F_{i}F_{j}F_{i}(P) ≃ F_{j}F_{i}F_{j}(P) for every indecomposable projective module P. We will show that for every indecomposable projective module P

F_{i}^{−1}F_{j}F_{i}(P) ≃ F_{j}F_{i}F_{j}^{−1}(P).

Claim

( P_{1} ; P_{2} ; ⋮ ; P_{i−1} ; P_{i} ; P_{i+1} ; ⋮ ; P_{j−1} ; P_{j} ; P_{j+1} ; ⋮ ; P_{n} ) ⟶^{F_{i}} ( P_{i} → P_{1} ; P_{i} → P_{2} ; ⋮ ; P_{i} → P_{i−1} ; P_{i} → 0 ; P_{i} → P_{i+1} ; ⋮ ; P_{i} → P_{j−1} ; P_{i} → P_{j} ; P_{i} → P_{j+1} ; ⋮ ; P_{i} → P_{n} ) ⟶^{F_{j}} ( P_{i} → P_{1} ; P_{i} → P_{2} ; ⋮ ; P_{i} → P_{i−1} ; P_{j} → P_{i} → 0 ; P_{i} → P_{i+1} ; ⋮ ; P_{i} → P_{j−1} ; P_{i} → 0 ; P_{i} → P_{j+1} ; ⋮ ; P_{i} → P_{n} ) ⟶^{F_{i}^{−1}} ( P_{1} ; P_{2} ; ⋮ ; P_{i−1} ; P_{j} → P_{i} → P_{i} ; P_{i+1} ; ⋮ ; P_{j−1} ; P_{i} ; P_{j+1} ; ⋮ ; P_{n} )

and

( P_{1} ; P_{2} ; ⋮ ; P_{i−1} ; P_{i} ; P_{i+1} ; ⋮ ; P_{j−1} ; P_{j} ; P_{j+1} ; ⋮ ; P_{n} ) ⟶^{F_{j}^{−1}} ( P_{1} → P_{j} ; P_{2} → P_{j} ; ⋮ ; P_{i−1} → P_{j} ; P_{i} → P_{j} ; P_{i+1} → P_{j} ; ⋮ ; P_{j−1} → P_{j} ; P_{j} ; P_{j+1} → P_{j} ; ⋮ ; P_{n} → P_{j} ) ⟶^{F_{i}} ( P_{1} → P_{j} ; P_{2} → P_{j} ; ⋮ ; P_{i−1} → P_{j} ; P_{i} → P_{i} → P_{j} ; P_{i+1} → P_{j} ; ⋮ ; P_{j−1} → P_{j} ; P_{i} → P_{j} ; P_{j+1} → P_{j} ; ⋮ ; P_{n} → P_{j} ) ⟶^{F_{j}} ( P_{1} ; P_{2} ; ⋮ ; P_{i−1} ; P_{j} → P_{i} → P_{i} ; P_{i+1} ; ⋮ ; P_{j−1} ; P_{i} ; P_{j+1} ; ⋮ ; P_{n} )

(1)

Proof of Claim : For all 1 ≤ i ≤ n, denote by δ_{i} : P_{i} → P_{i} where δ_{i}((i)) = (i), and for all 1 ≤ i, j ≤ n by μ_{ij} : P_{i} → P_{j}, the only map up to a scalar between P_{i} and P_{j} where

μ_{ij}(x) = x(i ⋯ j) for all x ∈ P_{i}.

Since F_{j} preserves cones, F_{j}(P_{i} → P_{j}) is isomorphic to

Cone(F_{j}(P_{i}) → F_{j}(P_{j})) = Cone ( \begin{matrix}F_{j}(P_{i}) = & P_{j} ⊗ _{j}P ⊗_{A} P_{i} & → & A ⊗_{A} P_{i} \\ & x ⊗ (j ⋯ i) & & x(j ⋯ i) \\ ↓ & ↓ & & ↓ \\ F_{j}(P_{j}) = & P_{j} ⊗ _{j}P ⊗_{A} P_{j} & → & A ⊗_{A} P_{j} \\ & x ⊗ (j ⋯ j) & & x(j ⋯ j) \end{matrix} )

≃ Cone ( \begin{matrix}P_{j} & →^{μ_{ji}} & P_{i} \\ x & & x(j ⋯ i) \\ ↓ _{δ_{j}} \\ P_{j} \\ x \end{matrix} )

≃ P_{i}[1].

For k ≠ j, F_{j}(P_{i} → P_{k}) is isomorphic to

Cone ( \begin{matrix}F_{j}(P_{i}) = & P_{j} ⊗ _{j}P ⊗_{A} P_{i} & → & A ⊗_{A} P_{i} \\ & x ⊗ (j ⋯ i) & & x(j ⋯ i) \\ ↓ & ↓ & & ↓ _{μ_{ik}} \\ F_{j}(P_{k}) = & P_{j} ⊗ _{j}P ⊗_{A} P_{k} & → & A ⊗_{A} P_{k} \\ & x ⊗ (j ⋯ k) & & x(j ⋯ k) \end{matrix} ) ≃ Cone ( \begin{matrix}P_{j} & →^{μ_{ji}} & P_{i} \\ x & & x(j ⋯ i) \\ ↓ _{δ_{j}} & & ↓ _{μ_{ik}} \\ P_{j} & →^{μ_{jk}} & P_{k} \\ x & & x(j ⋯ k) \end{matrix} )

≃ \begin{aligned}P_{j} & →^{μ_{ji}} & P_{i} \\ & _{δ_{j}} ↘ & ⊕ & ↘ _{μ_{ik}} \\ & & P_{j} & →^{μ_{jk}} & P_{k} \end{aligned}
≃ Cone(Cone(P_{j} → P_{i} ⊕ P_{j}) → P_{k})
≃ Cone ( Cone ( \begin{matrix}(P_{j} & →^{μ_{ji}} & P_{i}) \\ _{δ_{j}} ↓ \\ P_{j} \end{matrix} ) [−1] ⟶^{(μ_{ik},μ_{jk})} P_{k} )

≃ P_{i} ⟶^{μ_{ik}} P_{k}.

The complex F_{i}^{−1}(P_{i} → P_{k}) (k ≠ i) is isomorphic to

Cone ( \begin{matrix}F_{i}^{−1}(P_{i}) = & A ⊗_{A} P_{i} & → & P_{i} ⊗ _{i}P ⊗_{A} P_{i} \\ & x & & x(i ⋯ i) ⊗ (i) + x ⊗ (i ⋯ i) \\ ↓ & ↓ & & ↓ \\ F_{i}^{−1}(P_{k}) = & A ⊗_{A} P_{k} & → & P_{i} ⊗ _{i}P ⊗_{A} P_{k} \\ & x(i ⋯ k) & & x(i ⋯ i) ⊗ (i ⋯ k) \end{matrix} )

≃ Cone ( \begin{matrix}& & P_{i} \\ & & ↓ _{δ_{i}} \\ (P_{k} & →^{μ_{ki}} & P_{i}) \end{matrix} ) ≃ P_{k} (P_{i} lies in degree − 1).

The complex F_{i}^{−1}(P_{j} → P_{i} → 0) is isomorphic to

Cone ( \begin{matrix}F_{i}^{−1}(P_{j}) = & A ⊗_{A} P_{j} & → & P_{i} ⊗ _{i}P ⊗_{A} P_{j} \\ & x & & x(j ⋯ i) ⊗ (i ⋯ j) \\ ↓ & ↓ & & ↓ \\ F_{i}^{−1}(P_{i}) = & A ⊗_{A} P_{i} & → & P_{i} ⊗ _{i}P ⊗_{A} P_{i} \\ & x(j ⋯ i) & & x(j ⋯ i) ⊗ (i ⋯ i) \end{matrix} )

≃ Cone ( \begin{matrix}(P_{j} & →^{μ_{ji}} & P_{i}) \\ x & & x(j ⋯ i) \\ & & ↓ _{δ_{i}} \\ & & P_{i} \\ & & x(j ⋯ i) \end{matrix} ) ≃ P_{j} ⟶^{μ_{ji}} P_{i} ⟶^{δ_{i}} P_{i},

and the complex F_{i}(P_{i} → P_{j}) is isomorphic to

Cone ( \begin{matrix}F_{i}(P_{i}) = & P_{i} ⊗ _{i}P ⊗_{A} P_{i} & → & A ⊗_{A} P_{i} \\ & x ⊗ (i) + x ⊗ (i ⋯ i) & & x(i) \\ ↓ & ↓ & & ↓ _{μ_{ij}} \\ F_{i}(P_{j}) = & P_{i} ⊗ _{i}P ⊗_{A} P_{j} & → & A ⊗_{A} P_{j} \\ & x ⊗ (i ⋯ j) & & x(i ⋯ j) \end{matrix} )

≃ Cone ( \begin{matrix}P_{i} \\ x \\ _{δ_{i}} ↓ \\ (P_{i} & →^{μ_{ij}} & P_{j}) \\ x & & x(i ⋯ j) \end{matrix} ) ≃ P_{i} ⟶^{δ_{i}} P_{i} ⟶^{μ_{ij}} P_{j}

And finally, with a similar way, the complex F_{j}(P_{i} → P_{i} → P_{j}) (where P_{j} is in degree −1) is isomorphic to

F_{j}(Cone(P_{i} → (Cone(P_{i} → P_{j}))[−1])) ≃ Cone(F_{j}(P_{i}) → (F_{j}(P_{i} → P_{j}))[−1])
≃ Cone ( \begin{matrix}(P_{j} & →^{μ_{ji}} & P_{i}) \\ & & _{δ_{i}} ↓ \\ & & P_{i} \end{matrix} )

≃ P_{j} ⟶^{μ_{ij}} P_{i} ⟶^{δ_{i}} P_{i}.

This finishes the proof. □

4 Categorification of a braid homomorphism

Let B be a Brauer tree algebra associated to a line with n edges and no exceptional vertex. Denote by B(A_{n}) the Artin braid group on n strings generated by σ_{1}, ⋯, σ_{n} satisfying the relations

σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1} and σ_{i}σ_{j} = σ_{j}σ_{i} if |i − j| > 1.

In [16] Rouquier and Zimmermann obtained an action of B(A_{n}) on derived Picard group of B.

Theorem 4.1 (Rouquier and Zimmermann [16]) There is a group homomorphism

ψ_{n} : B(A_{n}) → T rP ic(B)
σ_{i} ↦ R_{i}

where R_{i} is the two-sided tilting complex 0 → Q_{i} ⊗ Hom_{k}(Q_{i}, k) → B → 0 and Q_{1}, ⋯ Q_{n} are the indecomposable projective modules of B. Moreover ψ_{2} is injective.

In [10] Khovanov and Seidel independently discovered this homomorphism ψ for similar algebras in a very different context.

Theorem 4.2 (Khovanov and Seidel [10]) The map ψ_{n} is injective for all n.

For simplicity, let us denote by ψ := ψ_{n}. Denote by R_{i} the corresponding functor R_{i} := R_{i}⊗_{B}−. By [16, Lemma 4.2], the image of the indecomposable projective modules Q_{j} of B under R_{i} are

R_{i}(Q_{j}) = \{ \begin{matrix}0 & → & Q_{i} & → & 0 & & & if i = j \\ 0 & → & Q_{i} & → & Q_{j} & → & 0 & if |i − j| = 1 \\ & & 0 & → & Q_{j} & → & 0 & if |i − j| > 1 \end{matrix}

where Q_{j} is in degree 0.

In the next subsections we will provide a link between B(K_{n}), B(A_{n}), A and B which will show that the action defined in Theorem 3.1 is unfaithful.

4.1 Braid homomorphism

Define

c_{n} := σ_{n} and c_{k} := σ_{k}^{−1}c_{k+1}σ_{k} for all 1 ≤ k ≤ n − 1.

Define a mapping

η : B(K_{n}) → B(A_{n})
a_{i}  ↦ c_{i}

Proposition 4.3 The mapping η is a surjective group homomorphism.

First we will need the following Lemma : Lemma 4.4 c_{j}σ_{i}^{−1} = σ_{i}^{−1}c_{j} for all 1 ≤ j ≤ n−1, 1 ≤ i ≤ n, i ≠ j−1, i ≠ j Proof If i > j, we have

c_{j}σ_{i}^{−1} = σ_{j}^{−1} ⋯ σ_{n−1}^{−1}σ_{n}σ_{n−1} ⋯ σ_{j}σ_{i}^{−1}
= σ_{j}^{−1} ⋯ σ_{n−1}^{−1}σ_{n}σ_{n−1} ⋯ σ_{i+1}σ_{i}σ_{i−1}σ_{i}^{−1}σ_{i−2} ⋯ σ_{j}
= σ_{j}^{−1} ⋯ σ_{n−1}^{−1}σ_{n}σ_{n−1} ⋯ σ_{i+1}σ_{i−1}^{−1}σ_{i}σ_{i−1}σ_{i−2} ⋯ σ_{j}
= σ_{j}^{−1} ⋯ σ_{i−1}^{−1}σ_{i}^{−1}σ_{i−1}^{−1}σ_{i+1}^{−1} ⋯ σ_{n−1}^{−1}σ_{n}σ_{n−1} ⋯ σ_{i+1}σ_{i}σ_{i−1}σ_{i−2} ⋯ σ_{j}
= σ_{j}^{−1} ⋯ σ_{i}^{−1}σ_{i−1}^{−1}σ_{i}^{−1}σ_{i+1}^{−1} ⋯ σ_{n−1}^{−1}σ_{n}σ_{n−1} ⋯ σ_{j}
= σ_{i}^{−1}c_{j}

If i < j, i ≠ j − 1,

c_{j}σ_{i}^{−1} = σ_{j}^{−1} ⋯ σ_{n−1}^{−1}σ_{n}σ_{n−1} ⋯ σ_{j}σ_{i}^{−1}
= σ_{i}^{−1}σ_{j}^{−1} ⋯ σ_{n−1}^{−1}σ_{n}σ_{n−1} ⋯ σ_{j}
= σ_{i}^{−1}c_{j}
□

Proof of Proposition 4.3 We need to show 1. σ_{n}c_{k}σ_{n} = c_{k}σ_{n}c_{k} for all 1 ≤ k ≤ n − 1.

2. c_{i}c_{j}c_{i} = c_{j}c_{i}c_{j} for all 1 ≤ i, j ≤ n − 1, i ≠ j.

3. σ_{i} is in the image of η for all 1 ≤ i ≤ n.

1. We will show this by induction. For k = n − 1,

σ_{n}c_{n−1}σ_{n} = σ_{n}σ_{n−1}^{−1}σ_{n}σ_{n−1}σ_{n}
= σ_{n}σ_{n}σ_{n−1}σ_{n}^{−1}σ_{n}
= σ_{n}σ_{n}σ_{n−1}.

c_{n−1}σ_{n}c_{n−1} = σ_{n−1}^{−1}σ_{n}σ_{n−1}σ_{n}σ_{n−1}^{−1}σ_{n}σ_{n−1}
= σ_{n−1}^{−1}σ_{n}σ_{n}^{−1}σ_{n−1}σ_{n}σ_{n}σ_{n−1}
= σ_{n}σ_{n}σ_{n−1}.

Assume that σ_{n}c_{j}σ_{n} = c_{j}σ_{n}c_{j}; we will show that σ_{n}c_{j−1}σ_{n} = c_{j−1}σ_{n}c_{j−1}.

σ_{n}c_{j−1}σ_{n} = σ_{n}σ_{j−1}^{−1}c_{j}σ_{j−1}σ_{n}
= σ_{j−1}^{−1}σ_{n}c_{j}σ_{n}σ_{j−1}
= σ_{j−1}^{−1}c_{j}σ_{n}c_{j}σ_{j−1}
= σ_{j−1}^{−1}c_{j}σ_{j−1}σ_{n}σ_{j−1}^{−1}c_{j}σ_{j−1}
= c_{j−1}σ_{n}c_{j−1}.

2. For i = n − 1, if j = n − 2,

c_{n−1}c_{n−2}c_{n−1} = σ_{n−1}^{−1}σ_{n}σ_{n−1}σ_{n−2}^{−1}σ_{n−1}^{−1}σ_{n}σ_{n−1}σ_{n−2}σ_{n−1}^{−1}σ_{n}σ_{n−1}
= σ_{n}σ_{n−1}σ_{n}^{−1}σ_{n−2}^{−1}σ_{n}σ_{n−1}σ_{n}^{−1}σ_{n−2}σ_{n}σ_{n−1}σ_{n}^{−1}
= σ_{n}σ_{n−1}σ_{n−2}^{−1}σ_{n−1}σ_{n−2}σ_{n−1}σ_{n}^{−1}
= σ_{n}σ_{n−1}σ_{n−2}^{−1}σ_{n−2}σ_{n−1}σ_{n−2}σ_{n}^{−1}
= σ_{n}σ_{n−1}σ_{n−1}σ_{n−2}σ_{n}^{−1}.

c_{n−2}c_{n−1}c_{n−2} = σ_{n−2}^{−1}σ_{n−1}^{−1}σ_{n}σ_{n−1}σ_{n−2}σ_{n−1}^{−1}σ_{n}σ_{n−1}σ_{n−2}^{−1}σ_{n−1}^{−1}σ_{n}σ_{n−1}σ_{n−2}
= σ_{n−2}^{−1}σ_{n}σ_{n−1}σ_{n}^{−1}σ_{n−2}σ_{n}σ_{n−1}σ_{n}^{−1}σ_{n−2}^{−1}σ_{n}σ_{n−1}σ_{n}^{−1}σ_{n−2}
= σ_{n−2}^{−1}σ_{n}σ_{n−1}σ_{n−2}σ_{n−1}σ_{n−2}^{−1}σ_{n−1}σ_{n}^{−1}σ_{n−2}
= σ_{n−2}^{−1}σ_{n}σ_{n−2}σ_{n−1}σ_{n−2}σ_{n−2}^{−1}σ_{n−1}σ_{n−2}σ_{n}^{−1}
= σ_{n}σ_{n−1}σ_{n−1}σ_{n−2}σ_{n}^{−1}.

If j ≠ n − 2, c_{n−1}c_{j}c_{n−1} =

= σ_{n−1}^{−1}σ_{n}σ_{n−1}(σ_{j}^{−1}σ_{j+1}^{−1} ⋯ σ_{n−3}^{−1}σ_{n−2}^{−1}σ_{n−1}^{−1}σ_{n}σ_{n−1}σ_{n−2}σ_{n−3} ⋯ σ_{j+1}σ_{j})σ_{n−1}^{−1}σ_{n}σ_{n−1}
= (σ_{j}^{−1}σ_{j+1}^{−1} ⋯ σ_{n−3}^{−1})σ_{n−1}^{−1}σ_{n}σ_{n−1}(σ_{n−2}^{−1}σ_{n−1}^{−1}σ_{n}σ_{n−1}σ_{n−2})σ_{n−1}^{−1}σ_{n}σ_{n−1}(σ_{n−3} ⋯ σ_{j})
= σ_{j}^{−1}σ_{j+1}^{−1} ⋯ σ_{n−3}^{−1}c_{n−1}c_{n−2}c_{n−1}σ_{n−3} ⋯ σ_{j}
= σ_{j}^{−1}σ_{j+1}^{−1} ⋯ σ_{n−3}^{−1}c_{n−2}c_{n−1}c_{n−2}σ_{n−3} ⋯ σ_{j},

c_{j}c_{n−1}c_{j} =

= σ_{j}^{−1}σ_{j+1}^{−1} ⋯ σ_{n−1}^{−1}σ_{n}σ_{n−1} ⋯ σ_{j}σ_{n−1}^{−1}σ_{n}σ_{n−1}σ_{j}^{−1}σ_{j+1}^{−1} ⋯ σ_{n−1}^{−1}σ_{n}σ_{n−1} ⋯ σ_{j}
= σ_{j}^{−1}σ_{j+1}^{−1} ⋯ σ_{n−3}^{−1}σ_{n−2}^{−1}σ_{n−1}^{−1}σ_{n}σ_{n−1}σ_{n−2}σ_{n−1}^{−1}σ_{n}σ_{n−1}σ_{n−2}^{−1}σ_{n−1}^{−1}σ_{n}σ_{n−1}σ_{n−2}σ_{n−3} ⋯ σ_{j}
= σ_{j}^{−1}σ_{j+1}^{−1} ⋯ σ_{n−3}^{−1}c_{n−2}c_{n−1}c_{n−2}σ_{n−3} ⋯ σ_{j}.

Now assume that c_{j}c_{k}c_{j} = c_{k}c_{j}c_{k} for all k > i. We will show that

c_{j}c_{i}c_{j} = c_{i}c_{j}c_{i}.

If j ≠ i + 1

c_{j}c_{i}c_{j} = c_{j}σ_{i}^{−1}c_{i+1}σ_{i}c_{j}
= σ_{i}^{−1}c_{j}c_{i+1}c_{j}σ_{i} by Lemma 4.4
= σ_{i}^{−1}c_{i+1}c_{j}c_{i+1}σ_{i}
= σ_{i}^{−1}c_{i+1}σ_{i}c_{j}σ_{i}^{−1}c_{i+1}σ_{i}
= c_{i}c_{j}c_{i}.

If j = i + 1

c_{i+1}c_{i}c_{i+1} = c_{i+1}σ_{i}^{−1}c_{i+1}σ_{i}c_{i+1}
= c_{i+1}σ_{i}^{−1}σ_{i+1}^{−1}c_{i+2}σ_{i+1}σ_{i}c_{i+1}
= σ_{i+1}^{−1}c_{i+2}σ_{i+1}σ_{i}^{−1}σ_{i+1}^{−1}c_{i+2}σ_{i+1}σ_{i}σ_{i+1}^{−1}c_{i+2}σ_{i+1}
= σ_{i+1}^{−1}c_{i+2}σ_{i}^{−1}σ_{i+1}^{−1}σ_{i}c_{i+2}σ_{i}^{−1}σ_{i+1}σ_{i}c_{i+2}σ_{i+1}
= σ_{i+1}^{−1}σ_{i}^{−1}c_{i+2}σ_{i+1}^{−1}c_{i+2}σ_{i+1}σ_{i}c_{i+2}σ_{i+1}
= σ_{i+1}^{−1}σ_{i}^{−1}c_{i+2}c_{i+1}c_{i+2}σ_{i}σ_{i+1}
= σ_{i+1}^{−1}σ_{i}^{−1}c_{i+1}c_{i+2}c_{i+1}σ_{i}σ_{i+1}
= σ_{i+1}^{−1}σ_{i}^{−1}σ_{i+1}^{−1}c_{i+2}σ_{i+1}c_{i+2}σ_{i+1}^{−1}c_{i+2}σ_{i+1}σ_{i}σ_{i+1}
= σ_{i}^{−1}σ_{i+1}^{−1}σ_{i}^{−1}c_{i+2}σ_{i+1}c_{i+2}σ_{i+1}^{−1}c_{i+2}σ_{i}σ_{i+1}σ_{i}
= σ_{i}^{−1}σ_{i+1}^{−1}c_{i+2}σ_{i}^{−1}σ_{i+1}c_{i+2}σ_{i+1}^{−1}σ_{i}c_{i+2}σ_{i+1}σ_{i}
= σ_{i}^{−1}σ_{i+1}^{−1}c_{i+2}σ_{i}^{−1}σ_{i+1}σ_{i}c_{i+2}σ_{i}^{−1}σ_{i+1}^{−1}σ_{i}c_{i+2}σ_{i+1}σ_{i}
= σ_{i}^{−1}σ_{i+1}^{−1}c_{i+2}σ_{i+1}σ_{i}σ_{i+1}^{−1}c_{i+2}σ_{i+1}σ_{i}^{−1}σ_{i+1}^{−1}c_{i+2}σ_{i+1}σ_{i}
= c_{i}c_{i+1}c_{i}.

3. We will show that

c_{k}c_{k+1}c_{k}^{−1} = c_{k+1}^{−1}c_{k}c_{k+1} = σ_{k}

for all 1 ≤ k ≤ n − 1. For k = n − 1,
c_{n−1}c_{n}c_{n−1}^{−1} = σ_{n−1}^{−1}σ_{n}σ_{n−1}σ_{n}σ_{n−1}^{−1}σ_{n}^{−1}σ_{n−1}
= σ_{n}σ_{n−1}σ_{n}^{−1}σ_{n}σ_{n−1}^{−1}σ_{n}^{−1}σ_{n−1}
= σ_{n−1}.

Assume c_{k+1}c_{k+2} = c_{k+2}σ_{k+1}. We will show that c_{k}c_{k+1} = c_{k+1}σ_{k}.

c_{k}c_{k+1} = σ_{k}^{−1}c_{k+1}σ_{k}c_{k+1}
= σ_{k}^{−1}c_{k+2}σ_{k+1}c_{k+2}^{−1}σ_{k}c_{k+2}σ_{k+1}c_{k+2}^{−1}
= σ_{k}^{−1}c_{k+2}σ_{k+1}σ_{k}σ_{k+1}c_{k+2}^{−1}
= σ_{k}^{−1}c_{k+2}σ_{k}σ_{k+1}σ_{k}c_{k+2}^{−1}
= c_{k+2}σ_{k+1}σ_{k}c_{k+2}^{−1}
= c_{k+1}c_{k+2}σ_{k}c_{k+2}^{−1}
= c_{k+1}σ_{k}.
□

Proposition 4.5 The homomorphism η is not injective.

Proof We have σ_{1}σ_{n} = σ_{n}σ_{1} but a_{1}a_{2}a_{1}^{−1}a_{n} ≠ a_{n}a_{1}a_{2}a_{1}^{−1} as we will show using the geometric representation of the Coxeter group of B(K_{n}).

Denote by W (K_{n}) the Coxeter group of B(K_{n}). Denote by d_{i} the image of a_{i} in W (K_{n}). By [7, section 5.3], W (K_{n}) acts on an n-dimensional vector space V with basis {v_{1}, ⋯, v_{n}}. Let us define a bilinear form (., .) : V × V → ℝ by

(v_{i}, v_{j}) = \{ \begin{matrix}−\frac{1}{2} & if i ≠ j, \\ 1 & if i = j. \end{matrix}

The action of W (K_{n}) is defined by d_{i} ↦ f_{i} : V → V where

f_{i}(x) = x − 2(v_{i}, x)v_{i}, x ∈ V.

Notice that

f_{i}(v_{i}) = v_{i} − 2(v_{i}, v_{i})v_{i} = v_{i} − 2v_{i} = −v_{i} and
f_{i}(v_{j}) = v_{j} − 2(v_{i}, v_{j})v_{i} = v_{i} + v_{j}.

Now if a_{1}a_{2}a_{1}^{−1}a_{n}a_{1}a_{2}^{−1}a_{1}^{−1}a_{n}^{−1} = e, then

d_{1}d_{2}d_{1}d_{n}d_{1}d_{2}d_{1}d_{n} = e and f_{1}f_{2}f_{1}f_{n}f_{1}f_{2}f_{1}f_{n} = id_{V};

but

f_{1}f_{2}f_{1}f_{n}f_{1}f_{2}f_{1}f_{n}(v_{n}) = f_{1}f_{2}f_{1}f_{n}f_{1}f_{2}f_{1}(−v_{n})
= f_{1}f_{2}f_{1}f_{n}f_{1}f_{2}(−v_{1} − v_{n})
= f_{1}f_{2}f_{1}f_{n}f_{1}(−2v_{2} − v_{1} − v_{n})
= f_{1}f_{2}f_{1}f_{n}(−2v_{1} − 2v_{2} − v_{n})
= f_{1}f_{2}f_{1}(−3v_{n} − 2v_{1} − 2v_{2})
= f_{1}f_{2}(−3v_{1} − 3v_{n} − 2v_{2})
= f_{1}(−4v_{2} − 3v_{1} − 3v_{n})
= (−4v_{1} − 4v_{2} − 3v_{n}) ≠ v_{n}, a contradiction.
□

4.2 Categorification

Let T be the direct sum of the following complexes of projective B-modules.

T_{1} : 0 → Q_{n} → Q_{n−1} → ⋯ → Q_{2} → Q_{1} → 0
T_{2} : 0 → Q_{n} → Q_{n−1} → ⋯ → Q_{2} → 0
⋮  ⋮
T_{n−1} : 0 → Q_{n} → Q_{n−1} → 0
T_{n} : 0 → Q_{n} → 0

where Q_{i} is in degree i − 1.

By [13, Theorem 4.2], using the stable equivalence between End_{D^{b}(B)}(T) and A and the result of Gabriel and Riedtmann [4], T is the tilting complex giving a derived equivalence between B and A. (We note that by computing the endomorphism ring of T directly with the method in [12], we can show that End_{D^{b}(B)}(T) ≃ A).

Denote by G the unique two-sided tilting complex of (A ⊗ B^{op})-modules associated to T (such a complex is unique up to isomorphism in D^{b}(A⊗B^{op}) by a theorem of Keller [8]). Denote by G the corresponding functor

G = G ⊗_{B} −.

It is easy to see that

G(T_{i}) = P_{i} for all 1 ≤ i ≤ n.  (2)

We will compute the image of R_{i} under this functor.

D^{b}(B)  →^{G}  D^{b}(A)
R_{i}⊗_{B}− ↓  ↓ _{G(R_{i})⊗_{A}−}
D^{b}(B)  →^{G}  D^{b}(A)

This will give the image of R_{i} in T rP ic(A) under the group isomorphism

Γ : T rP ic(B) →^{∼} T rP ic(A)

induced by G. From now on, when we do a multiplication of elements of T rP ic, we omit the tensor product symbol ⊗_{A} (F F′ := F ⊗_{A} F′).

Proposition 4.6 We have

Γ(R_{i}) = \{ \begin{matrix}F_{i}F_{i+1}F_{i}^{−1} & if 1 ≤ i ≤ n − 1, \\ F_{n} & if i = n. \end{matrix}

We need the following lemmas

Lemma 4.7 For all 1 ≤ i ≤ n − 1,

R_{i}(T_{j}) = \{ \begin{matrix}& T_{j} & if & |i − j| > 1 \\ & T_{i−1} & if & j = i − 1 \\ & T_{i+1} & if & j = i \\ Cone(Cone(T_{i} → T_{i+1}) → & T_{i+1}) & if & j = i + 1 \end{matrix}

Proof For all 1 ≤ j ≤ n − 1 denote by

\begin{aligned}ν_{j,j+1} : & Q_{j} & → & Q_{j+1} & and & ν_{j+1,j} : & Q_{j+1} & → & Q_{j} \\ & x & ↦ & x(j j + 1) & & & x & ↦ & x(j + 1 j) \end{aligned}

and for all 1 ≤ i ≤ n denote by ρ_{i} : Q_{i} → Q_{i} where ρ_{i}((i)) = (i).

For all 1 ≤ j ≤ n − 1 we have a triangle

T_{j} →^{f_{j}} T_{j+1} →^{g_{j}} Q_{j}[j] ⇝^{h_{j}} T_{j}[1]

where f_{j} = (ρ_{n}, ρ_{n−1}, ⋯, ρ_{j+1}, 0, ⋯, 0), ρ_{k} in degree k − 1; g_{j} = ν_{j+1,j} in degree j and 0 otherwise; and h_{j} = ρ_{j} in degree j and 0 otherwise.

Therefore,

Cone(T_{j+1}[−1] → Q_{j}[j − 1]) ≃ T_{j} and
Cone(T_{j} → T_{j+1}) ≃ Q_{j}[j].

For j > i + 1, we will prove the lemma by induction : if j = n,

R_{i}(T_{n}) = R_{i}(Q_{n}[n − 1]) = Q_{n}[n − 1] = T_{n}.

Suppose that R_{i}(T_{j+1}) = T_{j+1}. We will show that R_{i}(T_{j}) = T_{j}.

R_{i}(T_{j}) ≃ R_{i}(Cone(T_{j+1}[−1] → Q_{j}[j − 1]))
≃ Cone(R_{i}(T_{j+1})[−1] → R_{i}(Q_{j}[j − 1]))
≃ Cone(T_{j+1}[−1] → Q_{j}[j − 1]) ≃ T_{j}.  (3)

If j = i + 1,

R_{i}(T_{i+1}) = R_{i}(Cone(T_{i+2}[−1] → Q_{i+1}[i]))
≃ Cone(R_{i}(T_{i+2})[−1] → R_{i}(Q_{i+1}[i]))
≃ Cone(T_{i+2}[−1] → (Q_{i} → Q_{i+1})[i])
≃ Cone ( \begin{matrix}(Q_{n} → ⋯ → & Q_{i+3} & ⟶^{ν_{i+3,i+2}} & Q_{i+2}) \\ & ↓ _{0} & & ↓ _{ν_{i+2,i+1}} \\ & (Q_{i} & ⟶^{ν_{i,i+1}} & Q_{i+1}) \end{matrix} )

where Q_{i+1} lies in degree i

≃ ( \begin{matrix}Q_{n} → ⋯ → Q_{i+3} & → & Q_{i+2} \\ & ↘ _{0} & ⊕ & ↘ \\ & & Q_{i} & → & Q_{i+1} \end{matrix} )

≃ Cone ( \begin{matrix}& Q_{i} \\ & ↓ _{ν_{i,i+1}} \\ (Q_{n} → ⋯ → Q_{i+2} → & Q_{i+1}) \end{matrix} )

(since Hom_{B}(Q_{i+3}, Q_{i}) = 0)

≃ Cone(Q_{i}[i] → T_{i+1})
≃ Cone(Cone(T_{i} → T_{i+1}) → T_{i+1})

If j = i,

R_{i}(T_{i}) = R_{i}(Cone(T_{i+1}[−1] → Q_{i}[i − 1]))
≃ Cone(R_{i}(T_{i+1}[−1]) → R_{i}(Q_{i}[i − 1]))

≃ Cone (( \begin{matrix}Q_{n} → ⋯ → Q_{i+3} & → & Q_{i+2} \\ & ↘ & ⊕ & ↘ \\ & & Q_{i} & → & Q_{i+1} \end{matrix} ) → Q_{i}[i])

where Q_{i+1} lies in degree i − 1

≃ ( \begin{matrix}Q_{n} → ⋯ → Q_{i+3} & →^{ν_{i+3,i+2}} & Q_{i+2} \\ & _{0} ↘ & ⊕ & ↘ _{ν_{i+2,i+1}} \\ & & Q_{i} & →^{ν_{i,i+1}} & Q_{i+1} \\ & & & ↘ _{ρ_{i}} & ⊕ \\ & & & & Q_{i} \end{matrix} )

where Q_{i+1} lies in degree i

≃ Cone ((Q_{n} → ⋯ → Q_{i+3} → Q_{i+2}) ⟶^{ν_{i+2,i+1}} ( \begin{matrix}Q_{i} & ⟶^{ν_{i,i+1}} & Q_{i+1} \\ & _{ρ_{i}} ↘ & ⊕ \\ & & Q_{i} \end{matrix} ) [i])

≃ Cone (T_{i+2}[−1] → Cone ( \begin{matrix}Q_{i} & → & Q_{i+1} \\ ↓ \\ Q_{i} \end{matrix} ) [i − 1])
≃ Cone(T_{i+2}[−1] → Q_{i+1}[i]) ≃ T_{i+1}.

For j < i, by induction and the same argument as (3) we only need to show that

R_{i}(T_{i−1}) = T_{i−1}.

We have :

R_{i}(T_{i−1}) = R_{i}(Cone(T_{i}[−1] → Q_{i−1}[i − 2]))
≃ Cone(R_{i}(T_{i}[−1]) → R_{i}(Q_{i−1}[i − 2]))
≃ Cone(T_{i+1}[−1] → (Q_{i} → Q_{i−1})[i − 2])
≃ Cone ( \begin{matrix}(Q_{n} → ⋯ → & Q_{i+1}) \\ & ↓ _{ν_{i+1,i}} \\ & (Q_{i} & → Q_{i−1}) \end{matrix} )
where Q_{i−1} in degree i − 2
≃ T_{i−1}.
□

Lemma 4.8

R_{n}(T_{j}) = \{ \begin{matrix}Cone(T_{n} → T_{j}) & if j < n, \\ T_{n}[1] & if j = n. \end{matrix}

Proof For j = n, R_{n}(T_{n}) = R_{n}(Q_{n}[n − 1]) = Q_{n}[n] = T_{n}[1]. For j < n, we will show the result by induction.

R_{n}(T_{n−1}) = R_{n}(Cone(Q_{n} → Q_{n−1}))

≃ Cone(R_{n}(Q_{n}) → R_{n}(Q_{n−1}))
≃ Cone ( \begin{matrix}Q_{n} \\ _{ρ_{n}} ↓ \\ (Q_{n} & → & Q_{n−1}) \end{matrix} )
≃ Cone(T_{n} → T_{n−1})

Assume that R_{n}(T_{j+1}) = Cone(T_{n} → T_{j+1}). We will show that R_{n}(T_{j}) = Cone(T_{n} → T_{j}).

R_{n}(T_{j}) ≃ R_{n}(Cone(T_{j+1}[−1] → Q_{j}[j − 1]))
≃ Cone(R_{n}(T_{j+1}[−1]) → R_{n}(Q_{j}[j − 1]))
≃ Cone(Cone(T_{n} → T_{j+1})[−1] → Q_{j}[j − 1])
≃ Cone(T_{n} → Cone(T_{j+1}[−1] → Q_{j}[j − 1]))
≃ Cone(T_{n} → T_{j}).
□

Proof of Proposition 4.6 We have a commutative diagram

D^{b}(B) ⟶^{G} D^{b}(A)
↓ _{R_{i}}  ↓ _{W_{i}}
D^{b}(B) ⟶^{G} D^{b}(A)  (4)

where we denote

W_{i} := G ∘ R_{i} ∘ G^{−1}.

We will show that for all projective indecomposable A-modules P, 1. W_{i}(P) = F_{i}F_{i+1}F_{i}^{−1}(P) for all 1 ≤ i ≤ n − 1, and

2. W_{n}(P) = F_{n}(P).

1. Fix 1 ≤ i ≤ n−1. Recall that G(T_{j}) = P_{j} for all 1 ≤ j ≤ n. By Lemma

4.7 and the above commutative diagram (4),

W_{i}(P_{j}) = W_{i}(G(T_{j}))
= GR_{i}(T_{j})
= \{ \begin{matrix}& P_{j} & if & |i − j| > 1 or & j = i − 1, \\ & P_{i+1} & if & j = i, \\ P_{i} → P_{i+1} → & P_{i+1} & if & j = i + 1. \end{matrix}

From (1) (see Theorem 3.1) we get

F_{i}F_{i+1}F_{i}^{−1}(P_{j}) = \{ \begin{matrix}& P_{j} & if & |i − j| > 1 or & j = i − 1, \\ & P_{i+1} & if & j = i, \\ P_{i} → P_{i+1} → & P_{i+1} & if & j = i + 1. \end{matrix}

(5)

Therefore, we get W_{i} = F_{i}F_{i+1}F_{i}^{−1}.

2. By (2), Lemma 4.8 and Diagram (4),

W_{n}(P_{j}) = W_{n}G(T_{j})
= GR_{n}(T_{j})
= G(Cone(T_{n} → T_{j}))
= P_{n} → P_{j} for all 1 ≤ j < n,

and W_{n}(P_{n}) = W_{n}G_{n}(T_{n}) = GR_{n}(T_{n}) = G(T_{n}[1]) = P_{n}[1]. Hence, by Lemma 2.2, W_{n} = F_{n}. □

Proposition 4.9 Γ^{−1}(F_{n}) = R_{n} and

Γ^{−1}(F_{k}) = R_{k}^{−1}R_{k+1}^{−1} ⋯ R_{n−1}^{−1}R_{n}R_{n−1} ⋯ R_{k+1}R_{k}
= R_{n}R_{n−1} ⋯ R_{k+1}R_{k}R_{k+1}^{−1} ⋯ R_{n−1}^{−1}R_{n}^{−1}

Proof Define C_{n} := R_{n} and

C_{k} := R_{k}^{−1}C_{k+1}R_{k}, for all 1 ≤ k ≤ n − 1.

By Proposition 4.6, Γ(C_{n}) = F_{n}. Now assume that Γ(C_{k+1}) = F_{k+1}. We will show that Γ(C_{k}) = F_{k}.

Γ(C_{k}) = Γ(R_{k}^{−1}C_{k+1}R_{k})
= (F_{k}F_{k+1}^{−1}F_{k}^{−1})F_{k+1}(F_{k}F_{k+1}F_{k}^{−1})
= F_{k}F_{k+1}^{−1}F_{k}^{−1}F_{k}F_{k+1}F_{k}F_{k}^{−1} = F_{k}.
□

Corollary 4.10 The following diagram is commutative :

B(K_{n}) ⟶^{η} B(A_{n})
↓ _{φ}  ↓ _{ψ}
T rP ic(A) ⟶^{Γ^{−1}} T rP ic(B)

This defines a categorification of the group homomorphism η.

Corollary 4.11 Since η is not injective, the action of B(K_{n}) on T rP ic(A) is not faithful.

5 A faithful action

Denote by B(Ã_{n−1}) the affine braid group generated by h_{1}, ⋯, h_{n} subject to relations

h_{i}h_{i+1}h_{i} = h_{i+1}h_{i}h_{i+1} if 1 ≤ i ≤ n − 1,
h_{i}h_{j} = h_{j}h_{i}  if |i − j| ≠ 1, n − 1 and
h_{n}h_{1}h_{n} = h_{1}h_{n}h_{1}.

In [17] Schaps and Zakay-Illouz obtained an action of B(Ã_{n−1}) on the subgroup of T rP ic(A) generated by H_{1}, ⋯, H_{n} where H_{i} is the refolded tilting complex associated to the transposition (i i + 1). (See [17] for the definition).

Theorem 5.1 (Schaps and Zakay-Illouz [17]) There is a group homomorphism

ρ : B(Ã_{n−1}) → T rP ic(A)
h_{i}  ↦ H_{i}.

Remark In [17] the action is defined for Brauer star algebras in general.

Denote by H_{i} the corresponding functor H_{i} := H_{i} ⊗_{A} −. Using [17, Proposition 1] we get the image of indecomposable projective modules of A under H_{i} :

H_{i}(P_{j}) = \{ \begin{matrix}& & & & P_{j} & if & |i − j| > 1, \\ & & & & P_{i−1} & if & j = i − 1, \\ & & & & P_{i+1} & if & j = i, \\ P_{i} & → & P_{i+1} & → & P_{i+1} & if & j = i + 1. \end{matrix}

Comparing this with (5) we get

H_{i} = F_{i}F_{i+1}F_{i}^{−1} for all 1 ≤ i ≤ n − 1,  (6)

and it is easy to see from (1) that

H_{n} = F_{n}F_{1}F_{n}^{−1}.  (7)

5.1 Injective braid homomorphisms

Let B(B_{n}) be the braid group associated to Dynkin diagram of type B_{n} with generators b_{1}, b_{2}, ⋯, b_{n} subject to relations :

b_{i}b_{i+1}b_{i} = b_{i+1}b_{i}b_{i+1} if 1 ≤ i ≤ n − 1,
b_{i}b_{j} = b_{j}b_{i}  if |i − j| > 1,
b_{n−1}b_{n}b_{n−1}b_{n} = b_{n}b_{n−1}b_{n}b_{n−1}.

By [5], B(B_{n}) is also generated by τ, s_{1}, s_{2}, ⋯, s_{n} subject to relations :

s_{1}s_{n}s_{1} = s_{n}s_{1}s_{n},
s_{i}s_{i+1}s_{i} = s_{i+1}s_{i}s_{i+1} for 1 ≤ i ≤ n − 1,
s_{i}s_{j} = s_{j}s_{i} for |i − j| ≠ 1, n − 1,
τs_{i}τ^{−1} = s_{i−1} for 1 ≤ i ≤ n − 1,
τs_{1}τ^{−1} = s_{n},

and the subgroup of B(B_{n}) generated by s_{1}, ⋯, s_{n} is the affine braid group B(Ã_{n−1}). By [9], this shows that B(B_{n}) is a semidirect product of the infinite cyclic group generated by τ and B(Ã_{n−1}). Therefore B(Ã_{n−1}) injects into B(B_{n}) (see [9] for more details).

We can reformulate [5, Proposition 6] to get

τ = b_{n}b_{n−1} ⋯ b_{2},
s_{i} = b_{i} 1 ≤ i ≤ n − 1,
s_{n} = τb_{1}τ^{−1} = b_{n}b_{n−1} ⋯ b_{2}b_{1}b_{2}^{−1} ⋯ b_{n−1}^{−1}b_{n}^{−1},

such that the embedding of B(Ã_{n−1}) into B(B_{n}) is defined by the following map :

μ : B(Ã_{n−1}) → B(B_{n})
h_{i}  ↦ b_{i} 1 ≤ i ≤ n − 1,
h_{n}  ↦ b_{n}b_{n−1} ⋯ b_{2}b_{1}b_{2}^{−1} ⋯ b_{n−1}^{−1}b_{n}^{−1}.

Now define a group homomorphism

χ : B(B_{n}) → B(A_{n})
b_{i}  ↦ σ_{i} 1 ≤ i ≤ n − 1,
b_{n} ↦ σ_{n}^{2}.

By [3], χ is injective. Therefore we obtain an injective composition of injective group homomorphisms

χ ∘ μ : B(Ã_{n−1}) → B(A_{n})
h_{i}  ↦ σ_{i}, 1 ≤ i ≤ n − 1,
h_{n}  ↦ σ_{n}σ_{n}σ_{n−1} ⋯ σ_{2}σ_{1}σ_{2}^{−1} ⋯ σ_{n−1}^{−1}σ_{n}^{−1}σ_{n}^{−1}.

5.2 Another Categorification

By Proposition 4.6, Proposition 4.9, (6) and (7) we obtain

Γ^{−1}(H_{i}) = R_{i} for all 1 ≤ i ≤ n − 1,
Γ^{−1}(H_{n}) = Γ^{−1}(F_{n}F_{1}F_{n}^{−1})
= R_{n}R_{n}R_{n−1} ⋯ R_{2}R_{1}R_{2}^{−1} ⋯ R_{n−1}^{−1}R_{n}^{−1}R_{n}^{−1}.

Therefore, we get the following commutative diagram :

B(Ã_{n−1}) ⟶^{χ∘μ} B(A_{n})
↓ _{ρ}  ↓ _{ψ}
T rP ic(A) ⟶^{Γ^{−1}} T rP ic(B)

And since χ ∘ μ and ψ are injective, we obtain the following result :

Theorem 5.2 The action of B(Ã_{n−1}) on T rP ic(A) defined in Theorem 5.1 is faithful.

Remark We leave as an open question whether the action defined in [17] is faithful for Brauer star algebras with exceptional vertex. We also get a faithful action B(B_{n}) → T rP ic(A) by defining

B(B_{n}) → T rP ic(A),
b_{i} ↦ F_{i}F_{i+1}F_{i}^{−1} if 1 ≤ i ≤ n − 1,
b_{n} ↦ F_{n}F_{n}.

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Department of Mathematics, University of Leicester, University Road, LE1 7RH, Leicester, UK, email : idma1@le.ac.uk