Self-equivalences of the derived category of Brauer tree algebras with exceptional vertices

Alexander Zimmermann^{∗}

Faculté de Mathématiques et CNRS (FRE 2270) Université de Picardie; 33 rue St Leu; 80039 Amiens Cedex; France

                  electronic mail: Alexander.Zimmermann@u-picardie.fr
                 http://www.mathinfo.u-picardie.fr/alex/azim.html

to Mirela Stefanescu for her 60th birthday

Abstract

Let k be a field and A be a Brauer tree algebra associated with a Brauer tree with possibly non trivial exceptional vertex. In an earlier joint paper with Raphaël Rouquier we studied and defined the group T rP ic_{k}(Λ) of standard self-equivalences of the derived category of a k-algebra Λ. In the present note we shall determine a non trivial homomorphism of a group slightly bigger than the pure braid group on n + 1 strings to T rP ic_{k}(A). This is a generalization of the main result in the joint paper with Raphaël Rouquier. The proof uses the result with Raphaël Rouquier.

Introduction

Let A be a k-algebra for a commutative ring k. Suppose that A is projective as a k-module. An equivalence F : D^{b}(A) ⟶ D^{b}(A) of the derived category of bounded complexes of A modules to itself is called of standard type if there is a complex X in D^{b}(A⊗_{k} A^{op}) so that F is isomorphic to X ⊗_{A}^{𝕃} −. In an earlier paper with Raphaël Rouquier [10] we introduced the group T rP ic_{k}(A) of isomorphism classes of self-equivalences of standard type of the derived category D^{b}(A). In case A is a Brauer tree algebra for a Brauer tree with n + 1 vertices, n edges and no exceptional vertex, we obtained in [10] n self-equivalences F_{1}, F_{2}, ⋯, F_{n} of standard type so that F_{i} ∘ F_{j} ≃ F_{j} ∘ F_{i} if |i − j| ≥ 2 and F_{i} ∘ F_{j} ∘ F_{i} = F_{j} ∘ F_{i} ∘ F_{j} if not. In other words, the braid group B_{n+1} on n + 1 strings maps to T rP ic_{k}(A) by some homomorphism φ_{n} which identifies the natural braid generators with the self-equivalences above. Moreover, if n = 2 this morphism was shown to be injective with normal image of index 4 · |P ic_{k}(A)|. Khovanov and Seidel proved in [4] that the mapping φ_{n} is injective for all n. The question of the image remains open for n > 2.

The purpose of this note is to see what the construction in the paper [10] has to say for a Brauer tree algebra A of a Brauer tree with n edges, n+1 vertices and an exceptional vertex with multiplicity strictly bigger than 1. We shall see that we still get subgroups S_{n} of the standard braid group B_{n+1}, containing the pure braid group P_{n+1} of B_{n+1} and containing the parabolic subgroup B_{n} of B_{n+1}.

It should be noted that Schaps and Zakay-Illouz obtained in [11] a homomorphism of the braid group of the affine Dynkin diagram Ã_{n} to T rP ic_{k}(A). No statement about the kernel or the image is made. Our approach is completely independent from the approach of Schaps and Zakay-Illouz. One should observe that the set of homomorphisms of the braid group of the affine Dynkin diagram Ã_{n} to the braid group of the Dynkin diagram A_{m} for all n and m is infinite, as is the set of homomorphisms from the braid group of the diagram A_{m} to the braid group of the diagram Ã_{n}.

∗Mathematics Subject Classification: 20C05, 18E30, 16E40

Finally I should note that our notation is that of the monography [6] and that of the paper [10]. The present note follows an idea which was sketched in an example in [14] for the case n = 2. There the method was used to lift a self-equivalence of the derived category of a group algebra to a self-equivalence of the derived category of the group ring over a complete discrete valuation ring, which reduces to the algebra, subject to the additional condition that if the self-equivalence of the derived category of the algebra fixes the trivial module, then the lifted self-equivalence fixes the trivial lattice.

1 A brief review on self-equivalences of derived categories

(1.1) Let k be a commutative ring and let A be a k-algebra. Rickard proved in [7] that for a k-algebra B the derived category D^{b}(A) of bounded complexes of A-modules is equivalent as triangulated category to the derived category of bounded complexes of B-modules if and only if there is a so-called tilting complex T over A with endomorphism ring in the derived category isomorphic to B. A tilting complex over A is a bounded complex of finitely generated projective A-modules so that Hom_{D^{b}(A)}(T, T [n]) = 0 if n ≠ 0, and so that K^{b}(A−proj) is the smallest triangulated category of D^{b}(A) containing all direct summands of finite direct sums of T. Here K^{b}(A − proj) denotes the homotopy category of bounded complexes of finitely generated projective A-modules. Given an equivalence D^{b}(B) ⟶ D^{b}(A) the image of the rank 1 free B-module is a tilting complex.

(1.2) In case A is flat as k-module, there exists a complex X in D^{b}(A ⊗_{R} B^{op}) so that X ⊗_{B}^{𝕃} − : D^{b}(B) ⟶ D^{b}(A) is an equivalence (cf Keller [3] for the general case or Rickard [8] for A and B being projective as k-modules). The complex X is called a twosided tilting complex and the equivalence X ⊗_{B}^{𝕃} − is called an equivalence of standard type. Moreover, X ⊗_{B}^{𝕃} B is a tilting complex over A, and for any tilting complex T over A with endomorphism ring B there exists a twosided tilting complex X with T ≃ X ⊗_{B}^{𝕃} B (cf Keller [3]). Any two tilting complexes X_{1} and X_{2} with

X_{2} ⊗_{B}^{𝕃} B ≃ T ≃ X_{1} ⊗_{B}^{𝕃} B

differ by an automorphism of B (cf [10]). This automorphism of B is induced by possibly different identifications of B with the endomorphism ring of T.

(1.3) In case B and A are even projective as k-modules, a quasi-inverse of a standard equivalence between the derived categories of A and B is standard as well. Moreover, (cf [10]) if B = A, the set of isomorphism classes of self-equivalences of standard type of the derived category of bounded complexes of A-modules forms a group T rP ic_{k}(A). This group contains the ordinary Picard group P ic_{k}(A) as subgroup. P ic_{k}(A) is in general not normal in T rP ic_{k}(A).

(1.4) For the theory of representations of finite groups an important class of examples of algebras is the class of Brauer tree algebras (cf e.g. Feit [2] or in connection with derived equivalences [6]). Two Brauer tree algebras A_{1} and A_{2} defined over the same field and with respect to two Brauer trees with the same number of edges and vertices and with the same multiplicity of the exceptional vertex have equivalent derived categories. As consequence we get T rP ic_{k}(A_{1}) ≃ T rP ic_{k}(A_{2}), even though P ic_{k}(A_{1}) and P ic_{k}(A_{2}) are in general different.

If A is a Brauer tree algebra for a Brauer tree with n edges and n + 1 vertices and without exceptional vertex, in [10] Raphaël Rouquier and the author defined a homomorphism

φ_{n} : B_{n+1} ⟶ T rP ic_{R}(A).

The restriction of φ_{n} to any standard parabolic subgroup B_{3} is injective. Moreover, Khovanov and Seidel show in [4] that φ_{n} is always injective.

(1.5) We have to work explicitly with φ_{n} and so we shall give some details on its construction. By the above we may assume that the Brauer tree of A is a line with n edges. The projective indecomposable modules are numbered P_{1}, P_{2}, …, P_{n} so that Hom_{A}(P_{i}, P_{i+1}) ≠ 0 for all i ∈ {1, 2, … n − 1}. We use the convention that the vertices of the Brauer tree are numbered so that the projective P_{i} is supported by the ordinary characters with the numbers i and i − 1. Then, (cf [10]) for any i ∈ {1, 2, …, n} the complexes X_{i} defined by

… ⟶ 0 ⟶ P_{i} ⊗_{k} Hom_{A}(P_{i}, A) ⟶ A ⟶ 0 ⟶ …

are twosided tilting complexes with ismorphism class in T rP ic_{k}(A). They satisfy the relations

X_{i} ⊗_{A} X_{i+1} ⊗_{A} X_{i} ≃ X_{i+1} ⊗_{A} X_{i} ⊗_{A} X_{i+1}

and

X_{i} ⊗_{A} X_{j} ≃ X_{j} ⊗_{A} X_{i} .

This means that mapping the standard braid generators s_{i} of

B_{n+1} =< s_{1}, s_{2}, …, s_{n} | s_{i}s_{i+1}s_{i} = s_{i+1}s_{i}s_{i+1} ; s_{i}s_{j} = s_{j}s_{i}
∀i = 1, …, n − 1∀j = 2, …, n : |i − j| ≥ 2 >

to the self-equivalence induced by X_{i} defines a homomorphism of groups.

(1.6) We shall derive the tilting complex associated to X_{i}. If 2 ≤ i ≤ n − 1, then

X_{i} ⊗_{A} P_{j} ≃ \{ \begin{matrix}P_{j} & if & |j − i| ≥ 2 \\ P_{i}[1] & if & j = i \\ … ⟶ 0 ⟶ P_{i} ⟶ P_{j} ⟶ 0 ⟶ … & if & |j − i| = 1 \end{matrix}

For i = 1 one gets

X_{1} ⊗_{A} P_{j} ≃ \{ \begin{matrix}P_{j} & if & j ≥ 3 \\ P_{1}[1] & if & j = 1 \\ … ⟶ 0 ⟶ P_{1} ⟶ P_{2} ⟶ 0 ⟶ … & if & j = 2 \end{matrix}

For i = n one gets

X_{n} ⊗_{A} P_{j} ≃ \{ \begin{matrix}P_{j} & if & j ≤ n − 2 \\ P_{n}[1] & if & j = n \\ … ⟶ 0 ⟶ P_{n} ⟶ P_{n−1} ⟶ 0 ⟶ … & if & j = n − 1 \end{matrix}

The complexes

T_{i} := ⨁_{j=1}^{n} X_{i} ⊗_{A} P_{j}

are tilting complexes with endomorphism ring isomorphic to A.

2 Tilting with an exceptional vertex

(2.1) We study A(l), the Brauer tree algebra associated to a Brauer tree being a line with non trivial exceptional vertex associated to the vertex l for 0 ≤ l ≤ n. The indecomposable projective module of A(l) supported by the ordinary characters associated to the vertex i and i − 1 is called P_{i}(l). Then we can define complexes T_{i} as above, using the convention that if we do not make precise the differential, then the differential is a mapping with maximal possible image. This condition uniquely determines the differential. Then, the complexes T_{i} are still tilting complexes as is easily verified. The endomorphism ring is not isomorphic to A(l). Indeed, by [7] the endomorphism ring is again a Brauer tree algebra with the same exceptional multiplicity, and comparing the dimensions of the homomorphisms between the projective indecomposable modules of such an algebra with the dimensions of the homomorphisms of the indecomposable direct summands of T (i), we find easily

End_{D^{b}(A(l))}(T_{l}) ≃ A(l − 1) ∀l ∈ {1, …, n},
End_{D^{b}(A(l))}(T_{l+1}) ≃ A(l + 1) ∀l ∈ {0, …, n − 1}

and

End_{D^{b}(A(l))}(T_{i}) ≃ A(l)

for i ∉ {l, l + 1} and all l ∈ {0, …, n}.

(2.2) Let X be a complex in D^{b}(B_{1} ⊗_{k} B_{2}^{op}) for two k-algebras B_{1} and B_{2}. If now X ⊗_{B_{2}} B_{2} is a tilting complex over B_{1} with endomorphism ring B_{2} and if B_{1} ⊗_{B_{1}} X is a tilting complex over B_{2} with endomorphism ring B_{1}, then X is a twosided tilting complex. This lemma due to Rickard is proved in [12].

(2.3) The most easy case is where the tilting complexes T_{i} induce a self-equivalence. A twosided tilting complex X_{i} associated to the tilting complex T_{i} over A(l) is now

X_{i} : … ⟶ 0 ⟶ P_{i} ⊗_{k} Hom_{A(l)}(P_{i}, A(l)) ⟶ A(l) ⟶ 0 ⟶ …

if i ∉ {l, l + 1}.

(2.4) Instead of finding a twosided tilting complex X_{l} associated to the tilting complex T_{l} over A(l) we determine the projective indecomposable module X_{l} ⊗_{A(l)} P_{i} under the isomorphism X_{l}⊗_{A(l)}A(l) ≃ A(l−1). First, we have to get around the ambiguity in choosing X_{l}. For a Brauer tree algebra A associated to a line, let Aut_{0}(A) be the subgroup of the automorphism group Aut_{k}(A) of A which fixes each projective indecomposable module. Now, it is clear that Aut_{k}(A)/Aut_{0}(A) is cyclic of order 2. In fact, the quotient is generated by the graph automorphism of the Brauer tree, neglecting the exceptional vertex. This follows from the discussion in [10], but also from Frauke Bleher’s thesis [1]. Since the choice of a twosided tilting complex is unique up to an (outer) automorphism of the algebra and since we only ask for the images of the projective indecomposable modules, only this cyclic group of order 2 matters. This cyclic group of order 2 now corresponds to numbering the vertices in the graph from 0 to n or from n to 0. On the other hand, we may modify a given twosided tilting complex with tensoring by the A(n − l) − A(l)-bimodule _{α}A(l)_{1}, where α is the isomorphism A(n − l) ⟶ A(l) induced by the graph automorphism.

(2.5) Now, determining the image of the projective indecomposable modules is an easy task since we know that both algebras are Brauer tree algebras (cf [7]), associated to a line and the endomorphism algebra is isomorphic to A(l−1). If we compare the dimension of homomorphisms between the complexes X_{l} ⊗_{A(l)} P_{i} and the dimensions of the indecomposable projective A(l − 1)-modules, we find that in D^{b}(A(l − 1)) we have

X_{l} ⊗_{A(l)} P_{i} ≃ P_{i}

for all i ∈ {1, …, n}.

(2.6) Similarly arguments yield that a twosided tilting complex X_{l+1} associated to T_{l+1} over A(l) has the property

X_{l+1} ⊗_{A(l)} P_{i} ≃ P_{i}

in D^{b}(A(l + 1)) for all i ∈ {1, …, n}.

3 Deducing self-tilting complexes

We are now ready to determine a group induced by the tilting complexes X_{i} for i ∈ {1, …, n} mapping to T rP ic_{k}(A(l)) for any l.

(3.1) The braid group B_{n} maps onto the Weyl group 𝔖_{n} by adding the relations that the standard generators have order 2. Let π_{n} be this morphism. Let 𝔖_{n−1}(l) be the subgroup of 𝔖_{n} which fixes the letter l. Of course, 𝔖_{n−1}(l) ≃ 𝔖_{n−1}. Now, π_{n}^{−1}(𝔖_{n−1}(l)) is a subgroup of B_{n}.

(3.2) We observe that

D^{b}(∏_{l=0}^{n} A(l)) ≃ ∏_{i=0}^{n} D^{b}(A(l)) .

Since X_{i} is a complex of A(l) ⊗_{k} A(l)^{op}-modules if i ∉ {l, l + 1} and X_{l} is a complex of A(l) ⊗_{k} A(l + 1)^{op}-modules, while X_{l+1} is a complex of A(l + 1) ⊗_{k} A(l)^{op}-modules, an element s_{i} ∈ B_{n+1} acts on D^{b}(∏_{l=0}^{n} A(l)) as tensor product with the twosided tilting complex X_{i}^{∏} := ∏_{l=0}^{n} X_{i} in D^{b}((∏_{l=0}^{n} A(l))⊗_{k} (∏_{l=0}^{n} A(l))^{op}). Moreover, the tensor product with X_{i}^{∏} induces a permutation of the index set {0, …, n}.

(3.3) Hence, this way s_{i} permutes the l + 1 factors in the product ∏_{l=0}^{n} D^{b}(A(l)) as the involution (i − 1, i) permutes the set {0, …, n}. This is exactly the image of s_{i} under the mapping π_{n+1}.

Moreover, let w ∈ B_{n+1}. Then, the self-equivalence Ω induced by w on ∏_{l=0}^{n} D^{b}(A(l)) induces a self-equivalence Ω_{l} on the l-th factor if and only if the permutation π_{n+1}(w) of {0, 1, …, n} fixes l:

∏_{l=0}^{n} D^{b}(A(l)) ⟶^{Ω} ∏_{l=0}^{n} D^{b}(A(l))
↓  ↓
D^{b}(A(l))  ⟶^{Ω_{l}}  D^{b}(A(l))

Observe that if one defines a functor D^{b}(A(i)) ⟶ ∏_{l=0}^{n} D^{b}(A(l)) and composes Ω with it and the projection ∏_{l=0}^{n} D^{b}(A(l)) ⟶ D^{b}(A(i)), the result will not be an equivalence unless the permutation π_{n+1}(w) of {0, 1, …, n} fixes i.

(3.4) Now we have proved the main result.

Theorem 1 Let k be a field. Let A(l) be the Brauer tree algebra associated to a Brauer tree being a line with n edges, with exceptional vertex at the l-th vertex, and with exceptional multiplicity m ≥ 2. Let B_{n+1} be the (ordinary) braid group on n + 1 strings and let π_{n+1} : B_{n+1} ⟶ 𝔖_{n+1} be the canonical projection. Then, there is a group homomorphism

φ_{n+1} : π_{n+1}^{−1}(Stab_{𝔖_{n+1}}(l)) ⟶ T rP ic_{k}(A(l)) .

Moreover, s_{i} ∈ π_{n+1}^{−1}(Stab_{𝔖_{n+1}}(l)) for i ∈ {1, 2, …, l − 1, l + 2, l + 3, …, n} and then, the self-equivalence φ_{n+1}(s_{i}) is infinite cyclic.

Proof. Let w ∈ π_{n+1}^{−1}(Stab_{𝔖_{n+1}}(l)), then φ_{n+1}(w) will be defined to be the derived equivalence denoted by Ω_{l} in (3.3). Since Ω is standard, it is not difficult to show that Ω_{l} is standard as well.

Let |i − j| ≥ 2. Then, virtually the same computation as in [10] yields

X_{i}^{∏} ⊗_{∏ A(l)} X_{j}^{∏} ≃ X_{j}^{∏} ⊗_{∏ A(l)} X_{i}^{∏} .

Actually, this may be verified in each component separately.

Similarly, looking at each of the factors of the product category separately one gets

X_{i}^{∏} ⊗_{∏ A(l)} X_{i+1}^{∏} ⊗_{∏ A(l)} X_{i}^{∏} ≃ X_{i+1}^{∏} ⊗_{∏ A(l)} X_{i}^{∏} ⊗_{∏ A(l)} X_{i+1}^{∏}

as complex of

\begin{matrix}n \\ ∏ \\ ( & {l=0 \atop l∉\{i−1,i+1\}} & ) \end{matrix} A(l) ⊗_{k} A(l)^{op} ∏ A(i − 1) ⊗_{k} A(i + 1)^{op} ∏ A(i + 1) ⊗_{k} A(i − 1)^{op} -modules.

Hence, φ_{n} extends to a homomorphism φ_{n} : B_{n+1} ⟶ T rP ic_{k}(∏_{l=0}^{n} A(l)).

Let now w_{1}, w_{2} ∈ π_{n+1}^{−1}(Stab_{𝔖_{n+1}}(l)). By (3.3) one has φ_{n}(w_{1}) and φ_{n}(w_{2}) induce elements in T rP ic_{k}(A(l)). Moreover, φ_{n}(w_{1}w_{2}) = φ_{n}(w_{1})φ_{n}(w_{2}) is an easy consequence. The fact that φ_{n}(s_{i}) is infinite cyclic is easily seen to be true in T rP ic_{k}(∏_{l=0}^{n} A(l)) and as well in T rP ic_{k}(A(l)). This finishes the proof.

Corollary 3.1 Let A be a Brauer tree algebra associated to a Brauer tree with n edges and a non trivial exceptional vertex. Then, there is a group homomorphism

π_{n+1}^{−1}(Stab_{𝔖_{n+1}}(l)) ⟶ T rP ic_{k}(A) .

Under this group homomorphism the image of the remaining s_{i} is infinite cyclic.

Indeed, such an algebra is derived equivalent to the Brauer tree algebra A(0). The rest follows.

Remark 3.2 The p-adic analogue of a Brauer tree algebra is a Green order as defined by Roggenkamp in [9]. At the points (2.1) and (2.5) which were used in the proof of Theorem 1 we used that only a Brauer tree algebra with the same number of simple modules and with the same exceptional multiplicity can be derived equivalent to a Brauer tree algebra. This is not known for Green orders. Nevertheless, the tilting complexes T_{i} used in the proof are all of a certain type, which were studied in a joint paper with Steffen König (cf [5] and its generalization in [6, Chapter 4]). If one uses this paper at that point one sees that also there the endomorphism ring of such a tilting complex is a Green order again, associated to a Brauer tree being a line and with the same structural constants. Moreover, for these tilting complexes an explicit twosided tilting complex was determined in [12, 13]. The result holds in the same way for Green orders associated to a line with one exceptional vertex and so that the combinatorial data are the same for all the vertices except for the exceptional vertex. The result which takes the place of the main result of [10] is [15]. It is not difficult to fill the details for the proof of this statements. We leave this task to the reader.

Remark 3.3 φ_{n+1} maps any subgroup B_{3} =< s_{i}, s_{i+1} > of π_{n+1}^{−1}(Stab_{𝔖_{n+1}}(l)) injectively to T rP ic_{k}(A(l)). This follows from the method of proof in [10], where it is proved that a non trivial word in B_{3} is mapped to a non trivial complex in P_{i} and P_{i+1}. Here, we get the same result with the same proof for the image of P_{i} and P_{i+1} by this word.

References

[1] Frauke Bleher, Automorphismen von Gruppenringen und Blocktheorie, PhD thesis, Universität Stuttgart 1995

[2] Walter Feit, The representation theory of finite groups, North Holland 1982.

[3] Bernhard Keller, A remark on tilting theory and DG-algebras, manuscripta mathematica 79 (1993) 247-253.

[4] Michael Khovanov and Paul Seidel, Quivers, Floer Cohomology and Braid Group Actions, preprint http://xxx.lanl.gov/abs/math.QA/0006056

[5] Steffen König and Alexander Zimmermann, Tilting selfinjective algebras and Gorenstein orders, Oxford Quarterly Journal of Mathematics (2) 48 (1997) 351-361.

[6] Steffen König and Alexander Zimmermann, Derived equivalences for group rings, (Springer Lecture Notes in Mathematics 1685, 1998).

[7] Jeremy Rickard, Morita theory for derived categories. J. London Math. Soc. 39 (1989), 436–456.

[8] Jeremy Rickard, Derived equivalences as derived functors, J. London Math. Soc. 43 (1991), 37-48.

[9] Klaus W. Roggenkamp, Blocks of cyclic defect and Green orders, Comm. in Alg. 20 (6) (1992), 1715–1734.

[10] Raphaël Rouquier and Alexander Zimmermann, A Picard group for derived module categories, accepted in Proc. London Math. Soc.

[11] Mary Schaps and Evelyne Zakay-Illouz, Braid group action on the refolded tilting complex of the Brauer star algebra, preprint, november 2000.

[12] Alexander Zimmermann, Twosided tilting complexes for Green orders and Brauer tree algebras, Journal of Algebra 187 (1997), 446-473.

[13] Alexander Zimmermann, Twosided tilting complexes for Gorenstein orders, Journal of Algebra 209 (1998) 585-621

[14] Alexander Zimmermann, Cohomology of groups, abelian Sylow subgroups and splendid equivalences, Algebra Montpellier Announcements (electronic) 01-2000. http://www.emis.de/journals/AMA/index.html

[15] Alexander Zimmermann, Automorphisms of Green orders and their derived categories, 13 pages (2000) to appear in Algebras and Representation theory.