HOMOGENEOUS DEFORMATIONS OF BRAUER TREE ALGEBRAS

Mary Schaps and Evelyne Zakay-Illouz*

Abstract. Based on tilting theory, we demonstrate the existence of homogeneous deformations for the Brauer tree algebras, which are derived naturally from a tilting complex.

§1. Introduction.

This is the third and last of the papers based on the Ph.D thesis of the second author [9]. Though it is by far the shortest of the three, it was in fact the motivation for the study of certain “combinatorical” tilting complexes, undertaken in [7] and [8]. The connection between tilting theory and the Donald-Flanigan problem was first established in [1], based on Rickard’s paper [4].

The general subject is tilting complexes and deformations for blocks of cyclic defect group. The work was motivated by two problems: First, to find a Donald-Flanigan deformation of blocks of cyclic defect which would be more natural than that given in [6]. Second, to try to understand the combinatorics of the Brauer tree in terms of the tilting theory. Rickard had already shown that a connection existed in [3], where he constructed a tilting complex from a block of the “global” Brauer tree to the Brauer star. We chose to work in the opposite direction; to find tilting complexes for the Brauer star algebra from which one can “read off” the global Brauer tree directly.

Since not every tree is the Brauer tree of a group block, our results are not restricted to group blocks, and in fact, we will work with the larger class of alge-

Typeset by AMS-TEX 2 MARY SCHAPS AND EVELYNE ZAKAY-ILLOUZ

*Work done for a doctoral dissertation at Bar-Ilan University, and partially supported by the Bar-Ilan Research Authority.

bras called Brauer tree algebras, over fields of arbitrary characteristic. However, the main application is to modular group theory. The first paper corresponded to Chapters III and IV of the thesis, classifying two-restricted partial tilting complexes. The second paper, corresponding to Chapter V, demonstrated a one-to-one correspondence between Brauer trees with an additional structure, called a pointing, and equivalence classes of two-restricted tilting complexes modulo cyclic rotation of the Brauer star. This third paper, Chapter VI of the thesis, will give a “natural” deformation based on tilting theory.

Definition. Let e and m be natural numbers, with e > 1. Let K be any field containing a primitive e-th root of unity ξ. Let n̂ = em + 1. Let the cyclic group C_{e} = ⟨d⟩ act on the truncated polynomial ring A = K[x]/xn̂, d : x → ξx. The Brauer star algebra of type (e, m) is the skew group algebra

b = A[C_{e}]

in which d and x commute via the relation

d^{−1}xd = ξx.

The motivation for studying these algebras in depth lies in modular group representation theory, and we will not repeat that here, but will take the importance of these algebras for granted.

The algebra b has e distinct simple modules, corresponding to the idempotents

f_{i} = ∑_{j=0}^{e−1} ξ^{−ij}d^{j}, i = 1, …, e.

The corresponding indecomposable projective left modules will be denoted by

P_{i} = bf_{i}, i = 1, …, e.

HOMOGENEOUS DEFORMATIONS OF BRAUER TREE ALGEBRAS 3

We let {x^{s}f_{i}}_{s=0}^{em} be a basis for P_{i}, and define the following maps

ε : P_{i} → P_{i}  ε(f_{i}) = x^{e}f_{i}
h̃_{ij} : P_{i} → P_{j} h̃_{ij}(f_{i}) = x^{k}f_{i}, k ≡ j − i (mod e) 0 ≤ k < e.

For i ≠ j, we will denote h̃_{ij} by h_{ij}, and for i = j by id_{i} . The homomorphism ε^{m} : P_{i} → P_{i} will be called the socle map, for the obvious reason that it maps the top of P_{i} into its socle ⟨x^{em}f_{i}⟩. Definition. The homomorphism ε^{ℓ}h̃_{ij} : P_{i} → P_{j} will be called normal homogeneous of degree eℓ + k, where k ≡ j − i (mod e), 0 ≤ k < e.

A homomorphism of complexes will be called normal homogeneous if the map in each degree is normal homogeneous. Definition. A sequence of distinct indices, i_{1} → ⋯ → i_{r_{1}} is called short if h_{i_{r−1}i_{r}} ∘ ⋯ ∘ h_{i_{2}i_{3}} ∘ h_{i_{1}i_{2}} = h_{i_{1}i_{r}}. It is called long if h_{i_{r−1}i_{ir}} ∘ ⋯ ∘ h_{i_{1}i_{2}} = ε^{j}h_{i_{1}i_{r}}, j > 0.

We now consider complexes Q^{·} of projective modules. The notation Q^{·}[n] will indicate that the complex has been shifted n places to the left. Let R be any finite dimensional k-algebra. Definition. A complex Q^{·} of R-modules is called a tilting complex if

(1) Hom_{p}(Q^{·}, Q^{·}[n]) = 0 for n ≠ 0 in the homotopy category.

(2) For any indecomposable projective P, define the stalk complex to be the

complex

P^{·} : 0 → P → 0.

Then every such P^{·} is in the truncated category generated by the direct summands of Q^{·}.

Q^{·} satisfying (1) and (2) is called a tilting complex for R. If it satisfies (1), it is called a partial tilting complex.

4 MARY SCHAPS AND EVELYNE ZAKAY-ILLOUZ

Definition. A partial tilting complex Q^{·} for the Brauer star algebra B is called two-restricted if it is a direct sum of shifts of the indecomposable complexes

S_{i} : 0 → P_{i} → 0
T_{ij} : 0 → P_{i} → P_{j} → 0,

where the first non-zero term is in degree zero. The complexes S_{i}[n] and T_{ij}[n′] will be called elementary.

The main result of [9, III-IV] was a complete description of the set of tworestricted partial tilting complexes denoted by P T C_{2} and their endomorphisms. We quote the results in full. Theorem 1. If Q^{·} is in P T C_{2}, then:

(a) All elementary components of Q^{·} of the form S_{r}[n] must have their non-zero term in the same degree, which we will assume to be zero.

(b) If two elementary components of Q^{·} have a common term P_{r}, it must occur with the same degree in each.

(c) If T_{rs}[n] and S_{k} occur together in Q^{·} then r → k → s is long.

(d) If T_{rs}[n] and T_{tu}[n′] occur in Q^{·}, with r, s, t, u all distinct, then one of the following sequences must be short.

(1) r → s → t → u

(2) r → t → u → s

(3) r → s → u → t. The second theorem summarizes the normal homogeneous homomorphisms be-

tween the indecomposables. Theorem 2. Between two elementary complexes in P T C_{2}, there are chain maps not homotopic to zero in the following cases:

(i) From S_{i} to S_{j} there are m + δ_{ij} normal homogeneous maps. If we let

HOMOGENEOUS DEFORMATIONS OF BRAUER TREE ALGEBRAS 5

{j − i}_{e} represent the residue of j − i modulo e, then the maps have degree {j − i}_{e} + ke, 0 ≤ k ≤ m + δ_{ij}.

(ii) From T_{ij} to itself there are two normal homogeneous maps, the identity and

the socle map. (iii) Between two elementary components with one common index, there is a

normal homogeneous map of degree zero in one direction and of degree em in the opposite direction.

Finally, regarding distinct elementary complexes with a unique common index as in Theorem 2 (iii), we have more explicit information.

Theorem 3. (Local order theorem)

(1) If {S_{i_{1}}, …, S_{i_{r}}} are a set of S_{i} in Q· ∈ P T C_{2}, such that i_{1} → i_{2} → ⋯ → i_{r} is short, then we have homomorphisms h_{i_{k}i_{k+1}} : S_{i_{k}} → S_{i_{k+1}} such that

h_{i_{k−1}i_{k}} ∘ ⋯ ∘ h_{i_{k+1}i_{k+2}} ∘ h_{i_{k}i_{k+1}} = ε_{i_{k}} for k = 1, …, r.

(2) If {T_{s_{i}r}[1]}_{i=1}^{k} ∪ {T_{rt_{j}}}_{j=1}^{ℓ} are a set of irreducibles in Q^{·}, we can arrange the indices so that

s_{1} → ⋯ → s_{k} → r → t_{1} → ⋯ → t_{ℓ}

is short. All of the maps

T_{rt_{1}} → T_{rt_{2}} → ⋯ → T_{rt_{ℓ}} → T_{s_{1}r}[1] → ⋯ → T_{s_{k}r}[1]

will be identity on P_{r}, whereas T_{s_{k}r}[1] → T_{rt_{1}} will be ε^{m} on P_{r}. If S_{r} is in Q^{·}, then S_{r} can be inserted between T_{rt_{ℓ}} and T_{s_{1}r}[1], with identity maps in P_{r}.

6 MARY SCHAPS AND EVELYNE ZAKAY-ILLOUZ

§2. Deformation theory.

Let B_{0} be an algebra over an algebraically closed field k, and assume that B_{0} is basic, i.e., that B_{0}/ Rad B_{0} →^{∼} k^{r}. Let e_{1}, …, e_{r} be a complete set of orthogonal primitive idempotents. Let J_{0} = Rad(B_{0}). Let Q be the directed graph with r vertices corresponding to e_{1}, …, e_{r}, and m_{ij} = dim e_{i}(J_{0}/J_{0}^{2})e_{j} arrows from e_{i} to e_{j}. (This is the dual directed graph to the quiver.) B_{0} has a basis consisting of the paths from the path algebra k[Q] of Q, and the elements of this basis are said to respect the idempotent set e_{1}, …, e_{r}, because the basis contains the idempotent set, and every element in the basis lies in one of the components of the Peirce decomposition

B_{0} = ⨁_{i,j} e_{i}B_{0}e_{j}.

Let B be a flat deformation of B_{0} over k[t]. Since B is flat, every closed fiber

B_{a} = B ⊗_{k} k[t]/(t − a)

has the same underlying vector space V of dimension n. Let U be a vector bundle of rank n over Spec(k[t]). Definition. A deformation basis x̃_{1}, …, x̃_{n} is a set of sections x̃_{i} : Spec(k[t]) → U such that

(1) x̃_{i}(0) = x_{i}

(2) ẽ_{1}(a), …, ẽ_{r}(a) is a complete orthogonal set of idempotents in B_{a} for a ∈ k. (They will not generally still be primitive.)

(3) Each x̃_{ℓ}(a) lies in some ẽ_{i}(a)B_{a}ẽ_{j}(a).

(4) Each set {x̃_{i}(a)} is a basis for B_{a}.

In [5], it was proven that such a basis always exists if one is permitted to change the parameter space, but in practice it frequently exists over k[t]. For an interpretation of deformations with generic fiber separable as maximal orders, see [2].

HOMOGENEOUS DEFORMATIONS OF BRAUER TREE ALGEBRAS 7

Definition. If we can assign to each x̃_{i} a degree d_{i} ≥ 0 such that for every a ≠ 0,

x̃_{i}(1) → a^{−d_{i}} x̃_{i}(a)

is an isomorphism of k algebras, then {x̃_{i}} is called a homogeneous deformation basis and B is called a homogeneous jump deformation. Remark. It is easy to check that the degree of any idempotent is zero.

§3. Construction of a homogeneous deformation.

We can now prove our theorem. Theorem. Let T be a two-restricted tilting complex, and let B_{0} be the corresponding Brauer tree algebra B_{0} = End_{b}(T). There is a basis of B_{0} consisting of homogeneous endomorphisms mapping one elementary complex to another, and a lifting of this basis to a deformation basis of a homogeneous jump deformation, such that the degree of each basis element is the degree of the original map. Proof. In the local order theorem and Theorem 2, we saw that the endomorphisms were all generated by the endomorphisms mapping one elementary complex to another, between adjacent edges in counterclockwise order. At a non-exceptional vertex, these homomorphisms are all of degre zero, except for one that is of degree em. At the exceptional vertex, all endomorphisms are generated by S_{i} → S_{j} for j immediately after S_{i}, these being homogeneous of degree {j − i}_{e}. Every other homogeneous endomorphism of T is a composition of these, with degree equal to the sums of the degrees of the factors.

Let us establish some notation. Let u_{0} be the exceptional vertex, and let u be any ordinary, nonterminal vertex. Let v_{u} be the valence of u, i.e., the number of edges at u, and let v_{0} be the valence of u_{0}, Let K_{u} be the set of v_{u} edges at u, and assume that they are numbered counterclockwise beginning at the point {r_{1}, …, r_{v_{u}}}. Define

x_{r_{i}r_{i+1}}^{u}, i = 1, …, r_{v_{u}}

8 MARY SCHAPS AND EVELYNE ZAKAY-ILLOUZ

be the homomorphism from the complex of r_{i} to the complex of r_{i+1} defined in the local order theorem, where we understand that r_{v_{u}+1} = r_{1}. As was proven there, all the x_{r_{i}r_{i+1}}^{u} have degrees zero for i = 1, …, r_{v_{u}−1}, and x_{r_{vu} r_{1}} has degree em. For the exceptional vertex we similarly have {r_{1}, …, r_{v_{0}}}, and the degree of x_{r_{i}r_{i+1}}^{u_{0}} is {r_{i+1} − r_{i}}_{e}. In addition, for each elementary complex R of degree zero corresponding to an edge r, we have a homogeneous endomorphism F_{r}, which is the identity on R and zero elsewhere. If u_{0} is a terminal vertex and r is the unique edge on u_{0}, let x_{rr}^{u_{0}} be ε : S_{r} → S_{r}. The idempotents F_{r} and the arrows x_{rr′}^{u} determine the quiver of B_{0}, and generate the algebra B_{0}. Every homogeneous endomorphism of T is a composition of the arrows in some path in this quiver.

As was shown in [ES], in order to deform the algebra it suffices to deform the relations on the quiver in such a way that the dimension of the quotient remains fixed. In order to get a homogeneous deformation, we will want to use functions homogeneous in the arrows and in t. In the context of the Donald-Flanigan problem, we want to deform B_{0} to the separable k-algebra

B_{1} = ( ⨁_{u≠u_{0}} M_{v_{u}}(k)) ⊕ (⨁_{i=1}^{m} M_{v_{0}}(k)) .

More specifically, we will show that there is a k[t]-algebra B such that the closed fiber at t = 0 is isomorphic to B_{0} and the closed fiber B_{a} at t = a, a ≠ 0 is isomorphic to B_{1}. We will do this by finding elements F̃_{1}, …, F̃_{e} and {x_{r_{i}r_{i+1}}^{u}} in B_{1} ⊗_{k} k[t] such that when t = 1 they generate B_{1}, such that they are defined by multiplying an element of B_{1} by an appropriate power of t, and such that the relations between them reduce modulo(t) to the relations of B_{0}.

We begin by describing the relations of B_{0}. For any vertex u and any edge r_{i} at u, define

z_{r_{i}}^{u} = x_{r_{i−1}r_{i}}^{u} ∘ ⋯ ∘ x_{r_{i}r_{i+1}}^{u}.

If u is an ordinary vertex, then z_{r}^{u} is the endomorphism of the irreducible complex

HOMOGENEOUS DEFORMATIONS OF BRAUER TREE ALGEBRAS 9

R into itself which maps the projective P_{u} to its socle and, if R is of length 2, maps the other projective to zero. It has degree em. If u_{0} is the exceptional vertex, then z_{r_{i}}^{u_{0}} is ε_{r_{i}}, mapping S_{r_{i}} to S_{r_{i}}. It has degre e.

We have a fundamental relation that x_{r_{i}r_{i+1}}^{u} ∘ x_{r′_{j}r′_{j+1}}^{v} = 0 if u ≠ v. In addition, each edge of the Brauer graph determines one relation. Suppose r is an edge whose vertex in the direction of u_{0} is u and whose other vertex is w, where w may be a member of the set E of terminal vertices.

There are four kinds of relations depending on whether or not ω = u_{0} and whether or not ω ∈ E. Assume that r = r_{i} in K_{u} and if w ∉ E, then r = r′_{j} in K_{w}.

1) u ≠ u_{0}, w ∉ E z_{r}^{u} + z_{r}^{w} = 0

2) u ≠ u_{0}, w ∈ E x_{r_{i}r_{i+1}}^{u}z_{r_{i}}^{u} = 0

3) u = u_{0}, w ∉ E (z_{r_{i}}^{u_{0}})^{m} = z_{r′_{j}}^{w}

4) u = u_{0}, w ∈ E x_{r_{i}r_{i+1}}^{u_{0}}(z_{r_{i}}^{u_{0}})^{m} = 0

Our lifting of the generating set will be such that each fundamental relation will be preserved, and each of the deformations of type 1) – 4) will have homogeneous deformation.

In order to define our deformation B, let us give a basis for B_{1} in terms of matrix units. For each ordinary u, u ∉ E, the basis of M_{v_{u}}(k) will be {E_{r_{i}r_{j}}^{u}}_{r_{i},r_{j}∈K_{u}}. At the exceptional vertex, for each λ ∈ {1, …, m}, we have matrix units {E_{r_{i}r_{j}}^{λ}}_{r_{i},r_{j}∈k_{u_{0}}} . Let r be an edge with vertexes u, w, where u is closer to u_{0}, and suppose that r = r_{i} in K_{u} and r = r′_{j} in K_{v}. Let ξ_{1}, …, ξ_{m} be (em)-th roots of unity in k such that ξ_{1}^{e}, …, ξ_{m}^{e} are all distinct. We define the following elements of B_{1} ⊗ k[t] :

For u ≠ u_{0} \overline{F}_{r} = E_{rr}^{u} + E_{rr}^{w}

For u ≠ u_{0} \overline{x}_{r_{i}r_{i+1}}^{u} = \{ \begin{matrix}E_{r_{i}r_{i+1}}^{u} & r_{i} ≠ r_{v_{u}} \\ t^{em}E_{r_{vu} r_{1}} & r_{i} = r_{v_{u}} \end{matrix}

For u = u_{0} \overline{F}_{r} = ( ∑_{λ=1}^{m} E_{rr}^{λ}) + E_{rr}^{u}

For u = u_{0} \overline{x}_{r_{i}r_{i+1}}^{u_{0}} = t^{\{r_{i+1}−r_{i}\}_{e}} ∑_{λ=1}^{m} ξ_{λ}^{\{r_{i+1}−r_{i}\}_{e}}E_{r_{i}r_{i+1}}^{λ}

We first note that the degree of each element with respect to t is indeed the

10 MARY SCHAPS AND EVELYNE ZAKAY-ILLOUZ

degree of the corresponding element of B_{0}. Second, we note that it is not difficult to show that when we substitute a ≠ 0 for t we get a basis of B. We first separate the idempotents, by noting that if u ≠ u_{0}

\overline{z}_{r}^{u} = t^{em}E_{rr}^{u}, \overline{z}_{r}^{w} = t^{em}E_{rr}^{w}.

This also give us a deformation of the relations 1), namely

(\overline{1})  \overline{z}_{r}^{u} + \overline{z}_{r}^{w} = t^{em}\overline{F}_{r}

The idempotent E_{rr}^{w} of a terminal vertex is t^{−em}(t^{em}\overline{F}_{r} − \overline{z}_{r}^{u}), and since E_{r_{i}r_{i+1}}^{u} · E_{rr}^{w} = 0, the corresponding relation is

(\overline{2})  \overline{x}_{r_{i}r_{i+1}}^{u} \overline{z}_{r_{i}}^{u} = t^{em}\overline{x}_{r_{i}r_{i+1}}^{u}

If u = u_{0}, we separate the various E_{rr}^{λ} by noting that {(z_{rr}^{u_{0}})^{ℓ}}_{ℓ=0}^{m−1} is a set of vectors whose coefficients in E_{rr}^{λ} are Van der Monde determinants for {ξ_{1}^{e}, …, ξ_{m}^{e}}, and thus we can isolate each of the E_{rr}^{λ}, after which it is simple to get the various E_{r_{i}r_{j}}^{λ} by taking compositions of the \overline{x}_{r_{j}r_{j′}}^{u_{0}} . The corresponding relations are

(\overline{3})  u = u_{0}, w ∉ E, (\overline{z}_{r_{i}}^{u_{0}})^{m} = \overline{z}_{r_{j′}}^{w}
(\overline{4})  u = u_{0}, w ∈ E, \overline{x}_{r_{i}r_{i+1}}(\overline{z}_{r_{i}}^{u_{0}})^{m} = t^{em}\overline{x}_{r_{i}r_{i+1}}^{u_{0}}

By inspection we see that these relations reduce to the relations of B_{0} when t = 0.

References

1. Mejer C. and Schaps M., Separable deformations of blocks with abelian defect group and of derived equivalent global blocks, Canadian Mathematical Society Conference Proccedings 18 (1996), 505-518.

2. Michler G., Maximal orders and deformation of modular group algebras, Abelian Groups and Noncommutative Rings (L. Fuchs et al, eds.), Contemporary Math., vol. 130, Amer. Math. Soc., Providence, 1992, pp. 275-288.

3. Rickard J., Derived categories and stable equivalence, J. of Pure and Applied Algebra 61 (1989), 303-317.

4. Rickard J., Lifting theorems for tilting complexes, J. of Algebra 142 (1991), 383-393.

5. Schaps M., Deformations of finite dimensional algebras and their idempotents, Trans. AMS (1988), 843-856.

6. Schaps M., A modular version of Maschke’s theorem for groups with cyclic p-Sylow subgroup, J. of Algebra 163 (1994), 623-635.

7. M. Schaps and Zakay-Illouz E., Combinatorial partial tilting complexes for the Brauer star algebras, Proc. of the Conference on Representations of Algebras - Sao Paulo (to appear).

8. M. Schaps and Zakay-Illouz E., Pointed Brauer trees, preprint.

9. Zakay-Illouz E., The Green Correspondence between Separable Deformations, Ph.D. dissertation, Bar-Ilan University, 1999.

Department of Mathematics and Computer Science, Bar-Ilan University, 52900 Ramat-Gan, Israel

Department of Mathematics, Jordan Valley College, Jordan Valley, Israel