Homotopy, homology, and GL_{2}.

Vanessa Miemietz and Will Turner

June 23, 2008

Abstract We define weak 2-categories of finite dimensional algebras with bimodules, along with collections of operators 𝕆_{(c,x)} on these 2-categories. We prove that special examples 𝕆_{p} of these operators control all homological aspects of the rational representation theory of the algebraic group GL_{2}, over a field of positive characteristic. We prove that when x is a Rickard tilting complex, the operators 𝕆_{(c,x)} honour derived equivalences, in a differential graded setting. We give a number of representation theoretic corollaries, such as the existence of tight ℤ_{+}-gradings on Schur algebras S(2, r), and the existence of braid group actions on the derived categories of blocks of these Schur algebras.

Introduction.

Group theory has for a long time looked beyond itself for sources of intuition and understanding. Given that any topological space has a fundamental group, it is not surprising that one of those sources has been the theory of topological spaces. Homology groups, originally defined as topological invariants, have developed into the techniques of homological algebra, which have proved a powerful tool [4], and an intriguing source of questions in group representation theory [3].

We might expect more subtle topological invariants, such as homotopy groups, or invariants appearing in quantum field theory, to give rise to mathematical techniques, which also find application in group theory. One broad setting for homotopical algebra adopts a number of guises, which together form the subject of n-categories [5]. This setting also appears to be well adapted for the study of topological quantum field theories [1]. The existence of gauge theories suggests that n-categories should indeed play a role in group representation theory, at least for small n [2], [6], [24].

In this paper, we present an example in which the formalism of 2-category theory finds application in modular representation theory. We define weak 2categories, and collections of combinatorial operators on these 2-categories. Special examples of these operators control all homological aspects of the rational representation theory of GL_{2}, over a field of positive characteristic.

To be more precise, let F be a field of characteristic p > 0. We give the collection T of F -algebras with a bimodule the structure of a weak 2-category. To every object (c, x) of such a 2-category which is positively graded, we associate a 2-functor 𝕆_{(c,x)}. For a certain graded algebra c_{p}, of finite representation type, and a certain c_{p}-c_{p}-bimodule x_{p}, we thus obtain an operator 𝕆_{p}. Applying the operator 𝕆_{p}^{n} to the one-dimensional algebra F, with its trivial bimodule F, we obtain an algebra, denoted E_{n}, with a bimodule. We prove that every block of rational GL_{2}(F)-modules is equivalent to the category of modules over an inverse limit lim_{n} E_{n} of these algebras.

The operators 𝕆_{(c,x)} are obtained by twisting a tensor product with a bimodule. We perform a careful analysis of the effect of such twisted tensor products, in a differential graded setting. For example, we prove if x is a Rickard tilting complex, then the operator 𝕆_{(c,x)} respects derived equivalences.

Our theory links the combinatorially defined operator 𝕆_{p} with the representation theory of GL_{2}(F), and has a number of corollaries. For example, the Schur algebra S(2), which controls the category of polynomial representations of GL_{2}(F), has a tight ℤ_{+}-grading. This follows since the operator 𝕆_{p} is tightly ℤ_{+}-graded. Furthermore, derived categories of certain blocks of the Schur algebra S(2, r) admit an action of the braid group on p braids. This follows from the existence of a braid group action on the derived category of c_{p}, established by Khovanov-Seidel and Rouquier-Zimmermann, along with the analysis of a lift of the operator 𝕆_{p} to a differential graded setting.

We actually overturn a number of different algebras isomorphic to E_{n}. One such, denoted A_{n} in the paper, is a quiver algebra, modulo quadratic relations; a second, denoted B_{n}, is the basic algebra of a Schur algebra block; a third, C_{n}, is obtained via the iteration of a certain trivial extension operator; a fourth, D_{n}, is a quotient of the basic algebra of a group algebra of finite special linear group. We thus obtain a detailed understanding of the representation theory of all of these algebras.

The first section of this paper is combinatorial, and is concerned with establishing the isomorphism of various algebras, applying work of Koshita, and Erdmann-Henke. In the second section, we check out the 2-category theory involved in our constructions. In the third, we prove that the operators 𝕆_{(c,x)} respect derived equivalence in a differential graded setting. In the fourth, we study the special operators 𝕆_{p}. In the fifth, we draw the various strands of argument together, and present applications to the representation theory of GL_{2}(F).

1. Four algebras.

Let F be an algebraically closed field of characteristic p > 0.

Let n ≥ 0 be an integer. Suppose p ≥ 3.

We define four algebras, A_{n}, B_{n}, C_{n}, and D_{n}, eventually aiming to show they are all isomorphic:

1. We define an algebra A_{n} in terms of quivers and relations. Let Q′_{n} be the quiver whose set of vertices are all tuples (a_{0}, a_{1}, …, a_{n}) ∈ {0, …, p − 1}^{n+1} in which, for a tuple a = (a_{0}, a_{1}, …, a_{n}) and 1 ≤ i ≤ n, there is an arrow f_{i,±1}^{a} from a to (a_{0}, a_{1}, …, a_{i−2}, p − 2 − a_{i−1}, a_{i} ± 1, …, a_{n}) whenever a_{i−1} is not p − 1 and a_{i} ± 1 ∈ {0, …, p − 1}. Our desired quiver Q_{n} is the connected component of Q′_{n} containing (0, 0, …, 0). In order to avoid unnecessary supercripts, we will write f_{i,±1} := ∑ f_{i,±1}^{a} where the sum is over all vertices a such that f_{i,±1}^{a} is an arrow in Q_{n}. Then f_{i,±1}^{a} = f_{i,±1}a lies in the path algebra of Q_{n}. Our first algebra A_{n} is the path algebra of Q_{n} modulo the ideal I generated by the following relations:

(1) f_{i,ε_{1}}f_{j,ε_{2}}a − f_{j,ε_{2}}f_{i,ε_{1}}a if i ≠ j − 1, j, j + 1

(2) f_{i,ε}f_{i,ε}a for any i, ε and a

(3) f_{i,ε}f_{i,−ε}a if a_{i} = p − 1

(4) f_{i,1}f_{i,−1}a − f_{i,−1}f_{i,1}a if a_{i} > 0

(5) f_{i,ε_{1}}f_{i+1,ε_{2}}a − f_{i+1,ε_{2}}f_{i,−ε_{1}}a if a_{i} + ε_{1}, p − 2 − a_{i} − ε_{1} are both in {0, …, p − 1}, and a_{i−1}, a_{i} ≠ p − 1.

2. Let S(2) = ⊕_{r≥0}S(2, r) denote the Schur algebra associated to GL_{2}, namely the graded dual of the coalgebra of regular functions on the algebra of 2 × 2 matrices over F [11]. Let B_{n} be the basic algebra of a block of a Schur algebra S(2, r) with p^{n} simple modules, e.g. the principal block of S(2, p^{n+1} − 1).

3. We now recall the construction of an algebra C_{p}(A) where A is an associative algebra, and T is a self-dual bimodule over A, yielding another such algebra with a natural ℤ_{+}-grading [19]. In order to construct C_{p}(A) from A and T, we first define an algebra B as the infinite matrix algebra

( \begin{matrix}... & _{i−2}T_{i−1} & 0 & ⋯ \\ 0 & A_{i−1} & _{i−1}T_{i} & 0 & ⋯ \\ ⋯ & 0 & A_{i} & _{i}T_{i+1} & 0 & ⋯ \\ & ⋯ & 0 & A_{i+1} & _{i+1}T_{i+2} & 0 \\ & & & & A_{i+2} & ... \\ . & . & . \end{matrix} )

where all the A_{i} and _{i−1}T_{i} are isomorphic copies of A and T respectively. Denote by C(A) the trivial extension of B by the (B, B)-bimodule

B^{(∗)} = ⨁_{i∈ℤ} Hom_{F} (B1_{A_{i}}, F).

Then C_{p}(A) is the subquotient of C(A) given by

C_{p}(A) = η_{1}^{p+1}C(A)η_{1}^{p+1}/η_{1}^{p+1}C(A)1_{A_{p+1}}C(A)η_{1}^{p+1},

where η_{1}^{p+1} = ∑_{1≤i≤p+1} 1_{A_{i}}. The bimodule

X_{p}(A) = η_{1}^{p}C(A)η_{2}^{p+1}

admits the natural structure of a self-dual C_{p}(A)-C_{p}(A)-bimodule. The operator C_{p} : (A, T) ↦ (C_{p}(A), X_{p}(A)) acts on the collection of algebras with a self dual bimodule. If we apply the n-fold composition operator C_{p}^{n} to the pair (F, F), we obtain an algebra with a self-dual bimodule. We call the algebra obtained in this way C_{n}.

4. Let { \atop F SL(2, p^{m})} be the basic algebra of the group algebra of SL(2, p^{m}) over an algebraically closed field of characteristic p. This algebra has a presentation in terms of quiver and relations by Koshita ([16] ,[17], see also [20] [21]). We define our fourth algebra D_{n} to be a quotient of { \atop F SL(2, p^{n+2})}. Let e be the sum over all idempotents in Koshita’s quiver which correspond to vertices not in the connected component of (0, …, 0). Let f be the sum of idempotents corresponding to the vertices in the connected component of (0, …, 0), where the last entry in the n + 2-tuple labelling the vertex is nonzero. Then we define D_{n} :=

{ \atop F SL(2, p^{n+2})}/{ \atop F SL(2, p^{n+2})}(e + f){ \atop F SL(2, p^{n+2})}.

The main result of this section is

Theorem 1. All four algebras A_{n}, B_{n}, C_{n}, and D_{n} are isomorphic.

We prove the theorem via a sequence of lemmas and propositions.

Remark 2 The algebra A_{1} is the well-known quasihereditary algebra

c_{p} = \xymatrix{F (•^{1} \ar[r]^{ξ_{1}} & •^{2} \ar[l]^{η_{2}} \ar[r]^{ξ_{1}} & •^{3} \ar[l]^{η_{2}} & ⋯ & •^{p−1} \ar[r]^{ξ_{p−1}} & •^{p})/I \ar[l]^{η_{p−1}}} ,

where I = (ξ_{i+1}ξ_{i}, η_{i}η_{i+1}, ξ_{i}η_{i} −η_{i+1}ξ_{i+1}, ξ_{p−1}η_{p−1}), for 1 ≤ i ≤ p−2, and in the tuple notation vertex 1 corresponds to the tuple (a_{0}, a_{1}) = (0, 0), whilst vertex 2 corresponds to (a_{0}, a_{1}) = (p − 2, 1), vertex 3 corresponds to (a_{0}, a_{1}) = (0, 2), etc.

We can inductively define the quiver Q_{n} of A_{n} as the quiver

\xymatrix{ & Q_{n−2} \ar[d]^{h_{n}} \ar[r]^{g_{n−1}} & Q_{n−2} \ar[l]^{g_{n−1}} \ar[d]^{h_{n}} & ⋯ & Q_{n−2} \ar[d]^{h_{n}} \ar[r]^{g_{n−1}} & Q_{n−2} \ar[l]^{g_{n−1}} \ar[d]^{h_{n}} \ar[r]^{g_{n−1}} & Q_{n−2} \ar[l]^{g_{n−1}} \\ Q_{n−2} \ar[r]^{g_{n−1}} & Q_{n−2} \ar[l]^{g_{n−1}} \ar[u]^{h_{n}} \ar[d]^{h_{n}} \ar[r]^{g_{n−1}} & Q_{n−2} \ar[l]^{g_{n−1}} \ar[u]^{h_{n}} \ar[d]^{h_{n}} & ⋯ & Q_{n−2} \ar[u]^{h_{n,}} \ar[d]^{h_{n}} \ar[r]^{g_{n−1}} & Q_{n−2} \ar[l]^{g_{n−1}} \ar[u]^{h_{n}} \ar[d]^{h_{n}} \\  & Q_{n−2} \ar[u]^{h_{n}} \ar[r]^{g_{n−1}} & Q_{n−2} \ar[l]^{g_{n−1}} \ar[u]^{h_{n}} & ⋯ & Q_{n−2} \ar[u]^{h_{n}} \ar[r]^{g_{n−1}} & Q_{n−2} \ar[l]^{g_{n−1}} \ar[u]^{h_{n}} \ar[r]^{g_{n−1}} & Q_{n−2} \ar[l]^{g_{n−1}} \\  & ⋮ & ⋮ &  & ⋮ & ⋮ \\ Q_{n−2} \ar[r]^{g_{n−1}} & Q_{n−2} \ar[l]^{g_{n−1}} \ar[d]^{h_{n}} \ar[r]^{g_{n−1}} & Q_{n−2} \ar[l]^{g_{n−1}} \ar[d]^{h_{n}} & ⋯ & Q_{n−2} \ar[d]^{h_{n}} \ar[r]^{g_{n−1}} & Q_{n−2} \ar[l]^{g_{n−1}} \ar[d]^{h_{n}} \\  & Q_{n−2} \ar[u]^{h_{n}} \ar[r]^{g_{n−1}} & Q_{n−2} \ar[l]^{g_{n−1}} \ar[u]^{h_{n}} & ⋯ & Q_{n−2} \ar[u]^{g_{n}} \ar[r]^{g_{n−1}} & Q_{n−2} \ar[l]^{g_{n−1}} \ar[u]^{h_{n}} \ar[r]^{g_{n−1}} & Q_{n−2} \ar[l]^{g_{n−1}}}

where the above diagram has p rows, each containing p copies of Q_{n−2}; where h_{n} denotes a collection of arrows pointing from the vertices of one copy of Q_{n−2} to the vertices of a second copy of Q_{n−2}, so that the correspondence defined by these arrows is the identity map on vertices of Q_{n−2}; where each row of the diagram is inductively identified with a copy Q_{n−1}; and where g_{n} is identified with the collection of arrows pointing from one row to the next, therefore a union of p − 1 copies of h_{n−1}.

The sum f_{n,1} (resp. f_{n,−1}) of arrows in Q_{n} is equal to the sum of all arrows which point downwards (resp. upwards) in the diagram, whilst the sum f_{n−1,1} (resp. f_{n−1,−1}) is equal to the sum of arrows which point rightwards (resp. leftwards) in the diagram in odd rows, and leftwards (resp. rightwards) in even rows.

Relations R_{n} for the algebra A_{n} can also be defined inductively, as follows:

• Each copy of Q_{n−2} in the diagram is subject to relations R_{n−2}, defining a split subalgebra isomorphic to A_{n−2}.

• Each row Q_{n−1} of the diagram is subject to relations R_{n−1}, defining a split subalgebra isomorphic to A_{n−1}.

• Each column of the diagram is subject to relations R_{1}, defining a split subalgebra isomorphic to A_{n−2} ⊗ c_{p}, where the i^{th} vertex of c_{p} corresponds to a copy of Q_{n−2} in the i^{th} row.

• All squares commute.

• H_{n}x = xH_{n}, for every arrow x in Q_{n−2}, and every collection h_{n} in the diagram; where H_{n} denotes the sum of arrows in h_{n}; and where x is understood to lie in the subquiver at the source of h_{n}, on the left hand side of this relation, whilst x is understood to lie in the subquiver at the target of h_{n}, on the right hand side of this relation.

Lemma 3. A_{n} ≅ D_{n}.

Proof. This is true since the quiver and relations defining A_{n} are exactly obtained from Koshita’s quiver and relations for { \atop F SL(2, p^{n+2})}, taking first the connected component of (0, …, 0), and then factoring out all vertices of the form (a_{0}, a_{1}, …, a_{n}, a_{n+1}) with a_{n+1} > 0, before identifying the vertex (a_{0}, a_{1}, …, a_{n}, 0) with (a_{0}, a_{1}, …, a_{n}).

Lemma 4. There exists an epimorphism ϕ : D_{n} ↠ B_{n}.

Proof. We identify an n-tuple a = (a_{0}, a_{1}, …, a_{n}) ∈ {0, 1, …, p − 1}^{n} indexing irreducible modules for SL(2, p^{n}) as in Koshita’s notation [17] with the natural number λ_{a} := ∑_{0≤n−1} a_{i}p^{i} indexing a simple module for S(2, r). We first consider the first two algebras. Theorem 3.2 of an article of M. DeVisscher [8] states that there is an epimorphism

π : { \atop F SL(2, p^{n+2})} ↠ { \atop S(2, p^{n+2} − 1)} ⊕ { \atop S(2, p^{n+2} − 2)}

where for an algebra A, we denote its basic algebra by Ã. This epimorphism is defined via epimorphisms of projective indecomposable modules. Hence a set of primitive orthogonal idempotents \{e_{i}\} ⊂ { \atop F SL(2, p^{n+2})} maps to a set of primitive orthogonal idempotents \{ẽ_{i}\} ⊂ { \atop S(2, p^{n+2} − 1)} ⊕ { \atop S(2, p^{n+2} − 2)}. Only considering the quotient B_{n+1} of the Schur algebras, we get a surjection

π̃ : Λ_{n+1} := { \atop F SL(2, p^{n+2})}/{ \atop F SL(2, p^{n+2})}e{ \atop F SL(2, p^{n+2})} ↠ B_{n+1}

where e is the sum over all idempotents e_{i} such that ẽ_{i} ∉ B_{n+1}. In Koshita’s presentation of { \atop F SL(2, p^{n+2})} in terms of quiver and relations, the quotient

{ \atop F SL(2, p^{n+2})}/{ \atop F SL(2, p^{n+2})}e{ \atop F SL(2, p^{n+2})} is obtained by taking the connected

component of (0, …, 0). Now we have a surjection of B_{n+1} onto B_{n} given by factoring out B_{n+1}f̃B_{n+1} where f̃ = ∑_{ẽ_{i}∈B_{n},i≥p^{n+1}} ẽ_{i} = π(f). Since the image of Λ_{n+1}fΛ_{n+1} (where f = ∑_{e_{i}∈Λ_{n+1},i≥p^{n+1}} e_{i}) under π is contained in B_{n+1}f̃B_{n+1} we have an induced morphism between cokernels

ϕ : Λ_{n+1}/Λ_{n+1}fΛ_{n+1} ↠ B_{n+1}/B_{n+1}f̃B_{n+1} ≅ B_{n}.

Now, note that since (p^{n+2} ≥)i ≥ p^{n+1}, the coefficient of p^{n+1} in the p-adic expansion of i is nonzero, and the idempotents e and f in this theorem are defined exactly as in the definition of D_{n}. Therefore Λ_{n+1}/Λ_{n+1}fΛ_{n+1} ≅ D_{n} and ϕ is our desired epimorphism.

We claim that ϕ is an isomorphism. By Lemma 3 this is equivalent to showing that A_{n} ≅ B_{n}.

Proposition 5. A_{n} ≅ B_{n}.

Proof. We first note several things that are obvious from the relations for A_{n}.

• There exists a basis of A_{n} such that every path in Q_{n} is either equivalent to a element of this basis or zero in A_{n}. This is the case because relations in A_{n} are quadratic and either take the form a_{1}a_{2}, or a_{1}a_{2} − a′_{1}a′_{2}, for some arrows a_{1}, a_{2}, a′_{1}, a′_{2}.

• Any path of length r in the quiver Q_{n} defines defines an element of the path algebra which is equal to an element of the form f_{i_{1}ϵ_{1}} … f_{i_{r}ϵ_{r}}a, where a is the source vertex of the path. We will represent paths in this form whenever it is convenient to do so.

• If we fix c_{1}, … c_{r} ∈ [0, p − 1], and take the subalgebra generated by the vertices (a_{0}, …, a_{n−r}, c_{r}, …, c_{1}) and all arrows f_{i,ε} for i ≤ n − r between them, then there is a split embedding of A_{n−r} into A_{n} which takes (a_{0}, …, a_{n−r}) to (a_{0}, …, a_{n−r}, c_{r}, … c_{1}). We denote the image of this embedding by A_{n−r}^{c_{1},··· ,c_{r}}. The existence of a splitting follows because the relations are graded with respect to an n-coloring: i.e there is a ℤ_{+}^{n}-grading on A_{n} which descends from a ℤ_{+}^{n}-grading on the path algebra, giving f_{i,ε} degree (0, …, 0, 1, 0, …, 0) where 1 is in the ith position. We will use both notations P_{A_{n−r}^{c_{1},··· ,cr}} (a_{0}, …, a_{n−r}, c_{r}, … c_{1}) and P_{A_{n−r}^{c_{1},··· ,cr}} (a_{0}, …, a_{n−r}) for the projective indecomposable A_{n−r}^{c_{1},··· ,c_{r}}-module whose simple top is indexed by the vertex (a_{0}, …, a_{n−r}, c_{r}, … c_{1}). We denote the subquiver of Q_{n} which generates A_{n−r}^{c_{1},··· ,c_{r}} by Q_{A_{n−r}^{c_{1},··· ,cr}}.

• The projective indecomposable A_{n}-module corresponding to vertex a has simple composition factors which correspond to vertices whose last coordinate is either a_{n}, a_{n} + 1 or a_{n} − 1. Indeed, any path in Q_{n} from Q_{A_{n−1}^{an}} to Q_{A_{n−1}^{an±j}}, for j ≥ 2, contains at least j factors f_{n,±}, which by relations (1),(4) and (5) can be reduced to a path containing an expression f_{n,±}^{2}, which is zero by relation (2).

• Let c = a_{n}. Then P_{A_{n}}(a_{0}, …, a_{n−1}, c) is of the form

\begin{matrix}P \\ K^{+} K^{−} \\ L \end{matrix}  \begin{matrix}P \\ K^{+} \\ L \end{matrix}  \begin{matrix}P \\ K^{−} \end{matrix}  (1)

for 1 ≤ c ≤ p − 2, c = 0 and c = p − 1 respectively. Here the picture

\begin{matrix}P \\ K^{+} K^{−} \\ L \end{matrix} denotes a module filtration with top quotient P, with sub-

module L and with middle subquotient K^{+} ⊕ K^{−}. The other pictures should be interpreted similarly. The first two filtrations follow from relation (4) and the last one from relation (5), each applied in case i = n. The top quotient P has composition factors belonging to A_{n−1}^{c} -mod, and is isomorphic to the projective P_{A_{n−1}^{c}}(a)). The modules K^{±} have composition factors belonging to A_{n−1}^{c±1} -mod, whilst L again has composition factors belonging to A_{n−1}^{c} -mod.

The projective indecomposable B_{n}-module P_{B_{n}}(a) is a quotient of the projective indecomposable P_{A_{n}}(a), by Lemmas 3 and 4. We now recall some properties of B_{n}, from a previous article of ours [19], and from papers of Erdmann and Henke [12], [10].

• There is a quasi-hereditary quotient of B_{n},

B_{n−1}^{j} = ϵ_{j}B_{n}ϵ_{j}/ϵ_{j}B_{n}ϵ_{j+1}B_{n}ϵ_{j} ≅ B_{n−1},

where ϵ_{j} = ∑_{i>(j−1)p^{n}} ẽ_{i}, and the isomorphism to B_{n−1} takes an idempotent e_{i} to an idempotent e_{(j−1)p^{n}+i} A proof of this can be found in previous papers ([12], Theorem 5.1, [19], Proposition 29). By iteration, we obtain quotients B_{n−r}^{a_{n},a_{n−2},...,a_{n−r+1}} ≅ B_{n−r} of B_{n}. We remark that we do not yet know whether these quotients are split. As in the case of A_{n}, we use both notations P_{B_{n−r}^{an,··· ,a_{n−r+1}}} (a_{0}, …, a_{n}) and P_{B_{n−r}^{an,··· ,a_{n−r+1}}} (a_{0}, …, a_{n−r}) for

projective indecomposable B_{n−r}^{a_{n},··· ,a_{n−r+1}}-modules.

• The projective module P_{B_{n}}(a) has a filtration [19]

\begin{matrix}P_{B_{n−1}^{c}}(a) \\ T_{B_{n−1}^{c+1}}(ã) T_{B_{n−1}^{c−1}}(ã) \\ P_{B_{n−1}^{c}}(a)^{∗} \end{matrix}  \begin{matrix}P_{B_{n−1}^{c}}(a) \\ T_{B_{n−1}^{c+1}}(ã) \\ P_{B_{n−1}^{c}}(a)^{∗} \end{matrix}  \begin{matrix}P_{B_{n−1}^{c}}(a) \\ T_{B_{n−1}^{c−1}}(ã) \end{matrix}  (2)

for 1 ≤ c ≤ p − 2, c = 0 and c = p − 1 respectively. Here ã = (p − 1 − a_{i})_{(i=0,...,n−1)}. Indeed, the middle composition factors in these filtrations are given as the trace of all modules P_{B_{n}}(a_{0}, …, a_{n−1}, c ± 1) in P_{B_{n}}(a) ([19], Proposition 29), and isomorphic to the indecomposable tilting module T_{B_{n−1}^{c±1}}(ã) indexed by vertex ã = (p−1−a_{i})_{(i=0,...,n−1)} ([10], Proposition 25).

• T_{B_{n−1}^{c±1}}(ã) ≅ P_{B_{k}^{c±1,0,...,0}}(a_{0}, …, a_{k}), if k is the last non-zero entry in ã. See[10], Proposition 25. Another way of saying this is that, in the filtration (2) , P_{B_{n}}(a) has subquotients P_{B_{n−1}^{c}}(a), P_{B_{k}^{c±1,0,...,0}}(a_{0}, …, a_{k}) and (if c ≠ p − 1) P_{B_{n−1}^{c}}(a)∗, where k is the largest number less than n such that a_{k} < p − 1.

We now proceed by induction on n to show that A_{n} and B_{n} are isomorphic. This is trivial for n = 0, so assume that A_{n−1} ≅ B_{n−1}.

Then, for fixed a, P_{B_{n−1}^{c}}(a) ≅ P_{A_{n−1}^{c}}(a), so comparing filtrations (1) and (2) we conclude that K^{±} surjects onto T_{B_{n−1}^{c±1}}(ã), whilst L surjects onto P_{B_{n−1}^{c}}(a)^{∗}. It remains to show that these surjections are isomorphisms. Equivalently, it remains to show that K^{±} ≅ P_{A_{k}^{c±1,0,...,0}}(a_{0}, …, p − 2 − a_{k}), and L ≅ P_{A_{n−1}^{c}}(a)^{∗},

since by the inductive hypothesis A_{k}^{c±1,0,...,0} ≅ B_{k}^{c±1,0,...,0}. and A_{n−1}^{c} ≅ B_{n−1}^{c}.

We will only treat the case 0 < c < p − 1. The cases c = 0 and c = p − 1 are similar, but easier.

More facts about A_{n}.

• For a path p̃ = f_{i_{1},ε_{1}} ⋯ f_{i_{r},ε_{r}}a in Q_{n} with i_{j} < k for j = 1, …, r,

f_{k,ε}f_{i_{1},ε_{1}} ⋯ f_{i_{r},ε_{r}}a = f_{i_{1},ρ(ε_{1})} ⋯ f_{i_{r},ρ(ε_{r})}f_{k,ε}a  (3)

in A_{n}, where ρ(ε_{j}) = ε_{j} if i_{j} < k − 1 and ρ(ε_{j}) = −ε_{j} if i_{j} = k − 1. This is an immediate consequence of relations (1) and (5). We denote f_{i_{1},ρ(ε_{1})} ⋯ f_{i_{r},ρ(ε_{r})} by (p̃)^{ρ}

• There is a unique shortest path q in Q_{n} from a vertex a to the subquiver Q_{A_{n−1}^{c±1}}, and any other such path can be rewritten as a concatenation of this path with a path inside Q_{A_{n−1}^{c±1}}. Indeed, if a_{n−1} ≠ p − 1, then f_{n,±1}a is a path with target a′ = (a_{0}, …, p − 2 − a_{n−1}, c ± 1) in Q_{A_{n−1}^{c±1}}, of length 1, and this is obviously the shortest possible path. If a = (a_{0}, …, a_{k}, p − 1, …, p − 1, c), then we claim the shortest possible path to Q_{A_{n−1}^{c±1}} is

q := f_{n,±1}f_{n−1,−1}f_{n−2,−1} ⋯ f_{k+1,−1}a,

and the target of this path is a′ = (a_{0}, …, p − 2 − a_{k}, 0, …, 0, c ± 1). It can be easily checked that q is a path from a to a′. To see that this is the shortest path, consider the arrows f_{i,ε}^{b} in the quiver. Recall there is no arrow f_{i,ε}^{b} if b_{i−1} = p − 1. Therefore, walking along any path from a to a vertex with last coordinate different from c will successively alter coordinates a_{k+1}, …, a_{n−1} taking values p − 1, beginnining with the coordinate a_{k+1}. So when we represent our path, all the elements f_{n,±1}, f_{n−1,−1}, f_{n−2,−1}, …, f_{k+1,−1} will appear, in this order; in between two of these, say f_{n−r,−1} and f_{n−r−1,−1}, we will see elements f_{j,ε} for j ≤ n − r − 1. But by the previous bullet point, we can rewrite this path as p′q, for some p′. This shows that the path q is the shortest possible path from a to Q_{A_{n−1}^{c±1}}, and any other such path is equal in A_{n} to a concatenation of q with a path p′ inside Q_{A_{n−1}^{c±1}}.

We now proceed to show that K^{±} ≅ P_{A_{k}^{c±1,0,...,0}}(a_{0}, …, a_{k}).

• K^{±} has a simple top corresponding to vertex a′ and is hence a quotient of P_{A_{n−1}^{c±1}}(a′). This is a module-theoretic version of the statement in the previous bullet point that any path from a to Q_{A_{n−1}^{c±1}}, is equivalent to a concatenation of the path q with a path inside Q_{A_{n−1}^{c±1}}.

• K^{±} ≅ P_{A_{k}^{c±1,0,...,0}}(a_{0}, …, a_{k}). To see this, we establish that K^{±} is a quotient of P_{A_{k}^{c±1,0,...,0}}(a_{0}, …, a_{k}). This is sufficient since we have already established a surjection K^{±} ↠ P_{A_{k}^{c±1,0,...,0}}(a_{0}, …, a_{k}).

To see that K^{±} is a quotient of P_{A_{k}^{c±1,0,...,0}}(a_{0}, …, a_{k}), we show that f_{i,ε}q = 0, for k + 1 ≤ i < n. Indeed, apply f_{i,ε} to q, for k + 1 ≤ i < n. As the target of q has 0 in the i^{th} coordinate, the choice ε = −1 will yield zero. On the other hand, the choice ε = 1 gives

\begin{aligned}f_{i,1} & f_{n,±1}f_{n−1,−1}f_{n−2,−1} ⋯ f_{k+1,−1}a \\ & =^{(1)} f_{n,±1}f_{n−1,−1}f_{n−2,−1} ⋯ f_{i+2,−1}f_{i,1}f_{i+1,−1}f_{i,−1} ⋯ f_{k+1,−1}a \\ & =^{(5)} f_{n,±1}f_{n−1,−1}f_{n−2,−1} ⋯ f_{i+2,−1}f_{i+1,−1}f_{i,−1}f_{i,−1} ⋯ f_{k+1,−1}a \end{aligned} (4)

but this contains a term f_{i,−1}^{2} which is zero by relation (2), hence f_{i,1}q also vanishes. (The numbers on the equality indicate the relations used.) Therefore f_{i,ε}q is only non-zero for i ≤ k, which means that K^{±} factors over A_{k}^{c±1,0,...,0}, and is therefore a quotient of P_{A_{k}^{c±1,0,...,0}}(a_{0}, …, a_{k}), as desired.

As K^{±} ≅ P_{A_{k}^{c±1,0,...,0}}(a_{0}, …, a_{k}), we now know that the quotient of P_{A_{n}}(a) by L is isomorphic to the quotient of P_{B_{n}}(a) by P_{B_{n−1}^{c}}(a)^{∗}.

It remains to show that L ≅ P_{A_{n−1}^{c}}(a)^{∗}. Let us look at the composition factors.

• L has a simple head corresponding to vertex (a_{0}, …, p−2−a_{k}, 0, …, 0, p− 2, c), and is therefore a quotient of P_{A_{n−1}^{c}}(a_{0}, …, p − 2 − a_{k}, 0, …, 0, p − 2, c). Indeed, the shortest path from a to Q_{A_{n−1}^{c±1}} and back to Q_{A_{n−1}^{c}} is

q′ = f_{n,∓1}f_{n,±1}f_{n−1,−1}f_{n−2,−1} ⋯ f_{k+1,−1}a,

since it is obtained by multiplying the shortest path q from a to Q_{A_{n−1}^{c±1}} by f_{n,∓1}, only increasing the length by one. The target of q′ is (a_{0}, …, p − 2 − a_{k}, 0, …, 0, p − 2, c), and by the same argument as for K^{±}, L has a simple head as claimed.

• If a_{n−1} ≠ p − 1, then L ≅ P_{A_{n−1}^{c}}(a). Indeed, the path q′ given in the previous bullet point is just f_{n,∓1}f_{n,±1}a, with target a, hence L is a quotient of P_{A_{n−1}^{c}}(a). But P_{A_{n−1}^{c}}(a) is self-dual, as by the inductive hypothesis it is isomorphic to P_{B_{n−1}^{c}}(a_{0}, …, a_{n−1}), which (in case a_{n−1} ≠ p − 1) is self-dual, as is visible the filtrations for B_{n−1} given by Equation 2. We already know that L surjects onto P_{A_{n−1}^{c}}(a)^{∗} ≅ P_{A_{n−1}^{c}}(a), and so the claim follows.

From now on, we assume a_{n−1} = p − 1.

• We have a filtration of P_{A_{n−1}^{c}}(a)^{∗} with factors

\begin{matrix}P_{A_{k}^{(c,p−2,0,...,0)}}(a_{0}, …, a_{k−1}, p − 2 − a_{k}) \\ P_{A_{n−2}^{c,p−1}}(a)^{∗} \end{matrix} .

Indeed, by the inductive hypothesis A_{n−1} ≅ B_{n−1}, and so P_{A_{n−1}^{c}}(a)^{∗} has

a filtration with factors \begin{matrix}T_{A_{n−2}^{c,p−2}}(ã) \\ P_{A_{n−2}^{c,p−1}}(a)^{∗} \end{matrix} analogous to the filtration given in

Equation (2). But again T_{A_{n−2}^{c,p−2}}(ã) ≅ P_{A_{k}^{c,p−2,0,...,0)}}(a_{0}, …, a_{k−1}, p−2−a_{k}) ([10], Proposition 25), and by the inductive assumption that A_{n−1} ≅ B_{n−1}. We therefore have a filtration as claimed.

• The maximal quotient of L with composition factors in A_{n−2}^{c,p−2} is isomorphic to P_{A_{k}^{c,p−2,0,...,0}}(a_{0}, …, p − 2 − a_{k}). A proof of this runs similarly to the proof that K^{±} ≅ P_{A_{k}^{c±1,0,...,0}}(a_{0}, …, a_{k}), by noting that f_{i,ε} applied to (a_{0}, …, p − 2 − a_{k}, 0, …, 0, p − 2, c) yields zero, for k + 1 ≤ i ≤ n − 2. Indeed, for k + 1 ≤ i ≤ n − 2,

\begin{aligned}f_{i,1}f_{n,∓1}f_{n,±1}f_{n−1,−1}f_{n−2,−1} ⋯ f_{k+1,−1}a & = f_{i,ε}f_{n,∓1}q \\ & = f_{n,∓1}f_{i,ε}q = 0 \end{aligned}

by Equation (4) and f_{i,−1}(a_{0}, …, p−2−a_{k}, 0, …, 0, p−2, c) = 0, because a_{i} = 0.

• Those composition factors of L which are not in A_{n−2}^{c,p−2} -mod are all in A_{n−2}^{c,p−1} -mod, and the smallest submodule L′ containing all these factors is a quotient of P_{A_{n−2}^{c,p−1}}(a_{0}, …, p−2−a_{k}, 0, …, 0, p−2, p−1, c). This follows since L has a simple head (a_{0}, …, p−2−a_{k}, 0, …, 0, p−2, p−1, c), which is the target of f_{n−1,1}f_{n,∓1}q. Indeed, all composition factors of L are in A_{n−1}^{c} -mod. We therefore have to consider those composition factors which are in A_{n−2}^{c,j} -mod, for some j ≠ p − 2. We obtain paths in A_{n} corresponding to such factors by applying f_{n−1,±1} to p̃f_{n,∓1}q, for any path p̃ in Q_{A_{n−2}^{c,p−2}}. But such a path is equal to (p̃)^{ρ}f_{n−1,±1}f_{n,∓1}q in A_{n}, so we can equivalently apply f_{n−1,±1} directly to f_{n,∓1}q. Exploiting the commutation relations for A_{n}, and the fact that f_{n−1,±1}^{2} = 0, we obtain the claim.

• The smallest submodule of L′ containing all composition factors outside A_{n−3}^{c,p−1,p−2} -mod only has composition factors in A_{n−3}^{c,p−1,p−1}, and is a quotient of the projective P_{A_{n−2}^{c,p−1,p−1}}(a_{0}, …, p−2−a_{k}, 0, …, 0, p−2, p−1, p− 1, c). This follows since, by repeating the argument showing that the quotient of L with composition factors in A_{n−2}^{c,p−2} -mod is P_{A_{k}^{c,p−2,0,...,0}}(a_{0}, …, p−

2−a_{k}), we see that the quotient of L′ with composition factors in A_{n−3}^{c,p−1,p−2} is isomorphic to P_{A_{k}^{c,p−1,p−2,0,...,0}}(a_{0}, …, p − 2 − a_{k}). Then, as in the previous bullet point, the claim follows.

• L has a composition series with composition factors P_{0}, P_{1}, ..., P_{k}, L̂, where P_{1} ∈ A_{n−2}^{c,p−2} -mod, where P_{1} ∈ A_{n−3}^{c,p−1} -mod, where P_{2} ∈ A_{n−4}^{c,p−1,p−1} -mod,..., where L̂ is a quotient of P_{A_{k}^{c,p−1,...,p−1}}(a_{0}, …, a_{k}), and where a_{k} ≠ p − 1, for some k. This follows by iterating the arguments of previous bullet points.

• L̂ is isomorphic to P_{A_{k}^{c,p−1,...,p−1}}(a_{0}, …, a_{k}). First note that the projective module P_{A_{k}^{c,p−1,...p−1,p−1}}(a_{0}, …, a_{k}) is self-dual since a_{k} ≠ p − 1. Further-

more, since L surjects onto P_{B_{n}−1^{c}}(a)^{∗} ≅ P_{A_{n−1}^{c}}(a)^{∗}, by restriction, L̂ surjects onto P_{A_{k}^{c,p−1,...p−1,p−1}}(a_{0}, …, a_{k})^{∗}. Hence, by self-duality we have

surjections in either direction between the finite-dimensional spaces L̂ and P_{A_{k}^{c,p−1,...,p−1}}(a_{0}, …, a_{k}). These surjections are therefore isomorphisms.

Thanks to our belt of bullets, the surjection from L to P_{B_{n}−1^{c}}(a)^{∗} is an isomorphism, from which it follows that the algebras A_{n} and B_{n} are indeed isomorphic, as required.

We have now seen that the algebras A_{n}, B_{n} and D_{n} are all isomorphic. As a corollary, we obtain the following result (conjectured in [19]).

Lemma 6. B_{n} ≅ C_{n}.

Proof. As already observed, A_{n} has a ℤ_{+}^{n}-grading, in which f_{i,ε} lies in degree (0, …, 0, 1, 0, … 0), where the 1 is in the ith component. By Proposition 5, B_{n} is graded in the same way. We established previously that the graded ring grB_{n} of B_{n}, taken with respect to a certain filtration, is isomorphic to C_{n} ([19], Corollary 33). The ℤ_{+}^{n}-grading here is compatible with this filtration, which means to say that B_{n} ≅ grB_{n}. Therefore B_{n} ≅ C_{n}.

Looking the lemmas of this section together, it is visible that we have now completed the proof of Theorem 1.

Remark 7 Throughout this section, we have assumed that p ≥ 3. Here we state how to treat the case p = 2, which is slightly different, although analogous statements to those for p ≥ 3 hold.

We define an algebra A_{n} as follows: Let Q′_{n} be the quiver with vertices a = (a_{0}, … a_{n}), where all a_{i} ∈ {0, 1} and arrows f_{±1,i}^{a} (for 1 ≤ i ≤ n) point from a to a = (a_{0}, …, a_{i} ± 1, …, a_{n}), whenever a_{i} ± 1 ∈ {0, 1}. Let Q_{n} be the connected component in Q′_{n} containing (0, …, 0) and note that the coordinate a_{0} in this component is always zero, hence we can omit it. A_{n} is then given as the path algebra of Q_{n} modulo the following relations:

(i) f_{i,ε_{1}}f_{j,ε_{2}}a−f_{j,ε_{2}}f_{i,ε_{1}}a if i ≠ j −1, j, j +1 (and a_{i} +ε_{1}, a_{j} +ε_{2} ∈ {0, …, p− 1}, a_{i−1} = a_{j−1} = 0)

(ii) f_{i,1}f_{i,−1}a if a_{i} = 1 (and a_{i−1} = 0)

(iii) f_{i+1,ε}f_{i,−1}f_{i,1}a − f_{i,−1}f_{i,1}f_{i+1,ε}a (if a_{i} = a_{i−1} = 0)

(iv) f_{i,1}f_{i+1,ε}f_{i,−1}a if a_{i} = 1, a_{i−1} = 0.

Relation (i) corresponds with relation (1) from the case p ≥ 3, whilst relation (ii) corresponds with relation (3) from the case p ≥ 3. Relations (2) and (4) from the case p ≥ 3 are superfluous in case p = 2, and relations (iii) and (iv) given here replace the previous relation (5).

We denote by B_{n} a block of the Schur algebra S(2, r) with 2^{n} simple modules, eg. S(2, 2^{n} − 2).

The operator C_{2}, and the algebra C_{n} are defined in exactly the same way as in the case p ≥ 3.

We define D_{n} as in case p ≥ 3, only this time using Koshita’s quiver and relations from [16] for p = 2.

The same results hold as in case p ≥ 3, and we have

A_{n} ≅ B_{n} ≅ C_{n} ≅ D_{n}.

The proofs are entirely analogous.

2. Homotopical constructions.

Here we define a weak 2-category T of algebras with a bimodule, and define operators 𝕆_{(c,x)} on T . An introduction to weak 2-categories can be found in a note by Leinster [18].

We define a 2-category T with the following data:

• a collection Ob T whose elements are pairs \underline{A} := (A,_{A}T_{A}), where A is a finite-dimensional algebra, and _{A}T_{A} is an (A, A)-bimodule (these elements are the 0-cells);

• for each pair (\underline{A}, \underline{B}) a category T (\underline{A}, \underline{B}) as follows:

– objects (1-cells of T ) are pairs (M, ϕ_{M}) where M = _{A}M_{B} is an (A, B)-bimodule and ϕ_{M} : _{A}T_{A} ⊗_{A} M → M ⊗_{B} _{B}T_{B} is an(A, B)- bimodule isomorphism;

– morphisms (2-cells of T ) (M, ϕ_{M}) → (N, ϕ_{N}) are (A, B)-bimodule morphisms f : M → N such that the diagram

\xymatrix{_{A}T_{A} ⊗_{A} M \ar[r]^{ϕ_{M}} \ar[d]^{1⊗f} & M ⊗_{B} _{B}T_{B} \ar[d]^{f⊗1} \\ _{A}T_{A} ⊗_{A} N \ar[r]^{ϕ_{N}} & N ⊗_{B} _{B}T_{B}}

commutes.

• functors

T (\underline{A}, \underline{B}) × T (\underline{B}, \underline{C}) → T ((\underline{A}, \underline{C})

taking pairs of 1-cells ((M, ϕ_{M}), (M̃, ϕM̃)) to (M ⊗M̃, (ϕ_{M} ⊗1)∘(1⊗ϕM̃)), and pairs of 2-cells (f, g) to f ⊗g; furthermore a 1-cell I_{A} which is just the bimodule _{A}A_{A} together with the obvious isomorphism ϕ_{A} : A ⊗_{A} _{A}T_{A} → AT_{A} ⊗_{A} A.

Since tensor products of bimodules are associative, up to isomorphism, and the associativity isomorphisms obey the relevant pentagon axiom, T indeed forms a weak 2-category.

Now let (A, T) be an object of T and let (c, x) be another object of T , such that c = ⊕_{j≥0}c^{(j)} is a ℤ_{+}-graded algebra, and x = ⊕_{j≥0}x^{(j)} is a ℤ_{+}-graded bimodule over c.

Definition 8. The algebra c(A, T) is the vector space c(A, T) := ⨁_{j≥0}c^{(j)} ⊗_{F} T^{j},

where c^{(j)} is the jth graded component of c, and the formal expression T^{j} denotes the j-fold tensor product over A of T with itself. The multiplication is given by

(c_{1} ⊗ t_{1} ⋯ t_{j})(c_{2} ⊗ t_{j+1} ⋯ t_{j+k}) = (c_{1}c_{2} ⊗ t_{1} ⋯ t_{j} ⋯ t_{j+k}),

for c_{1} ∈ c^{(j)}, c_{2} ∈ c^{(k)} and t_{1} ⋯ t_{j} = t_{1} ⊗ ⋯ ⊗ t_{j} for t_{i} ∈ T.

Note that c(A, T) is clearly an associative algebra, namely the subalgebra of homogeneous tensors inside c ⊗_{F} A[T], where A[T] denotes the tensor algebra of T over A.

We call c(A, T) a twisted tensor product, since it resembles a tensor product c ⊗ A, twisted by the bimodule T.

Analogously, we define the bimodule x(A, T) := ⨁_{j≥0}x^{(j)} ⊗_{F} T^{j}, where x^{(j)}

is the jth homogeneous component of x. The left and right c(A, T)-actions on x(A, T) are given by

(c_{1} ⊗ t_{1} ⋯ t_{j})(x_{1} ⊗ t_{j+1} ⋯ t_{j+k})(c_{2} ⊗ t_{j+k+1} ⋯ t_{j+k+l}) =
(c_{1}x_{1}c_{2} ⊗ t_{1} ⋯ t_{j} ⋯ t_{j+k+l}),

for c_{1} ∈ c^{(j)}, c_{2} ∈ c^{(l)}, x_{1} ∈ x^{(k)} and t_{i} ∈ T. This oviously defines a bimodule action.

We define (c, x) ∗ (A, T) := (c(A, T), x(A, T)). The product ∗, defined on graded objects of T , is not associative. We write 𝕆_{(c,x)} for the operator (c, x)∗−, defined on objects of T .

We now claim that this operator is natural on a categorical level, namely we have the following lemma.

Lemma 9. 𝕆_{(c,x)} lifts to an endo-2-functor of T .

Proof. We have the following data:

• 𝕆_{(c,x)} defines a correspondence on 0-cells, taking \underline{A} = (A,_{A}T_{A}) to (c, x) ∗ \underline{A} := (c(A, T), x(A, T)) as defined above; • (𝕆_{(c,x)})_{\underline{A},\underline{B}} is a functor

T (\underline{A}, \underline{B}) → T (𝕆_{(c,x)}(\underline{A}), 𝕆_{(c,x)}(\underline{B}))
(M, ϕ_{M}) ↦ (c(A, T) ⊗_{A} M, 𝕆_{(c,x)}(ϕ_{M}))
(f : M → N) ↦ (1 ⊗ f : c(A, T) ⊗ M →^{1⊗f} c(A, T) ⊗ N);

We first need to check that c(A, T) ⊗_{A} M is indeed an (c(A, T), c(B, T))- bimodule. However, due to the isomorphism ϕ_{M} we have and isomorphism A[T] ⊗_{A} M ≅ M ⊗_{B} B[T], hence c ⊗_{F} A[T] ⊗_{A} M ≅ M ⊗_{B} B[T] ⊗_{F} c and this is a graded isomorphism. Hence it restricts to c(A, T) ⊗_{A} M ≅ M ⊗_{B} c(B, T).

Next we need to define

𝕆_{(c,x)}(ϕ_{M}) : x(A, T) ⊗_{c(A,T)} (c(A, T) ⊗_{A} M) → (M ⊗_{B} c(B, T)) ⊗_{c(B,T)} x(B, T).

This is done in the same way. We take the isomorphism A[T] ⊗_{A} M → M ⊗_{B} B[T], and tensor it with x to obtain an isomorphism x ⊗_{F} A[T] ⊗_{A} M → M ⊗_{B} B[T] ⊗_{F} x which is graded, and hence restricts to the homogeneous part x(A, T) ⊗_{c(A,T)} (c(A, T) ⊗_{A} M) ≅ x(A, T) ⊗_{A} M → M ⊗_{B} x(B, T) ≅ (M ⊗_{B} c(B, T)) ⊗_{c(B,T)} x(B, T).

It remains to check that the diagram

\xymatrix{x(A, T) ⊗_{c(A,T)} (c(A, T) ⊗_{A} M) \ar[r]^{𝕆_{(c,x)}(ϕ_{M})} \ar[d]^{1⊗1⊗f} & (M ⊗_{B} c(B, T)) ⊗_{c(B,T)} x(B, T) \ar[d]^{f⊗1⊗1} \\ x(A, T) ⊗_{c(A,T)} (c(A, T) ⊗_{A} N) \ar[r]^{ϕ_{N}} & (N ⊗_{B} c(B, T)) ⊗_{c(B,T)} x(B, T)}

commutes. This follows by the same argument as above from commutativity of the diagram

\xymatrix{A[T] ⊗_{A} M \ar[r]^{A[ϕ_{M}]} \ar[d]^{1⊗f} & M ⊗_{B} B[T] \ar[d]^{f⊗1} \\ A[T] ⊗_{A} N \ar[r]^{ϕ_{N}} & N ⊗_{B} B[T],}

which follows directly from ϕ_{N} ∘ (1 ⊗ f) = (f ⊗ 1) ∘ ϕ_{M}.

• For \underline{A}, \underline{B}, \underline{C} ∈ T , we have two functors from the category T (\underline{A}, \underline{B}) × T (\underline{B}, \underline{C}) to T (𝕆_{(c,x)}(\underline{A}), 𝕆_{(c,x)}(\underline{C})), obtained by applying the functors 𝕆_{(c,x)}, and composition, in different orders. There is a natural transformation η_{\underline{A}\underline{B}\underline{C}} between these functors.

Indeed, we define

η_{\underline{A}\underline{B}\underline{C}}(M, N) : c(A, T) ⊗_{A} M ⊗_{c(B,T)} c(B, T) ⊗_{B} N → c(A, T) ⊗_{A} M ⊗_{B} ⊗N,

to be the natural isomorphism obtained by contracting along the isomorphism M ⊗_{c(B,T)} c(B, T) ≅ M, for (M, ϕ) ∈ T (\underline{A}, \underline{B}), and (N, ψ) ∈ T (\underline{B}, \underline{C}).

The natural transformation η_{\underline{A}\underline{B}\underline{C}} is obviously compatible with identity morphisms, and therefore 𝕆_{(c,x)} is a 2-functor, as required.

Example 10 Let c_{p} be the quasi-hereditary algebra with p simple modules, defined in the first chapter (see Remark 2). This algebra has various incarnations, such as A_{1}, B_{1}, C_{1}, and D_{1}. In particular, applying the operator C_{p} to the trivial algebra F with its trivial bimodule F, we obtain the algebra c_{p}, along with a c_{p}-c_{p}-bimodule, which we denote x_{p}.

Looking at its definition via quiver and relations (see Remark 2), we see that c_{p} = ⊕_{i,j∈ℤ}c_{p}^{(i),(j)} is ℤ_{+}^{2}-graded, with arrows ξ in degree (1, 0) and arrows η in degree (0, 1). Summing these gradings, we obtain a ℤ_{+}-grading on c. We define the operator 𝕆_{p} on T to be the operator 𝕆_{(c_{p},x_{p})}, defined with respect to this grading. Thus,

𝕆_{p}(A, T) = (c_{p}, x_{p}) ∗ \underline{A} = (c_{p}(\underline{A}), x_{p}(\underline{A})).

3. Twisted tensor products.

Here, we define twisted tensor products in a differential graded setting. We show that, under favourable conditions, taking a twisted tensor product respects derived equivalences.

We first need to introduce some standard notation. In an additive category with an object x, we denote by add x the closure under direct sums and summands. If x is an object in a triangulated category, x -perf denotes the closure under direct sums, summands, shifts and extensions.

If c, d are finite dimensional algebras, whose bounded derived categories are triangle equivalent, then there exists a Rickard tilting complex _{c}x_{d}, for which the pair of adjoint functors (x⊗_{d}^{L} −, RHom_{c}(x, −)) induces the equivalence [22]. Here, by a Rickard tilting complex, we mean a complex of (c, d)-bimodules, projective on both sides, such that the right multiplication morphism d → RHom_{c}(x, x) and the left multiplication morphism c → RHom_{d}(x, x) are quasiisomorphisms.

We work in this section with finite dimensional graded algebras and dg algebras. We also work with finite dimensional graded dg algebras. These are algebras α with a bigrading α = ⊕_{i,j∈ℤ}α^{(i),(j)}, equipped with a differential d of degree (0, 1), which obeys the Leibnitz rule with respect to the ℤ-grading obtained by forgetting the i-coordinate. Therefore, with respect to the ℤ-grading obtained by forgetting the i-coordinate, α is a dg algebra, whilst with respect to the grading obtained by forgetting the j-coordinate, α is an ordinary graded algebra. We call the grading obtained by forgetting the i-coordinate the homological grading, whilst we call the grading obtained by forgetting the j-coordinate the standard grading.

We denote a shift by k in the standard grading by ⟨k⟩. We thus work with the convention that (M⟨k⟩)^{(i)} := M^{(i+k)} for a standardly graded object M = ⊕_{i∈ℤ}M^{(i)}.

If α is graded dg algebra, whilst m and n are graded dg α-modules, we denote by Hom_{α}(m, n) the collection of all α-module homomorphisms between m and n, and grHom_{α}(m, n) the collection of α-module homomorphisms which respect the standard grading. Thus Hom_{α}(m, n) = ⊕_{i∈ℤ} Hom_{α}^{(i)}(m, n) is a direct sum of chain complexes Hom_{α}^{(i)}(m, n) = grHom_{α}(m, n⟨j⟩). We denote by D_{dg}(α) the derived category of all dg α-modules, where the dg structure is taken with respect to the homological grading.

We now discuss twisted tensor products in the world of dg-algebras.

Let c and d be finite dimensional ℤ_{+}-graded algebras. Let_{c}x_{d} be a positively graded complex of (c, d)-bimodules. Let also a be a finite dimensional algebra, and _{a}t_{a} a complex of (a, a)-bimodules.

Definition 11. We define a graded dg-algebra c(a, t) as the vector space c(a, t) := ⨁_{j≥0}c^{(j)} ⊗_{F} t^{j}, where c^{(j)} is the jth graded component of c, and the formal ex-

pression t^{j} denotes the j-fold tensor product over a of t with itself. The multiplication is given by

(c_{1} ⊗ t_{1} ⋯ t_{j})(c_{2} ⊗ t_{j+1} ⋯ t_{j+k}) = (c_{1}c_{2} ⊗ t_{1} ⋯ t_{j} ⋯ t_{j+k}),

for c_{1} ∈ c^{(j)}, c_{2} ∈ c^{(k)}, and t_{1} ⋯ t_{j} = t_{1} ⊗ ⋯ ⊗ t_{j} for t_{i} ∈ t. The differential is the total differential on t^{j} within each standard homogeneous piece.

As before, this is clearly an associative algebra. To check that this is indeed a dg-algebra, consider the dg-algebra c⊗_{F} a[t], and consider c(a, t) as the subalgebra which consists of homogeneous tensors. It then follows immediately that this is a dg-subalgebra since it is closed under multiplication and the differential.

Note that we have two independent gradings here, the standard grading coming from c and the homological grading coming from the complex t. So in fact c(a, t) is a graded dg-algebra.

Definition 12. We define the complex x(a, t) := ⨁_{j≥0}x^{(j)} ⊗_{F} t^{j}, where x^{(j)} is the

jth graded component of x and as such a complex in its own right. The differential on x(a, t) is the total differential on t^{j} within each standard homogeneous piece.

The complex x(a, t) admits a c(a, t)-action, given by the obvious multiplication

(c_{1} ⊗ t_{1} ⋯ t_{j})(x_{1} ⊗ t_{j+1} ⋯ t_{j+k}) = (c_{1}x_{1} ⊗ t_{1} ⋯ t_{j} ⋯ t_{j+k}),

for c_{1} ∈ c, x_{1} ∈ x^{(k)} and t_{i} ∈ t. This is indeed a graded dg-algebra action, since it is a restriction of the graded dg algebra action of c ⊗_{F} a[t] on x ⊗_{F} a[t].

The algebra d(a, t) acts similarly on the right of x(a, t). By construction, left and right actions commute, making x(a, t) into a graded (c(a, t), d(a, t))-dg-bimodule.

The main theorem of this section is the following:

Theorem 13. Suppose that x and t are Rickard tilting complexes. Then the dg bimodule x(a, t) induces a triangle equivalence

\xymatrix{D_{dg}(c(a, t)) \ar[r]^{RHom(x(a,t),−)} & D_{dg}(d(a, t)) \ar[l]^{x(a,t)⊗_{d(a,t)}^{L}−}}

Two finite dimensional algebras are derived equivalent if, and only if, there is a Rickard tilting complex inducing that equivalence [22]. So roughly speaking, this theorem says that if x and t induce derived equivalences, so does x(a, t). In particular, if _{c}x_{c} is a Rickard tilting complex, then 𝕆_{(c,x)} respects derived equivalences.

To prove the theorem, we apply B. Keller’s Morita theory for differential graded algebras. Indeed, by Lemma 3.10 of Keller’s ICM article [14], to prove Theorem 13, it is sufficient to show that _{c(a,t)}x(a, t) ∈ c(a, t) -perf, that _{c(a,t)} is a generator of D_{dg}(c(a, t)), and that the natural morphism

d(a, t) → Hom_{c(a,t)}(x(a, t), x(a, t))

is a quasi-isomorphism. We do this in a sequence of steps.

Lemma 14. If two positively graded complexes x and y are quasi-isomorphic as graded complexes, then x(a, t) and y(a, t) are quasi-isomorphic complexes.

Proof. Since we assume our quasi-isomorphism ϕ to be graded, we can write it as a sum of quasi-isomorphisms ϕ_{i} (of vector spaces), where ϕ_{i} is maps x in standard degree i to y in standard degree i. Then we can define a map ϕ_{i}(a, t) : x(a, t)^{(i)} ≅ x^{(i)} ⊗ t^{i} → y(a, t)^{(i)} ≅ y^{(i)} ⊗ t^{i} to be ϕ_{i} ⊗ id. Since we’re tensoring over F this will remain a quasi-isomorphism, so x(a, t) and y(a, t) are quasi-isomorphic in every degree and hence altogether.

We now compare the structure of Hom_{c(a,t)}(x(a, t), x(a, t)) to the structure of Hom_{c}(x, x). As a c-module x is the direct sum of projectives in possibly shifted degrees, i.e. x = ⨁_{1≤k≤r}ce_{λ_{k}}⟨−i_{k}⟩ for i_{k} > 0. The degree j component

Hom_{c}^{(j)}(x, x) – where we take the grading with respect to the grading of c and ignore that coming from the complex structure – is then given by

Hom_{c}^{(j)}( ⨁_{1≤k≤r}ce_{λ_{k}}⟨−i_{k}⟩, ⨁_{1≤l≤r}ce_{λ_{l}}⟨−i_{l}⟩)

= grHom_{c}( ⨁_{1≤k≤r}ce_{λ_{k}}⟨−i_{k}⟩, ⨁_{1≤l≤r}ce_{λ_{l}}⟨j − i_{l}⟩)

≅ ⨁_{1≤k≤r} ⨁_{1≤l≤r} grHom_{c}(ce_{λ_{k}}, ce_{λ_{l}}⟨j − i_{l} + i_{k}⟩).

Since the ce_{λ_{k}} are projective,

⨁_{1≤k≤r} ⨁_{1≤l≤r} grHom_{c}(ce_{λ_{k}}, ce_{λ_{l}}⟨j − i_{l} + i_{k}⟩)

≅ ⨁_{1≤k≤r} ⨁_{1≤l≤r} grHom_{c^{(0)}}(ce_{λ_{k}}^{(0)}, (ce_{λ_{l}}⟨j − i_{l} + i_{k}⟩)^{(0)})

= ⨁_{1≤k≤r} ⨁_{1≤l≤r} Hom_{c^{(0)}}(ce_{λ_{k}}^{(0)}, ce_{λ_{l}}).

Now we consider the degree j component of Hom_{c(a,t)}(x(a, t), x(a, t)).

Lemma 15.

Hom_{c(a,t)}^{(j)}(x(a, t), x(a, t)) ≅ ⨁_{1≤k≤r} ⨁_{1≤l≤r} Hom_{a}(t^{i_{k}}, Hom_{c^{(0)}}((ce_{λ_{k}})^{(0)}, (ce_{λ_{l}}))⊗t^{j+i_{k}})

Proof. Define ẽ:= e ⊗ 1_{A} for an idempotent e ∈ c. Then, by definition, x(a, t) decomposes as x(a, t) := ⨁_{1≤k≤r}c(a, t)ẽ_{λ_{k}}⟨−i_{k}⟩ ⊗ t^{i_{k}} as a c(a, t)-module. Hence

Hom_{c(a,t)}^{(j)}(x(a, t), x(a, t))

≅ Hom_{c(a,t)}^{(j)}( ⨁_{1≤k≤r}c(a, t)ẽ_{λ_{k}}⟨−i_{k}⟩ ⊗_{a} t^{i_{k}}, ⨁_{1≤l≤r}c(a, t)ẽ_{λ_{l}}⟨−i_{l}⟩ ⊗_{a} t^{i_{l}})

= grHom_{c(a,t)}( ⨁_{1≤k≤r}c(a, t)ẽ_{λ_{k}}⟨−i_{k}⟩ ⊗_{a} t^{i_{k}}, ⨁_{1≤l≤r}c(a, t)ẽ_{λ_{l}}⟨j − i_{l}⟩ ⊗_{a} t^{i_{l}})

≅ grHom_{c(a,t)}( ⨁_{1≤k≤r}c(a, t)ẽ_{λ_{k}} ⊗_{a} t^{i_{k}}, ⨁_{1≤l≤r}c(a, t)ẽ_{λ_{l}}⟨j − i_{l} + i_{k}⟩ ⊗_{a} t^{i_{l}})

≅ Hom_{a}(t^{i_{k}}, grHom_{c(a,t)}( ⨁_{1≤k≤r}c(a, t)ẽ_{λ_{k}}, ⨁_{1≤l≤r}c(a, t)ẽ_{λ_{l}}⟨j − i_{l} + i_{k}⟩ ⊗_{a} t^{i_{l}}))

≅ ⨁_{1≤k≤r} ⨁_{1≤l≤r} Hom_{a}(t^{i_{k}}, grHom_{c(a,t)}(c(a, t)ẽ_{λ_{k}}, c(a, t)ẽ_{λ_{l}}⟨j − i_{l} + i_{k}⟩ ⊗_{a} t^{i_{l}}))

≅ ⨁_{1≤k≤r} ⨁_{1≤l≤r} Hom_{a}(t^{i_{k}}, grHom_{c(a,t)}(c(a, t)ẽ_{λ_{k}}, c(a, t)ẽ_{λ_{l}}⟨j − i_{l} + i_{k}⟩) ⊗_{a} t^{i_{l}})

where the last isomorphism is by virtue of c(a, t)ẽ_{λ_{k}} being projective as a c(a, t)- module. By the same argument,

grHom_{c(a,t)}(c(a, t)ẽ_{λ_{k}}, c(a, t)ẽ_{λ_{l}}⟨j − i_{l} + i_{k}⟩)
≅ grHom_{c(a,t)^{(0)}}(c(a, t)ẽ_{λ_{k}}^{(0)}, (c(a, t)ẽ_{λ_{l}}⟨j − i_{l} + i_{k}⟩)^{(0)})
= Hom_{c(a,t)^{(0)}}((c(a, t)ẽ_{λ_{k}})^{(0)}, (c(a, t)ẽ_{λ_{l}}))
≅ Hom_{c^{(0)}⊗a}((ce_{λ_{k}})^{(0)} ⊗ a, (ce_{λ_{l}}) ⊗ t^{j−i_{l}+i_{k}})
≅ Hom_{c^{(0)}}((ce_{λ_{k}})^{(0)}, (ce_{λ_{l}})) ⊗ Hom_{a}(a, t^{j−i_{l}+i_{k}})
≅ Hom_{c^{(0)}}((ce_{λ_{k}})^{(0)}, (ce_{λ_{l}})) ⊗ t^{j−i_{l}+i_{k}}.

Note that grHom_{c(a,t)}(c(a, t)ẽ_{λ_{k}}, c(a, t)ẽ_{λ_{l}}⟨j −i_{l} +i_{k}⟩) is only nonzero if i_{l} −i_{k} − j > 0 since x is concentrated in positive degrees, hence we only obtain positive tensor powers of t.

Inserting the last equation back into Hom_{c(a,t)}^{(j)}(x(a, t), x(a, t)) yields

Hom_{c(a,t)}^{(j)}(x(a, t), x(a, t)) ≅ ⨁_{1≤k≤r} ⨁_{1≤l≤r} Hom_{a}(t^{i_{k}}, Hom_{c^{(0)}}((ce_{λ_{k}})^{(0)}, (ce_{λ_{l}}))⊗t^{j+i_{k}})

where the left a-module structure of the second argument is just that of t^{j+i_{k}}. This proves the lemma.

Lemma 16. The (a, a)-bimodule, Hom_{a}(t^{i}, t^{j}) is quasi-isomorphic to the bimodule Hom_{a}(t^{i−j}, a), if i > j and to Hom_{a}(a, t^{j−i}) if j ≥ i.

Proof. Since t is a Rickard tilting complex, we have a quasi-isomorphism a → Hom_{a}(t, t). For i = 0, or j = 0, the claim is trivial, so assume inductively that it holds for Hom_{a}(t^{i−1}, t^{j−1}). Then

Hom_{a}(t^{i}, t^{j}) = Hom_{a}(t ⊗_{a} t^{i−1}, t ⊗_{a} t^{j−1})
≅ Hom_{a}(t^{i−1}, Hom_{a}(t, t ⊗_{a} t^{j−1}))
≅ Hom_{a}(t^{i−1}, Hom_{a}(t, t) ⊗_{a} t^{j−1})

the last isomorphism depending on the fact that all modules occurring in t are projective. Again, since all modules are projectives on both sides, both functors − ⊗_{a} t^{j−1} and Hom_{a}(t^{i−1}, −) are exact, hence the quasi-isomorphism a → Hom_{a}(t, t) remains a quasi-isomorphism under composition with both functors, and we obtain the desired quasi-isomorphism Hom_{a}(t^{i−1}, a ⊗_{a} t^{j−1}) → Hom_{a}(t^{i−1}, Hom_{a}(t, t) ⊗_{a} t^{j−1}), which completes the induction.

Remark 17 Note that the composite quasi-isomorphism Υ

Hom_{a}(t^{i−1}, a ⊗_{a} t^{j−1}) → Hom_{a}(t^{i−1}, Hom_{a}(t, t) ⊗_{a} t^{j−1})
→ Hom_{a}(t^{i}, t^{j})

is given by the explicit expression

\begin{aligned}(t_{1} ⋯ t_{i−1} ↦ 1 ⊗ f(t_{1} ⋯ t_{i−1})) & ↦ (t_{1} ⋯ t_{i−1} ↦ (t_{i} ↦ t_{i}) ⊗ f(t_{1} ⋯ t_{i−1})) \\ & ↦ (t′_{1} ⋯ t′_{i} ↦ t′_{1}f(t′_{2} ⋯ t′_{i})). \end{aligned}

Lemma 18. x(a, t) ∈ c(a, t) -perf.

Proof. We know that all summands of x are projective c-modules, possibly shifted by a positive degree. Hence x(a, t) is an iterated extension of projective c(a, t)-modules and projective c(a, t)-modules, tensored with further powers of t. However, as t ∈ a -perf, P ⊗_{a} t for a projective c(a, t)-module P is itself a complex of projective c(a, t)-modules, hence in c(a, t) -perf. So, x(a, t) is a finite iterated extension of elements in c(a, t) -perf, hence in c(a, t) -perf itself.

Lemma 19. x(a, t) generates D_{dg}(c(a, t)).

Proof. First note that c(a, t) generates D_{dg}(c(a, t)), i.e.

if Hom_{D_{dg}(c(a,t))}(c(a, t), M[i]) = 0 for all i ∈ ℤ, then M = 0.

Note also that under the tilting equivalence between c and d, c corresponds to Hom_{c}(x, c), and x corresponds to d. Hence, since Hom_{c}(x, c) is in d -perf, c is also in x -perf.

We now show that c(a, t) ∈ x(a, t) -perf. Consider an x-resolution x̃ of c and let j be the largest positive integer such that some x_{k} ∈ add x appears in x̃ as x_{k}⟨−j⟩. Then c⟨j⟩ has a resolution by objects of add x and non-negative shifts of those. Now by the same argument as in Lemma 18 we obtain that c(a, t)⊗_{a} t^{j} is in x(a, t) -perf (c(a, t) ⊗_{a} t^{j} is an iterated extension of some x_{i} ∈ add x(a, t) and some x_{i} ⊗_{a} t^{m} for x_{i} ∈ add x(a, t) and m > 0, the latter being in x(a, t) -perf, since t is a finite complex of projectives). Now t is a tilting complex, hence a is in t -perf, say with a resolution t̃. Then c(a, t) ⊗ t^{j−1} ⊗_{a} t̃ ∈ x(a, t) -perf is a resolution of c(a, t)⊗t^{j−1} ⊗_{a} a (since c(a, t)⊗t^{j−1} ⊗_{a} − is exact) and inductively we finally obtain that c(a, t) is in x(a, t) -perf as desired.

Now we can consider an extension x̃ : 0 → x_{d} → ⋯ → x_{0} → 0 which is quasi-isomorphic to c(a, t) in the 0-component, and where all x_{i} ∈ add x(a, t). If Hom_{D_{dg}(c(a,t))}(x(a, t), Y [i]) = 0 for all i ∈ ℤ for some Y, then by induction on the length of the iterated extension, Hom_{D_{dg}(c(a,t))}(x̃, Y [i]) = 0. Indeed, there is a triangle

x_{d} ⟶ x̃ ⟶ x̃^{≤d−1} ⇝

where x̃^{≤d−1} = (0 → x_{d−1} → ⋯ → x_{0} → 0). From the long exact sequence we get by applying Hom_{D_{dg}(c(a,t))}(−, Y [i]) and the inductive hypothesis that Hom_{D_{dg}(c(a,t))}(x_{d}, Y [i]) = 0 = Hom_{D_{dg}(c(a,t))}(x̃^{≤d−1}, Y [i]), we obtain the claim. But since x̃ is quasi-isomorphic to c(a, t) this implies that Hom_{D_{dg}(c(a,t))}(c(a, t), Y [i]) = 0 for all i ∈ ℤ, hence Y = 0. This shows that x(a, t) generates D_{dg}(c(a, t)).

Proof of Theorem 13.

First note that Lemma 18 and Lemma 19 show that we can apply [14], Theorem 3.10, to see that c(a, t) is derived equivalent to Hom_{c(a,t)}(x(a, t), x(a, t)). Now we proceed to show that the natural map d(a, t) → Hom_{c(a,t)}(x(a, t), x(a, t)) coming from the action of d(a, t) on x(a, t) is a quasi-isomorphism.

Define the vector space Hom_{c}(x, x)(a) as follows:

Hom_{c}^{(j)}(x, x)(a) := \{ \begin{matrix}Hom_{c}^{(j)}(x, x) ⊗ t^{j} & if j ≥ 0 \\ Hom_{c}^{(j)}(x, x) ⊗ Hom(t^{−j}, a) & if j < 0 \end{matrix} .

Note that the graded quasi-isomorphism d → Hom_{c}(x, x) lifts to a quasiisomorphism d(a, t) → Hom_{c}(x, x)(a, t), since tensoring over F with t^{j} is an exact functor.

Now we wish to construct a quasi-isomorphism

β : Hom_{c}(x, x)(a, t) → Hom_{c(a,t)}(x(a, t), x(a, t)).

For an element f_{j} ⊗t_{1} ⋯ t_{j}, or f_{j} ⊗h, where f_{j} ∈ Hom_{c}^{(j)}(x, x), and t_{1} ⋯ t_{j} ∈ t^{j} if j ≥ 0, and where h ∈ Hom_{a}(t^{−j}, a) if j < 0, we define

β(f_{j} ⊗ t_{1} ⋯ t_{j}) = (x_{k} ⊗ t′_{1} ⋯ t′_{k} ↦ f_{j}(x_{k}) ⊗ t′_{1} ⋯ t′_{k}t_{1} ⋯ t_{j}),
β(f_{j} ⊗ h) =

(x_{k} ⊗ t′_{1} ⋯ t′_{k} ↦ \{ \begin{matrix}0 & if − j > k \\ f_{j}(x_{k}) ⊗ t′_{1} ⋯ t′_{k−(−j)}h(t′_{k−(−j)+1} … t′_{k}) & if − j ≤ k \end{matrix} )

The image under β is clearly a collection of endomorphisms of x(a, t) which commute with the action of c(a, t) on the left (since the parts coming from t^{j} or Hom_{a}(t^{j}, a) act from the right and c(a, t) by multiplication on the left). Plugging the iterated version of the explicit map Υ given in Remark 17 into the formula given in Lemma 15, we obtain exactly our map β. Lemma 16 then implies that β is a quasi-isomorphism.

Composing the quasi-isomorphism d(a, t) → Hom_{c}(x, x)(a, t) with β takes d_{l} ⊗ t_{1} ⋯ t_{l} ∈ d(a, t)^{(l)} to

(right action of d_{l}) ⊗ t_{1} ⋯ t_{l} ∈ Hom_{c}^{(l)}(x, x) ⊗ t^{l},

then to

(right action of d_{l}) ⊗ (right multiplication with t_{1} ⋯ t_{l}),

an element of Hom_{c(a,t)}(x(a, t), x(a, t)) This element is also the image of d_{l} ⊗ t_{1} ⋯ t_{l} ∈ d(a, t)^{(l)} under the canonical map γ : d(a, t) → Hom_{c(a,t)}(x(a, t), x(a, t)) coming from the action of d(a, t) on x(a, t). Hence γ is the composition of two quasi-isomorphisms and therefore a quasi-isomorphism itself. This completes the proof of the theorem.

4. The operators 𝕆_{p}.

Recall 𝕆_{p} is the endo-2-functor of T defined by

𝕆_{p}(A, T) = (c_{p}, x_{p}) ∗ \underline{A} = (c_{p}(\underline{A}), x_{p}(\underline{A})).

Here we compare the operator 𝕆_{p} with the operator C_{p}, and consider braid group actions relative to these operators.

Quasi-hereditary algebras were invented by Cline, Parshall, and Scott [7], whilst their tilting theory was developed by Ringel. We refer to Donkin’s book for an account of the general theory [9].

Let U be the sub-2-category of T , consisting of pairs (A, T), where A is a quasi-hereditary algebra and T is a self-dual tilting bimodule.

Proposition 20. The weak 2-category U is stable under C_{p} and 𝕆_{p}. We have 𝕆_{p}(\underline{A}) ≅ C_{p}(\underline{A}), for \underline{A} ∈ U.

Proof. In previous work, we established that U is stable under C_{p} ([19], Corollary 20). It therefore suffices to show that, whenever A is a quasi-hereditary algebra with a self-dual tilting bimodule T,

(c_{p}(A, T), x_{p}(A, T)) ≅ (C_{p}(A), X_{p}(A)).

To see this, we identify the sum of copies of A ⊂ C_{p}(A) with c_{p}^{0} ⊗ A ⊂ c_{p}(A, T); we identify the sum of copies of T ⊂ C_{p}(A) with c_{p}^{(1,0)} ⊗T ⊂ c(A, T); we identify the sum of copies of T^{∗} ⊂ C_{p}(A) with c_{p}^{(0,1)} ⊗ T ⊂ c(A, T) by self-duality; we identify the sum of copies of A^{∗} ⊂ C_{p}(A) with c_{p}^{(1,1)} ⊗T^{2} ⊂ c(A, T), via the dual of the isomorphism

T^{2∗} = Hom_{F} (T ⊗_{A} T, F) ≅ Hom_{A}(T, Hom(T, F)) ≅ Hom_{A}(T, T) ≅ A.

Note the isomorphism Hom_{A}(T, T) ≅ A holds since T is a tilting bimodule for A. By functorality, the resulting isomorphism of vector spaces C_{p}(A) ≅ c(A, T) is an algebra isomorphism. Similarly, we define an isomorphism of bimodules X_{p}(A) ≅ x(A, T), identifying the sum of copies of A ⊂ X_{p}(A) with x_{p}^{(0,0)} ⊗ A ⊂ x(A, T); the sum of copies of T ⊂ X_{p}(A) with x_{p}^{(1,0)} ⊗ T ⊂ x(A, T); the sum of copies of T^{∗} ⊂ X_{p}(A) with x_{p}^{(0,1)} ⊗ T ⊂ x(A, T); and the sum of copies of A^{∗} ⊂ X_{p}(A) with x_{p}^{(1,1)} ⊗ T^{2} ⊂ x(A, T).

Since the pair (F, F) consists of a Ringel self-dual algebra with a self-dual tilting bimodule, upon writing E_{n} for the algebra component E_{n} of 𝕆_{p}^{n}(F, F), we obtain the following corollary:

Corollary 21. The algebra C_{n} is isomorphic to E_{n}.

Lemma 22. Suppose that (A, T) ∈ U, and t ↠ T is a projective bimodule resolution of T. Then the natural algebra homomorphism c_{p}(A, t) ↠ c_{p}(A, T) is a quasi-isomorphism.

Proof. Since all modules appearing in t are projective as bimodules, tensoring with t is exact. Hence t ⊗_{A} t is a resolution of T ⊗_{A} t. But again since all modules appearing in t are projective as bimodules and hence filtered by standard modules, tensoring with T over A is exact on t (see [9]), hence t ⊗_{A} t is a resolution of T ⊗_{A} T. Since c_{p} is concentrated in degrees 0, 1, and 2, we deduce the homology of c_{p}(A, t) is indeed isomorphic to c_{p}(A, T).

Let

Y_{i} : ⋯ 0 → c_{p}e_{i} ⊗ e_{i}c_{p} → c_{p} → 0 ⋯,

where e_{i} is the primitive idempotent corresponding to one of the vertices 1, …, p− 1 in the quiver of c_{p}. By a theorem of Khovanov-Seidel ([15], Proposition 2.4) and Rouquier-Zimmermann [23], tensoring with this complex induces a selfequivalence of D^{b}(c_{p} -mod).

Theorem 23. Let (a, t) ∈ T , where a is an associative algebra, and _{a}t_{a} is a Rickard tilting complex of bimodules. Let 1 ≤ i ≤ p − 1. There is a derived self-equivalence

s_{i} : D_{dg}(c_{p}(a, t)) ⟶^{Y_{i}(a,t)⊗^{L}−} D_{dg}(c_{p}(a, t)).

The derived equivalences s_{i} satisfy braid relations s_{i}s_{i+1}s_{i} ≃ s_{i+1}s_{i}s_{i+1} and s_{i}s_{j} ≃ s_{j}s_{i}, for |i − j| ≥ 1.

Proof. We apply the theory of previous section 3 to c_{p}. The role of the algebra c and d is played by the algebra c_{p} and that of the complex x by the Rickard tilting complex Y_{i}. It follows from Theorem 13, that we have a derived equivalence s_{i} as stated.

We now prove the braid relations. Define Y′_{i} := 0 → c_{p} → c_{p}e_{i} ⊗e_{i}c_{p}⟨−1⟩ → 0, where c_{p} is in homological degree zero. Then Y_{i} ⊗_{c_{p}} Y′_{i} and Y′_{i} ⊗_{c_{p}} Y_{i} are both quasi-isomorphic to c_{p}, as graded complexes (see [15], Proposition 2.4).

Furthermore, Y_{i} ⊗_{c_{p}} Y_{i+1} ⊗_{c_{p}} Y′_{i} is quasi-isomorphic to Y′_{i+1} ⊗_{c_{p}} Y_{i} ⊗_{c_{p}} Y_{i+1}, again as graded complexes (see the proof [15], Theorem 2.5). From this it follows that Y_{i} ⊗_{c_{p}} Y_{i+1} ⊗_{c_{p}} Y_{i} is quasi-isomorphic to Y_{i+1} ⊗_{c_{p}} Y_{i} ⊗_{c_{p}} Y_{i+1} (by tensoring both complexes on the right with Y_{i} and on the left with Y_{i+1}). Now note that, in the setup of Definition 12, it is clear that for two bimodules x and y, x(a, t) ⊗_{c(a,t)} y(a, t) is isomorphic to (x ⊗_{c} y)(a, t). Therefore s_{i}s_{i+1}s_{i} is represented by (Y_{i} ⊗ Y_{i+1} ⊗ Y_{i})(a, t), and s_{i+1}s_{i}s_{i+1} is represented similarly. Now Lemma 14 implies the Theorem.

Corollary 24. If (A, T) ∈ U, then the derived category D^{b}(c_{p}(A, T) -mod) admits a weak braid group action.

Proof. This follows from Theorem 23. Let t denote a projective bimodule resolution of _{A}T_{A}. Since A is quasi-hereditary, it has finite global dimension [7]. Therefore, since T is a tilting module, t is a Rickard tilting complex. We have quasi-isomorphism c_{p}(A, t) → c_{p}(A, T) by Lemma 22, which induces an equivalence between D(c_{p}(A, T) -mod) and D_{dg}(c_{p}(A, T)) (see [13], Proposition 6.2). The self-equivalence s_{i} therefore defines a self-equivalence s_{i} of D(c_{p}(A, T) -mod), under which compact objects map to compact objects. Since compact objects in D(c_{p}(A, T) -mod) can be identified with c_{p}(A, T) -perf, by an observation of Rickard, which can in turn be identified with D^{b}(c_{p}(A, T) -mod) because c_{p}(A, T) has finite global dimension, the self-equivalence s_{i} restricts to a self-equivalence of D^{b}(c_{p}(A, T) -mod). By Theorem 23, these equivalences satisfy braid relations, defining a weak braid group action on D^{b}(c_{p}(A, T) -mod).

Example 25 As an example, let us describe the dg-algebra c_{2}(a, t) where a ≅ c_{2}, and t is a resolution of x_{2}. The algebra c_{2} is a 5-dimensional algebra with basis {e_{1}, e_{2}, ξ, η, ξη}, where the underlying quiver is

\xymatrix{•^{1} \ar[r]^{ξ} & •^{2}. \ar[l]^{η}}

The idempotents are in degree 0, ξ and η are in degree 1 and ξη is therefore in degree 2. The tilting module of c_{2} is given by c_{2}e_{1} ⊕ c_{2}e_{1}/ Rad c_{2}e_{1}. This is isomorphic to the cone of a complex

t : ⋯ 0 → c_{2} → c_{2}e_{1} ⊗ e_{1}c_{2} → 0...,

which is obviously projective on both sides. To obtain c_{2}(c_{2}, t) from c_{2}, we replace the idempotents e_{1} and e_{2} with copies of c_{2}, we replace ξ and η with copies of the complex t and ξη with t ⊗_{c_{2}} t. A dg bimodule representing the self-equivalence s_{1} of D_{dg}(c_{2}(c_{2}, t)) is given by the total complex of the complex of complexes

...0 → c_{2}(c_{2}, t)(e_{1} ⊗ 1) ⊗_{c_{2}} (e_{1} ⊗ 1).c_{2}(c_{2}, t) → c_{2}(c_{2}, t) → 0....

5. Applications to the representation theory of GL_{2}.

We gather here a number of results concerning the representations of GL_{2}, which follow from our theory.

Recall the category of polynomial representations of GL_{2} is equivalent to the category of finite dimensional modules over the Schur algebra S(2) [11].

A ℤ_{+}-grading is said to be tight if its degree zero component is semisimple.

Theorem 26. The Schur algebra S(2) admits a tight ℤ_{+}-grading.

Proof. Every block of the Schur algebra S(2) is Morita equivalent to the quotient of a block B_{n} by an ideal B_{n}iB_{n}, for some n ≥ 0 and some idempotent i. Since A_{n} is tightly graded by definition, B_{n} is tightly graded by Theorem 1. The theorem follows.

Recall E_{n} was defined to be the algebra component of 𝕆_{p}^{n}(F, F). As we have a natural algebra homomorphism from c_{p} to F, which sends vertices 2, .., p to zero, and vertex 1 to 1_{F}, we have an inherited algebra homomorphism from the algebra component of 𝕆_{p}(A, T) to A, for any (A, T) ∈ T . We thus have a directed sequence of algebra homomorphisms.

E_{0} ← E_{1} ← E_{2} ← ...

The operator 𝕆_{p} controls the rational representation theory of GL_{2}:

Theorem 27. Every block of rational representations of GL_{2} is equivalent to the category of lim_{n} E_{n}-modules.

Proof. As we noted in a previous article, every block of representations of GL_{2} is equivalent to the category of lim_{n} B_{n}-modules [19]. By Theorem 1 and Corollary 21, we have B_{n} ≅ E_{n}. The theorem follows.

Theorem 28. The derived category of every block of the Schur algebra with p^{r} irreducible modules admits a weak action of the braid group Br_{p−1}.

Proof. Every block of the Schur algebra with p^{r} irreducible modules is Morita equivalent to B_{n}, hence to E_{n} by Theorem 1. But E_{n} is the algebra component c_{p}(A, T) of 𝕆_{p}(A, T), for some (A, T) ∈ U, by Proposition 20. Therefore, the derived category of E_{n} admits a weak braid group action, by Corollary 24

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