Cell 2-representations of finitary 2-categories
aa r X i v : . [ m a t h . R T ] N ov CELL -REPRESENTATIONS OF FINITARY -CATEGORIES VOLODYMYR MAZORCHUK AND VANESSA MIEMIETZ
Abstract.
We study 2-representations of finitary 2-categories with involutionand adjunctions by functors on module categories over finite dimensional al-gebras. In particular, we define, construct and describe in detail (right) cell2-representations inspired by Kazhdan-Lusztig cell modules for Hecke alge-bras. Under some natural assumptions we show that cell 2-representations arestrongly simple and do not depend on the choice of a right cell inside a two-sided cell. This reproves and extends the uniqueness result on categorificationof Kazhdan-Lusztig cell modules for Hecke algebras of type A from [MS]. Introduction and description of the results
The philosophy of categorification, which originated in work of Crane and Frenkel(see [Cr, CF]) some fifteen years ago, is nowadays usually formulated in termsof 2-categories. A categorification of an algebra (or category) A is now usuallyunderstood as a 2-category A , whose decategorification is A . Therefore a naturalproblem is to “upgrade” the representation theory of A to a 2-representation theoryof A . The latter philosophy has been propagated by Rouquier in [Ro1, Ro2] basedon the earlier development in [CR].Not much is known about the 2-category of 2-representations of an abstract 2-category. Some 2-representations of 2-categories categorifying Kac-Moody algebraswere constructed and studied in [Ro2]. On the other hand, there are many examplesof 2-representations of various 2-categories in the literature, sometimes without anexplicit emphasis on their 2-categorical nature, see for example [Kv, St, KMS, MS,KhLa] and references therein. A different direction of the representation theory ofcertain classes of 2-categories was investigated in [EO, EGNO].2-categorical philosophy also appears, in a disguised form, in [Kh]. In this articlethe author defines so-called “categories with full projective functors” and consid-ers “functors naturally commuting with projective functors”. The former can beunderstood as certain “full” 2-representations of a 2-category and the latter asmorphisms between these 2-representations.The aim of the present article is to look at the study of 2-representations of abstract2-categories from a more systematic and more abstract prospective. Given analgebra A there are two natural ways to construct A -modules. The first way is tofix a presentation for A and construct A -modules using generators and checkingrelations. The second way is to look at homomorphisms between free A -modulesand construct their cokernels. Rouquier’s approach to 2-representation theory from[Ro1, Ro2] goes along the first way. In the present article, we try the second one.Our main object of study is what we call a fiat category C , that is a (strict) 2-category with involution which has finitely many objects, finitely many isomor-phism classes of indecomposable 1-morphisms, and finite dimensional spaces of 2-morphisms that are also supposed to contain adjunction morphisms. Our 2-setup isdescribed in detail in Section 2. In Section 3 we study principal -representations offiat categories, which are analogues of indecomposable projective modules over an Date : May 2, 2019. algebra. We give an explicit construction of principal 2-representations and provea natural analogue of the universal property for them. Adding up all principalrepresentations we obtain the regular C -bimodule, which gives rise to an abelian2-category ˆ C enveloping the original category C . The category ˆ C is no longer fiat,but has the advantage of being abelian. We show that every 2-representation of C extends to a 2-representation of ˆ C in a natural way.Inspired by Kazhdan-Lusztig combinatorics (see [KaLu]), in Section 4 we define,for every fiat category C , the concepts of left, right and two-sided cells and cell 2-representations associated with right cells. We expect cell representations to be themost natural candidates for “simple” 2-representations, whatever the latter couldpossibly mean (which is still unclear). We describe the algebraic structure of modulecategories on which a cell 2-representation operates and determine homomorphismsfrom a cell 2-representation. We also study in detail the combinatorial structure oftwo natural classes of cells, which we call regular and strongly regular. These turnout to have particularly nice properties and appear in many natural examples.Section 5 is devoted to the study of the local structure of cell 2-representations. Weshow that the essential part of cell 2-representations is governed by the action of1-morphisms from the associated two-sided cell and describe algebraic propertiesof cell 2-representations in terms of the cell combinatorics of this two-sided cell.In Section 6 we define and study the notions of cyclicity and strong simplicity for2-representations. A 2-representation is called cyclic if it is generated, in the 2-categorical sense of categories with full projective functors in [Kh], by some object M . This means that the natural map from ˆ C to our 2-representation, sendingF to F M is essentially surjective on objects and surjective on morphisms. A 2-representation is called strongly simple if it is generated, in the 2-categorical sense,by any simple object. We show that all cell 2-representations are cyclic and provethe following main result: Theorem 1.
Let C be a fiat category. Then, under some natural technical as-sumptions, every cell -representation of C associated with a strongly regular rightcell is strongly simple. Moreover, under the same assumptions, every two cell -representations of C associated with strongly regular right cells inside the sametwo-sided cell are equivalent. Finally, in Section 7 we give several examples. The prime example is the fiat cate-gory of projective functors acting on the principal block (or a direct sum of some,possibly singular, blocks) of the BGG category O for a semi-simple complex finitedimensional Lie algebra. This example is given by Kazhadan-Lusztig combinatoricsand our cells coincide with the classical Kazhdan-Lusztig cells. As an applicationof Theorem 1 we reprove, extend and strengthen the uniqueness result on cate-gorification of Kazhdan-Lusztig cell modules for Hecke algebras of type A from[MS]. We also present another example of a fiat category C A given by projectiveendofunctors of the module category of a weakly symmetric self-injective finite di-mensional associative algebra A . We show that the latter example is “universal” inthe sense that, under the same assumptions as mentioned in Theorem 1, every cell2-representation of a fiat category gives rise to a 2-functor to some C A . Acknowledgments.
The first author was partially supported by the SwedishResearch Council. A part of this work was done during a visit of the second authorto Uppsala University, which was supported by the Faculty of Natural Science ofUppsala University. The financial support and hospitality of Uppsala Universityare gratefully acknowledged. We thank Catharina Stroppel and Joseph Chuang forstimulating discussions.
ELL 2-REPRESENTATIONS 3
2. 2 -setup
Notation.
For a 2-category C , objects of C will be denoted by i , j and so on.For i , j ∈ C , objects of C ( i , j ) (1-morphisms of C ) will be called F , G and so on.For F , G ∈ C ( i , j ), morphisms from F to G (2-morphisms of C ) will be written α, β and so on. The identity 1-morphism in C ( i , i ) will be denoted i and the identity2-morphism from F to F will be denoted id F . Composition of 1-morphisms will bedenoted by ◦ , horizontal composition of 2-morphisms will be denoted by ◦ andvertical composition of 2-morphisms will be denoted by ◦ . We often abbreviateid F ◦ α and α ◦ id F by F( α ) and α F , respectively.For the rest of the paper we fix an algebraically closed field k . As we will oftenconsider categories C ( i , j ), we will denote the morphism space between X and Y insuch a category by Hom C ( i , j ) ( X, Y ) to avoid the awkward looking C ( i , j )( X, Y ).2.2.
Finitary -categories and -representations. In what follows, by a 2 -category we always mean a strict bicategory for thecorresponding non-strict structure. Note that any bicategory is biequivalent to a2-category (see, for example, [Le, 2.3]).We define a 2-category C to be k -finitary provided that(I) C has finitely many objects;(II) for every i , j ∈ C the category C ( i , j ) is a fully additive (i.e. karoubian) k -linear category with finitely many isomorphism classes of indecomposableobjects, moreover, horizontal composition of 1-morphisms is biadditive;(III) for every i ∈ C the object i ∈ C ( i , i ) is indecomposable.From now on C will always be a k -finitary 2-category.Denote by R k the 2-category whose objects are categories equivalent to module cat-egories of finite-dimensional k -algebras, 1-morphisms are functors between objects,and 2-morphisms are natural transformations of functors. We will understand a2 -representation of C to be a strict 2-functor from C to R k . By [Le, 2.0], 2-representations of C , together with strict 2-natural transformations (i.e. morphismsbetween 2-representation, given by a collection of functors) and modifications (i.e.morphisms between strict 2-natural transformations, given by natural transforma-tions between the defining functors), form a strict 2-category, which we denote by C - mod . For simplicity we will identify objects in C ( i , j ) with their images under a2-representation (i.e. we will use module notation). Example 2.
Consider the algebra D := C [ x ] / ( x ) of dual numbers. It is easyto check that the endofunctor F := D ⊗ C − of D -mod satisfies F ◦ F ∼ = F ⊕ F.Therefore one can consider the 2-category S defined as follows: S has one object i := D -mod; 1-morphisms of S are all endofunctors of i which are isomorphicto a direct sum of copies of F and the identity functor; 2-morphisms of S are allnatural transformations of functors. The category S is a C -finitary 2-category. Itcomes together with the natural representation (the embedding of S into R C ).2.3. Path categories associated to C ( i , j ) . For i , j ∈ C let F , F , . . . , F r be acomplete list of pairwise non-isomorphic indecomposable objects in C ( i , j ). Denoteby C i , j the full subcategory of C ( i , j ) with objects F , F , . . . , F r . As C is k -finitary,the path algebra of C i , j is a finite dimensional k -algebra. There is a canonicalequivalence between the category C op i , j -mod and the category of modules over thepath algebra of C i , j . Example 3.
For the category S from Example 2 the category S ( i , i ) has twoindecomposable objects, namely i and F. Realizing exact functors on D -modas D -bimodules, the functor i corresponds to the bimodule D and the functorF corresponds to the bimodule D ⊗ C D . Let α : D ⊗ C D → D be the unique VOLODYMYR MAZORCHUK AND VANESSA MIEMIETZ morphism such that 1 ⊗ β : D → D ⊗ C D be the unique morphism suchthat 1 ⊗ x + x ⊗
1; and γ : D ⊗ C D → D ⊗ C D be the unique morphism suchthat 1 ⊗ ⊗ x − x ⊗
1. Then it is easy to check that the category C i , i is givenby the following quiver and relations: • α * * γ $ $ • β j j , γ = − ( βα ) , ( αβ ) = 0 ,αγ = γβ = 0 . -categories with involution. If C is a k -finitary 2-category, then an involu-tion on C is a lax involutive object-preserving anti-automorphism ∗ of C . A finitary2-category C with involution ∗ is said to have adjunctions provided that for any i , j ∈ C and any 1-morphism F ∈ C ( i , j ) there exist 2-morphisms α : F ◦ F ∗ → j and β : i → F ∗ ◦ F such that α F ◦ F( β ) = id F and F ∗ ( α ) ◦ β F ∗ = id F ∗ . A k -finitary2-category with an involution and adjunctions will be called a fiat category . Example 4.
The category S from Example 2 is easily seen to be a fiat category.3. Principal -representations -representations P i . Let C be a finitary 2-category. For i , j ∈ C denoteby C ( i , j ) the category defined as follows: Objects of C ( i , j ) are diagrams of theform F α / / G , where F , G ∈ C ( i , j ) are 1-morphisms and α is a 2-morphism.Morphisms of C ( i , j ) are equivalence classes of diagrams as given by the solid partof the following picture:F α / / β (cid:15) (cid:15) G β ′ (cid:15) (cid:15) ξ x x F ′ α ′ / / G ′ , F , F ′ , G , G ′ ∈ C ( i , j ) , modulo the ideal generated by all morphisms for which there exists ξ as shown bythe dotted arrow above such that α ′ ξ = β ′ . As C is finitary category, the category C ( i , j ) is abelian and equivalent to C op i , j -mod, see [Fr].For i ∈ C define the 2-functor P i : C → R k as follows: for j ∈ C set P i ( j ) = C ( i , j ). Further, for k ∈ C and F ∈ C ( j , k ) left horizontal composition with (theidentity on) F defines a functor from C ( i , j ) to C ( i , k ). We define this functor to be P i (F). Given a 2-morphism α : F → G, left horizontal composition with α gives anatural transformation from P i (F) to P i (G). We define this natural transformationto be P i ( α ). From the definition it follows that P i is a strict 2-functor from C to R k . The 2-representation P i is called the i -th principal C .For i , j ∈ C and a 1-morphism F ∈ C ( i , j ) we denote by P F the projective object0 → F of C ( i , j ).3.2. The universal property of P i .Proposition 5. Let M be a -representation of C and M ∈ M ( i ) . ( a ) For j ∈ C define the functor Φ M j : C ( i , j ) → M ( j ) as follows: Φ M j sendsa diagram F α / / G in C ( i , j ) to the cokernel of M ( α ) M . Then Φ M =(Φ M j ) j ∈ C is the unique morphism from P i to M sending P i to M . ( b ) The correspondence M Φ M is functorial.Proof. Claim (a) follows directly from 2-functoriality of M . To prove claim (b) let f : M → M ′ . Choose now any F , G ∈ C ( i , j ) and α : F → G. Applying M to ELL 2-REPRESENTATIONS 5 F α −→ G gives M (F) M ( α ) −→ M (G). Applying the latter to M f −→ M ′ yields thecommutative diagram M (F) M M (F) f / / M ( α ) M (cid:15) (cid:15) M (F) M ′ M ( α ) M ′ (cid:15) (cid:15) M (G) M M (G) f / / M (G) M ′ . This commutative diagram implies that { M (F) f : F ∈ C ( i , j ) } extends to a naturaltransformation from Φ M j to Φ M ′ j and claim (b) follows. (cid:3) Connections to categories with full projective functors.
Denote by C i the full 2-subcategory of C with object i . Restricting P i to i defines a (unique)principal 2-representation of C i . As C is finitary, the identity i is indecomposableand hence so is the projective object P i . By definition, for any F , G ∈ C i ( i , i )the evaluation mapHom C i ( i , i ) (F , G) → Hom C ( i , i ) (F ◦ P i , G ◦ P i )is surjective (and, in fact, even bijective). Therefore the category C ( i , i ) withthe designated object P i and endofunctors P i (F), F ∈ C i ( i , i ), is a categorywith full projective functors in the sense of [Kh]. The notion of functors naturallycommuting with projective functors in [Kh] corresponds to morphisms between 2-representations of C i in our language. It might be worth pointing out that [Kh]works in the setup of bicategories (without mentioning them).Similarly, for every j ∈ C and any F , G ∈ C ( i , j ) the evaluation mapHom C ( i , j ) (F , G) → Hom C ( i , j ) (F ◦ P i , G ◦ P i )is surjective (and, in fact, even bijective).3.4. The regular bimodule.
For i , j , k ∈ C and any 1-morphism F ∈ C ( k , i ) theright horizontal composition with (the identity on) F gives a functor from C ( i , j )to C ( k , j ). For any 1-morphisms F , G ∈ C ( k , i ) and a 2-morphism α : F → Gthe right horizontal composition with α gives a natural transformation between thecorresponding functors. This turns C ( · , · ) into a 2-bimodule over C . This bimoduleis called the regular bimodule .3.5. The abelian envelope of C . Because of the previous subsection, it is naturalto expect that one could turn C into a 2-category with the same set of objects as C . Unfortunately, we do not know how to do this as it seems that C contains “toomany” objects (and hence only has the natural structure of a bicategory). Instead,we define a biequivalent 2-category ˆ C as follows: Objects of ˆ C are objects of C .To define 1-morphisms of ˆ C consider the regular 2-bimodule C ( · , · ) over C just asa left 2-representation. Let R be the 2-category with same objects as C and suchthat for i , j ∈ C the category R ( i , j ) is defined as the category of all functorsfrom M k ∈ C C ( k , i ) to M k ∈ C C ( k , j ), where morphisms are all natural transformationsof functors. We are going to define ˆ C as a 2-subcategory of R .The regular bimodule 2-representation of C is a 2-functor from C to R (which is theidentity on objects). As usual, for every i , j ∈ C and any F ∈ C ( i , j ) we will denotethe image of F under this 2-functor also by F. We define 1-morphisms in ˆ C ( i , j ) asfunctors in R ( i , j ) of the form Coker( α ), where α is a 2-morphism from F to G forsome F , G ∈ C ( i , j ). We define 2-morphisms in ˆ C ( i , j ) as natural transformations VOLODYMYR MAZORCHUK AND VANESSA MIEMIETZ between the corresponding cokernel functors coming from commutative diagramsof the following form, where all solid arrows are 2-morphisms in C :F α / / ξ ′ (cid:15) (cid:15) G ξ (cid:15) (cid:15) proj / / / / Coker( α ) (cid:15) (cid:15) F ′ α ′ / / G ′ proj / / / / Coker( α ′ ) Lemma 6. ( a ) 1 -morphisms in ˆ C are closed with respect to the usual compositionof functors in R . ( b ) 2 -morphisms in ˆ C are closed with respect to both horizontal and vertical com-positions in R .Proof. Let i , j , k ∈ C , F , G ∈ C ( i , j ), F ′ , G ′ ∈ C ( j , k ) and F α / / G ,F ′ α ′ / / G ′ be some 2-morphisms. Then the interchange law for the 2-category C yields that the following diagram is commutative:F ′ ◦ F F ′ ( α ) / / α ′ F (cid:15) (cid:15) F ′ ◦ G α ′ G (cid:15) (cid:15) G ′ ◦ F G ′ ( α ) / / G ′ ◦ GThis means that Coker( α ′ ) ◦ Coker( α ) = Coker(( α ′ G , G ′ ( α ))) , where ( α ′ G , G ′ ( α )) is given by the following diagram:(F ′ ◦ G) ⊕ (G ′ ◦ F) ( α ′ G , G ′ ( α )) / / G ′ ◦ G . This implies claim (a).That 2-morphisms are closed with respect to vertical composition follows directlyfrom the definitions. To see that 2-morphisms are closed with respect to horizontalcomposition, consider the following two commutative diagrams in C :F α / / ξ (cid:15) (cid:15) G η (cid:15) (cid:15) F ′ α ′ / / G ′ and F β / / ξ (cid:15) (cid:15) G η (cid:15) (cid:15) F ′ β ′ / / G ′ These diagrams induce 2-morphisms between the corresponding cokernels. Thehorizontal composition of these two morphisms is induced by the following commu-tative diagram: F ◦ G ⊕ G ◦ F α G2 , G ( β )) / / ξ ◦ η η ◦ ξ (cid:15) (cid:15) G ◦ G η ◦ η (cid:15) (cid:15) F ′ ◦ G ′ ⊕ G ′ ◦ F ′ α ′ G ′ , G ′ ( β ′ )) / / G ′ ◦ G ′ . This proves claim (b) and completes the proof. (cid:3)
ELL 2-REPRESENTATIONS 7
From Lemma 6 it follows that ˆ C is a 2-subcategory of R . From the construction italso follows that for any i , j ∈ C the categories C ( i , j ) and ˆ C ( i , j ) are equivalent.Furthermore, directly from the definitions we have: Lemma 7.
There is a unique full and faithful -functor i : C → ˆ C such that forany i , j ∈ C , F , G ∈ C ( i , j ) and α : F → G we have i (F) = Coker( 0 / / F ) and i ( α ) is induces by / / (cid:15) (cid:15) F α (cid:15) (cid:15) / / G . As usual, the 2-functor i induces the restriction 2-functor ˆ i : ˆ C - mod → C - mod . Forthe opposite direction we have: Theorem 8.
Every -representation of C extends to a -representation of ˆ C .Proof. Let M ∈ C - mod . Abusing notation we will denote the extension of M toa 2-representation of ˆ C also by M . Let i , j ∈ C , F , G ∈ C ( i , j ) and α : F → G.Then for Coker( α ) ∈ ˆ C ( i , j ) we define M (Coker( α )) as Coker( M ( α )).To define M on 2-morphisms in ˆ C , let F ′ , G ′ ∈ C ( i , j ), α ′ : F ′ → G ′ , β : F → F ′ and β ′ : G → G ′ are such that the diagramΓ := F α / / β (cid:15) (cid:15) G β ′ (cid:15) (cid:15) F ′ α ′ / / G ′ is commutative. Then a typical 2-morphism γ in ˆ C is induced by Γ. Applying M induces the commutative solid part of the following diagram: M (F) M ( α ) / / M ( β ) (cid:15) (cid:15) M (G) M ( β ′ ) (cid:15) (cid:15) proj / / ______ Coker( M ( α )) ξ (cid:15) (cid:15) (cid:15) (cid:15) (cid:31)(cid:31)(cid:31) M (F ′ ) M ( α ′ ) / / M (G ′ ) proj / / / / ______ Coker( M ( α ′ ))Because of the commutativity of the solid part, the diagram extends uniquely toa commutative diagram by the dashed arrows as shown above. Directly from theconstruction it follows that M becomes a 2-representation of ˆ C . (cid:3) Because of Theorem 8 it is natural to call ˆ C the abelian envelope of C . In whatfollows we will always view 2-representation of C as 2-representation of ˆ C via theconstruction given by Theorem 8.4. Cells and cell -representations of fiat categories From now on we assume that C is a fiat category.4.1. Orders and cells.
Set C = ∪ i , j C i , j . Let i , j , k , l ∈ C , F ∈ C i , j and G ∈ C k , l .We will write F ≤ R G provided that there exists H ∈ C ( j , l ) such that G occurs asa direct summand of H ◦ F (note that this is possible only if i = k ). Similarly, wewill write F ≤ L G provided that there exists H ∈ C ( k , i ) such that G occurs as adirect summand of F ◦ H (note that this is possible only if j = l ). Finally, we willwrite F ≤ LR G provided that there exists H ∈ C ( k , i ) and H ∈ C ( j , l ) such thatG occurs as a direct summand of H ◦ F ◦ H . The relations ≤ L , ≤ R and ≤ LR arepartial preorders on C . The map F F ∗ preserves ≤ LR and swaps ≤ L and ≤ R . VOLODYMYR MAZORCHUK AND VANESSA MIEMIETZ
For F ∈ C the set of all G ∈ C such that F ≤ R G and G ≤ R F will be called the right cell of F and denoted by R F . The left cell L F and the two-sided cell LR F are defined analogously. We will write F ∼ R G provided that G ∈ R F and define ∼ L and ∼ LR analogously. These are equivalence relations on C . If F ≤ L G andF L G, then we will write F < L G and similarly for < R and < LR . Example 9.
The 2-category S from Example 2 has two right cells { i } and { F } ,which are also left cells and thus two-sided cells as well.4.2. Annihilators and filtrations.
Let M be a 2-representation of C . For any i ∈ C and any M ∈ M ( i ) consider the annihilator Ann C ( M ) := { F ∈ C : F M = 0 } of M . The set Ann C ( M ) is a coideal with respect to ≤ R in the sense that F ∈ Ann C ( M )and F ≤ R G implies G ∈ Ann C ( M ). The annihilator Ann C ( M ) := \ M Ann C ( M ) of M is a coideal with respect to ≤ LR .Let I be a coideal in C with respect to ≤ LR . For every i ∈ C denote by M I ( i )the Serre subcategory of M ( i ) generated by all simple modules L such that I ⊂
Ann C ( L ). Lemma 10.
By restriction, M I is a -representation of C .Proof. We need to check that M I is stable under the action of elements from C . If L is a simple module in M I ( i ) and F ∈ C i , j , then for any G ∈ I the 1-morphismG ◦ F is either zero or decomposes into a direct sum of 1-morphisms in I (as I isa coideal with respect to ≤ LR ). This implies G ◦ F L = 0. Exactness of G impliesthat G K = 0 for any simple subquotient K of F L . The claim follows. (cid:3) Assume that for any i ∈ C we fix some Serre subcategory N ( i ) in M ( i ) suchthat for any j ∈ C and any F ∈ C ( i , j ) we have F N ( i ) ⊂ N ( j ). Then N ( i ) is a2-representation of C by restriction. It will be called a Serre -subrepresentation of M . For example, the 2-representation M I constructed in Lemma 10 is a Serre2-subrepresentation of M . Proposition 11. ( a ) For any coideal I in C with respect to ≤ LR we have M I = M Ann C ( M I ) . ( b ) For any Serre -subrepresentation N of M we have Ann C ( N ) = Ann C ( M Ann C ( N ) ) . Proof.
We prove claim (b). Claim (a) is proved similarly. By definition, for every i ∈ C we have N ( i ) ⊂ M Ann C ( N ) ( i ). This implies Ann C ( M Ann C ( N ) ) ⊂ Ann C ( N ).On the other hand, by definition Ann C ( N ) annihilates M Ann C ( N ) , so Ann C ( N ) ⊂ Ann C ( M Ann C ( N ) ). This completes the proof. (cid:3) Proposition 11 says that
I 7→ M I and N Ann C ( N ) is a Galois correspondencebetween the partially ordered set of coideals in C with respect to ≤ LR and thepartially ordered set of Serre 2-subrepresentations of M with respect to inclusions.4.3. Annihilators in principal -representations. Let i ∈ C . By construction,for j ∈ C isomorphism classes of simple modules in P i ( j ) are indexed by C i , j . ForF ∈ C i , j we denote by L F the unique simple quotient of P F . Lemma 12.
For F , G ∈ C the inequality F L G = 0 is equivalent to F ∗ ≤ L G .Proof. Without loss of generality we may assume G ∈ C i , j and F ∈ C j , k . ThenF L G = 0 if and only if there is H ∈ C i , k such that Hom C ( i , k ) ( P H , F L G ) = 0. Using P H = H P i and adjunction we obtain0 = Hom C ( i , k ) ( P H , F L G ) = Hom C ( i , j ) (F ∗ ◦ H P i , L G ) . ELL 2-REPRESENTATIONS 9
This inequality is equivalent to the claim that P G = G P i is a direct summand ofF ∗ ◦ H P i , that is G is a direct summand of F ∗ ◦ H. The claim follows. (cid:3)
Lemma 13. ( a ) For F , G , H ∈ C the inequality [F L G : L H ] = 0 implies H ≤ R G . ( b ) For G , H ∈ C such that H ≤ R G there exists F ∈ C such that [F L G : L H ] = 0 .Proof. Without loss of generality we may assume(1) G ∈ C i , j , F ∈ C j , k and H ∈ C i , k . Then [F L G : L H ] = 0 is equivalent to Hom C ( i , k ) ( P H , F L G ) = 0. Similarly toLemma 12 we obtain that G must be a direct summand of F ∗ ◦ H. This means thatH ≤ R G, proving (a).To prove (b) we note that H ≤ R G implies existence of F ∈ C such that G is adirect summand of F ∗ ◦ H. We may assume that F , G and H are as in (1). Then,by adjunction, we have0 = Hom C ( i , k ) (F ∗ P H , L G ) = Hom C ( i , j ) ( P H , F L G ) , which means that [F L G : L H ] = 0. This completes the proof. (cid:3) Corollary 14.
Let F , G , H ∈ C . If L F occurs in the top or in the socle of H L G ,then F ∈ R G .Proof. We prove the claim in the case when L F occurs in the top of H L G , the othercase being analogous. As [H L G : L F ] = 0, we have F ≤ R G by Lemma 13. On theother hand, by adjunction, L G occurs in the socle of H ∗ L F . Hence [H ∗ L F : L G ] = 0and thus we have G ≤ R F by Lemma 13. The claim follows. (cid:3)
Lemma 15.
For any F ∈ C i , j there is a unique (up to scalar) nontrivial homo-morphism from P i to F ∗ L F . In particular, F ∗ L F = 0 .Proof. Adjunction yieldsHom C ( i , i ) ( P i , F ∗ L F ) = Hom C ( i , j ) ( P F , L F ) ∼ = k and the claim follows. (cid:3) Serre -subrepresentations of P i . Let I be an ideal in C with respect to ≤ R , i.e. F ∈ I and F ≥ R G implies G ∈ I . For i , j ∈ C define P I i ( j ) as theSerre subcategory of P i ( j ) generated by L F for F ∈ C i , j ∩ I . Then from Lemma 13it follows that P I i is a Serre 2-subrepresentation of P i and that every Serre 2-subrepresentation of P i arises in this way. For F ∈ I ∩ C i , j we denote by P I F themaximal quotient of P F in P I i ( j ). The module P I F is a projective cover of L F in P I i ( j ). Since P I i is a 2-subrepresentation of P i , for F ∈ C we haveF P I i = ( P I F , F ∈ I ;0 , otherwise . From the definition we have that 2-morphisms in C surject onto homomorphismsbetween the various P I F . The natural inclusion i I : P I i → P i is a morphism of2-representations, given by the collection of exact inclusions i I j : P I i ( j ) → P i ( j ).Note that C \ I is a coideal in C with respect to ≤ R . Hence for any 2-representation M of C we have the corresponding Serre 2-subrepresentation M C\I of M . Proposition 16 (Universal property of P I i ) . Let M a -representation of C . ( a ) For any morphism
Φ : P I i → M we have Φ( P I i ) ∈ M C\I ( i ) . ( b ) Let M ∈ M C\I ( i ) . For j ∈ C let Φ M j : P I i ( j ) → M ( j ) be the unique rightexact functor such that for any F ∈ C ( i , j ) we have Φ M j : P I F M (F) M. Then Φ M = (Φ M j ) j ∈ C : P i → M is the unique morphism sending P I F to M . ( c ) The correspondence M Φ M is functorial.Proof. Claim (a) follows from the fact that
C \ I ⊂
Ann C ( P I i ). Mutatis mutandis,the rest is Proposition 5. (cid:3) Right cell -representations. Fix i ∈ C . Let R be a right cell in C suchthat R ∩ C i , j = ∅ for some j ∈ C . Proposition 17. ( a ) There is a unique submodule K = K R of P i which has thefollowing properties: ( i ) Every simple subquotient of P i /K is annihilated by any F ∈ R . ( ii ) The module K has simple top L G R for some G R ∈ C and F L G R = 0 forany F ∈ R . ( b ) For any F ∈ R the module F L G R has simple top L F . ( c ) We have G R ∈ R . ( d ) For any F ∈ R we have F ∗ ≤ L G R and F ≤ R G ∗R . ( e ) We have G ∗R ∈ R .Proof. Let F ∈ R . Let further j ∈ C be such that F ∈ R ∩ C i , j . Then themodule F P i is a nonzero indecomposable projective in C ( i , j ). Hence F does notannihilate P i and thus there is at least one simple subquotient of P i which isnot annihilated by F. Let K be the minimal submodule of P i such that everysimple subquotient of P i /K is annihilated by F. As Ann C ( P i /K ) is a coidealwith respect to ≤ R , the module P i /K is annihilated by every G ∈ R . Similarly,we have that for any simple subquotient L in the top of K and for any G ∈ R wehave G L = 0. This implies that K does not depend on the choice of F ∈ R . Then(ai) is satisfied and to complete the proof of (a) we only have to show that K hassimple top.Applying F to the exact sequence K ֒ → P i ։ P i /K we obtain the exact sequenceF K ֒ → F P i ։ F P i /K. As F P i /K = 0, we see that F K ∼ = F P i is an indecomposable projective andhence has simple top. Applying F to the exact sequence rad K ֒ → K ։ top K weobtain the exact sequence F rad K ֒ → F K ։ F top K. As F K has simple top by the above, we obtain that F top K has simple top. Byconstruction, top K is semi-simple and none of its submodules are annihilated byF. Therefore top K is simple, which implies (a) and also (b).For F ∈ R , the projective module P F surjects onto the nontrivial module F L G R by the above. Hence L F occurs in the top of F L G R and thus (c) follows fromCorollary 14. For F ∈ R we have F L G R = 0 and hence (d) follows from Lemma 12.From (d) we have G R ≤ R G ∗R . Assume that G ∗R
6∈ R and let ˜ R be the right cellcontaining G ∗R . By Lemma 15 we have G ∗R L G R = 0, which implies that K R ⊂ K ˜ R .If K R = K ˜ R , then L G R = L G ˜ R and hence R = ˜ R , which implies (e). If K R ( K ˜ R ,then from (ai) we have G R L G ˜ R = 0. As Ann C ( L G ˜ R ) is a coideal with respect to ≤ R , it follows that L G ˜ R is annihilated by G ∗R . This contradicts (aii) and hence (e)follows. The proof is complete. (cid:3) ELL 2-REPRESENTATIONS 11
For simplicity we set L = L G R . For j ∈ C denote by D R , j the full subcategoryof P i ( j ) with objects G L , G ∈ R ∩ C i , j . As each G L is a quotient of P G and 2-morphisms in C surject onto homomorphisms between projective modules in P i ( j )(see Subsection 3.3), it follows that 2-morphisms in C surject onto homomorphismsbetween the various G L . Lemma 18.
For every F ∈ C and G ∈ R , the module F ◦ G L is isomorphic to adirect sum of modules of the form H L , H ∈ R .Proof. Any H occurring as a direct summand of F ◦ G satisfies H ≥ R G. On theother hand, H L = 0 implies H ∗ ≤ L G R by Lemma 12. This is equivalent toH ≤ R G ∗R . By Proposition 17(e), we have G ∗R ∈ R . Thus H ∈ R , as claimed. (cid:3) Lemma 19.
For every F , H ∈ R ∩ C i , j we have dim Hom C ( i , j ) (F L, H L ) = [H L : L F ] . Proof.
Let k denote the multiplicity of G R as a direct summand of H ∗ ◦ F. Thenfor the right hand side we have[H L : L F ] = dim Hom C ( i , j ) (F P i , H L )(by adjunction) = dim Hom C ( i , i ) (H ∗ ◦ F P i , L )= k. At the same time, by adjunction, for the left hand side we have(2) dim Hom C ( i , j ) (F L, H L ) = dim Hom C ( i , i ) (H ∗ ◦ F L, L ) . From Proposition 17(b) it follows that the right hand side of (2) is at least k . Onthe other hand,dim Hom C ( i , j ) (F L, H L ) ≤ Hom C ( i , j ) (F P i , H L ) = [H L : L F ] = k, which completes the proof. (cid:3) For F ∈ R consider the short exact sequence(3) Ker F ֒ → P F ։ F L, given by Proposition 17(b). SetKer R , j = M F ∈R∩C i , j Ker F , P R , j = M F ∈R∩C i , j P F , Q R , j = M F ∈R∩C i , j F L. Lemma 20.
The module
Ker R , j is stable under any endomorphism of P R , j .Proof. Let F , H ∈ R ∩ C i , j and ϕ : P F → P H be a homomorphism. It is enough toshow that ϕ (Ker F ) ⊂ Ker H . Assume this is false. Composing ϕ with the naturalprojection onto Q H we obtain a homomorphism from P F to Q H which does notfactor through Q F . However, the existence of such homomorphism contradictsLemma 19. This implies the claim. (cid:3) Now we are ready to define the cell 2-representation C R of C corresponding to R .Define C R ( j ) to be the full subcategory of P i ( j ) which consists of all modules M admitting a two step resolution X → X ։ M , where X , X ∈ add( Q R , j ). Lemma 21.
The category C R ( j ) is equivalent to D op R , j - mod .Proof. Consider first the full subcategory X of P i ( j ) which consists of all modules M admitting a two step resolution X → X ։ M , where X , X ∈ add( P R ). By[Au, Section 5], the category X is equivalent to End C op i , j ( P R , j ) op -mod. By Lemma 20,the algebra End C op i , j ( Q R , j ) is the quotient of End C op i , j ( P R , j ) by a two-sided ideal. Itis easy to see that the standard embedding of End C op i , j ( Q R , j ) op -mod ∼ = D op R , j -modinto X coincides with C R ( j ). The claim follows. (cid:3) Theorem 22 (Construction of right cell 2-representations) . Restriction from P i defines the structure of a -representation of C on C R .Proof. From Lemma 18 it follows that for any F ∈ C j , k we have F C R ( j ) ⊂ C R ( k ).The claim follows. (cid:3) The 2-representation C R constructed in Theorem 22 is called the right cell -representation corresponding to R . Note that the inclusion of C R into P i is onlyright exact in general. Example 23.
Consider the category S from Example 2. For the cell repre-sentation C { i } we have G { i } = i , which implies that C { i } ( i ) = C -mod; C { i } (F) = 0 and C { i } ( f ) = 0 for f = α, β, γ . For the cell representation C { F } we have G { F } = F, which implies that C { F } ( i ) = D -mod, C { F } (F) = F and C { F } ( f ) = f for f = α, β, γ .4.6. Homomorphisms from a cell -representation. Consider a right cell R and let i ∈ C be such that G R ∈ C i , i . Let further F ∈ C ( i , i ) and α : F → G R besuch that P i ( α ) : F P i → G R P i gives a projective presentation of L G R . Theorem 24.
Let M be a -representation of C . Denote by Θ = Θ M R the cokernelof M ( α ) . ( a ) The functor Θ is a right exact endofunctor of M ( i ) . ( b ) For every morphism Ψ from C R to M we have Ψ( L G R ) ∈ Θ( M ( i )) . ( c ) For every M ∈ Θ( M ( i )) there is a unique morphism Ψ M from C R to M givenby a collection of right exact functors such that Ψ M sends L G R to M . ( d ) The correspondence M Ψ M is functorial in M in the image Θ( M ( i )) of Θ .Proof. Both M (F) and M (G R ) are exact functors as C is a fiat category and M isa 2-functor. The functor Θ is the cokernel of a homomorphism between two exactfunctors and hence is right exact by the Snake lemma. This proves claim (a). Claim(b) follows from the definitions.To prove claims (c) and (d) choose M ∈ Θ( M ( i )) such that M = Θ N for some N ∈ M ( i ). Consider the morphism Φ N given by Proposition 5. As Φ N is amorphism of 2-representations, Φ N ( L G R ) = Θ N = M . The restriction Ψ M of Φ N to C R is a morphism from C R to M . Now the existence parts of (c) and (d) followfrom Proposition 5. To prove uniqueness, we note that, for every j ∈ C , everyprojective in C R ( j ) has the form F L G R for some F ∈ C ( i , j ) and every morphismbetween projectives comes from a 2-morphism of C (see Subsection 4.5). As anymorphism from C R to M is a natural transformation of 2-functors, the value of thistransformation on L G R uniquely determines its value on all other modules. Thisimplies the uniqueness claim and completes the proof. (cid:3) A canonical quotient of P i associated with R . Fix i ∈ C . Let R be aright cell in C such that R ∩ C i , j = ∅ for some j ∈ C . Denote by ∆ R the uniqueminimal quotient of P i such that the composition K R ֒ → P i ։ ∆ R is nonzero. Proposition 25.
For every F ∈ R the image of the a unique (up to scalar) nonzerohomomorphism ϕ : P i → F ∗ L F is isomorphic to ∆ R .Proof. The existence of ϕ is given by Lemma 15. Let Y denote the image of ϕ .Assume that F ∈ C i , j . For X ∈ { P i , ∆ R } we have, by adjunction,Hom C ( i , i ) ( X, F ∗ L F ) = Hom C ( i , j ) (F X, L F ) = k as F X is a nontrivial quotient of P F (see Proposition 17). By construction, Y hassimple top isomorphic to L i and, by the above, the latter module occurs in F ∗ L F with multiplicity one. Since ∆ R also has simple top isomorphic to L i , it follows ELL 2-REPRESENTATIONS 13 that the image of any nonzero map from ∆ R to Y covers the top of Y and hence issurjective. To complete the proof it is left to show that the image of any nonzeromap from ∆ R to Y is injective.By construction, L G R is the simple socle of ∆ R . Let N denote the cokernel of L G R ֒ → ∆ R . Similarly to the previous paragraph, we haveHom C ( i , i ) ( N, F ∗ L F ) = Hom C ( i , j ) (F N, L F ) = 0since all composition factors of N are annihilated by F (by Proposition 17(ai)).The claim follows. (cid:3) We complete this section with the following collection of useful facts:
Lemma 26. ( a ) For any F , G ∈ C i , j we have [F ∗ L G : L i ] = 0 if and only if F = G . ( b ) For any F ∈ C we have F ∼ LR F ∗ .Proof. Using adjunction, we haveHom C ( i , i ) ( P i , F ∗ L G ) = Hom C ( i , j ) ( P F , L G ) = ( k , F = G;0 , otherwise , which proves (a).To prove (b) let R be the right cell containing F. Then we have F ∼ R G R andhence F ∗ ∼ L G ∗R . At the same time G R ∼ R G ∗R by Proposition 17(e). Claim (b)follows and the proof is complete. (cid:3) Regular cells.
We denote by ⋆ the usual product of binary relations. Lemma 27.
We have ≤ LR = ≤ R ⋆ ≤ L = ≤ L ⋆ ≤ R .Proof. Obviously the product of ≤ R and ≤ L (in any order) is contained in ≤ LR .On the other hand, for F , G ∈ C we have F ≤ LR G if and only if there exist H , K ∈ C such that G occurs as a direct summand of H ◦ F ◦ K. This means that there is adirect summand L of H ◦ F such that G occurs as a direct summand of L ◦ K. Bydefinition, we have F ≤ R L and L ≤ R G. This implies that ≤ LR is contained in ≤ R ⋆ ≤ L and hence ≤ LR coincides with ≤ R ⋆ ≤ L . Similarly ≤ LR coincides with ≤ L ⋆ ≤ R and the claim of the lemma follows. (cid:3) A two-sided cell Q is called regular provided that any two different right cells inside Q are not comparable with respect to the right order. From Lemma 26(b) it followsthat Q is regular if and only if any two different left cells inside Q are not comparablewith respect to the left order. A right (left) cell is called regular if it belongs toa regular two-sided cell. An element F is called regular if it belongs to a regulartwo-sided cell. Proposition 28 (Structure of regular two-sided cells) . Let Q be a regular two-sidedcell. ( a ) For any right cell R in Q and left cell L in Q we have L ∩ R 6 = ∅ . ( b ) Let ∼ Q R and ∼ Q L denote the restrictions of ∼ R and ∼ L to Q , respectively. Then Q × Q = ∼ Q R ⋆ ∼ Q L = ∼ Q L ⋆ ∼ Q R .Proof. If F ∈ R and G ∈ L , then there exist H , K ∈ C such that G occurs as adirect summand of H ◦ F ◦ K. This means that there exists a direct summand N ofH ◦ F such that G occurs as a direct summand of N ◦ K. Then N ≥ R F and N ≤ L G.As F ∼ LR G it follows that N ∈ Q . Since Q is regular, it follows that N ∈ R andN ∈ L proving (a).To prove (b) consider F , G ∈ Q . By (a), there exist H ∈ Q such that H ∼ L F andH ∼ R G. Similarly, there exist K ∈ Q such that K ∼ R F and K ∼ L G. Then wehave (F , G) = (F , H) ⋆ (H , G) and (F , G) = (F , K) ⋆ (K , G) proving (b). (cid:3)
For a regular right cell R the corresponding module ∆ R has the following property. Proposition 29.
Let R be a regular right cell and M the cokernel of L G R ֒ → ∆ R .Then for any composition factor L F of M we have F < R G R and F < L G R .Proof of Proposition 29. Let F ∈ C be such that L F is a composition factor of M .As ∆ R is a submodule of G ∗R L G R (by Proposition 25), from Lemma 13(a) it followsthat F ≤ R G R .Consider I := { H ∈ C : H ≤ R G R } . Then I is an ideal with respect to ≤ R . Assumethat G R ∈ C i , i and consider the 2-subrepresentation P I i of P i . Then F ∈ C i , i and∆ R ∈ P I i ( i ). Using adjunction, we have:0 = Hom C ( i , i ) (F P I i , ∆ R ) = Hom C ( i , i ) ( P I i , F ∗ ∆ R ) . This yields F ∗ ∆ R = 0. The module F ∗ ∆ R on the one hand belongs to P I i ( i ) (byLemma 13(a)), on the other hand is a quotient of F ∗ P i (as ∆ R is a quotient of P i and F ∗ is exact). The module F ∗ P i has simple top L F ∗ . This implies F ∗ ≤ R G R by Lemma 13(a) and thus F ≤ L G ∗R ∈ R (see Proposition 17(e)).This leaves us with two possibilities: either F LR G R , in which case we have bothF < R G R and F < L G R , as desired; or F ∼ LR G R , in which case we have bothF ∼ L G R and F ∼ R G R since R is regular. In the latter case we, however, haveG ∗R L F = 0 by Lemma 12, which contradicts Proposition 17(ai). This completesthe proof. (cid:3) A two-sided cell Q is called strongly regular if it is regular and for every left cell L and right cell R in Q we have |L ∩ R| = 1. A left (right) cell is strongly regular ifit is contained in a strongly regular two-sided cell. Proposition 30 (Structure of strongly regular right cells) . Let R be a stronglyregular right cell. Then we have: ( a ) G R ∼ = G ∗R . ( b ) If F ∈ R satisfies F ∼ = F ∗ , then F = G R . ( c ) If F ∈ R and G ∼ L F is such that G ∼ = G ∗ , then G L F = 0 and every simpleoccurring both in the top and in the socle of G L F is isomorphic to L F .Proof. Claim (a) follows from the strong regularity of R and Proposition 17(e).Claim (b) follows directly from the strong regularity of R .Let us prove claim (c). That G L F = 0 follows from Lemma 12. If some L H occursin the top of G L F = 0 then, using adjunction and G ∼ = G ∗ , we get G L H = 0. Thelatter implies G ∼ L H by Lemma 12. At the same time H ∼ R F by Corollary 14.Hence H = F because of the strong regularity of R . This completes the proof. (cid:3) The -category of a two-sided cell The quotient associated with a two-sided cell.
Let Q be a two-sided cellin C . Denote by I Q the 2-ideal of C generated by F and id F for all F LR Q .In other words, for every i , j ∈ C we have that I Q ( i , j ) is the ideal of C ( i , j )consisting of all 2-morphisms which factor through a direct sum of 1-morphisms ofthe form F, where F LR Q . Taking the quotient we obtain the 2-category C / I Q . Lemma 31.
Let
R ⊂ Q be a right cell. Then I Q annihilates the cell -represen-tation C R . In particular, C R carries the natural structure of a -representation of C / I Q .Proof. This follows from the construction and Lemma 12. (cid:3)
The construction of C / I Q is analogous to constructions from [Be, Os]. ELL 2-REPRESENTATIONS 15
The -category associated with Q . Denote by C Q the full 2-subcategoryof C / I Q , closed under isomorphisms, generated by the identity morphisms i , i ∈ C , and F ∈ Q . We will call C Q the 2 -category associated to Q . This categoryis especially good in the case of a strongly regular Q , as follows from the followingstatement: Proposition 32.
Assume Q is a strongly regular two-sided cell in C . Then Q remains a two-sided cell for C Q .Proof. Let F ∈ Q . Denote by G the unique self-adjoint element in the right cell R of F. The action of G on the cell 2-representation C R is nonzero and hence G = 0,when restricted to C R .Further, by Proposition 17(b), G L G has simple top L G . Using Proposition 17(b)again, we thus get F ◦ G L G = 0, implying F ◦ G = 0, when restricted to C R . Butthe restriction of F ◦ G decomposes into a direct sum of some H ∈ Q , which are inthe same right cell as G and in the same left cell as F. Since Q is strongly regular,the only element satisfying both conditions is F. This implies that, when restrictedto C R , F occurs as a direct summand of F ◦ G, which yields F ≥ R G in C Q .Now consider the functor F ∗ ◦ F. Since F = 0, when restricted to C R , by adjunctionwe have F ∗ ◦ F = 0, when restricted to C R , as well. The functor F ∗ ◦ F decomposesinto a direct sum of functors from
R ∩ R ∗ = { G } . This implies G ≥ R F in C Q andhence R remains a right cell in C Q . Using ∗ we get that all left cells in Q remain leftcells in C Q . Now the claim of the proposition follows from Proposition 28(b). (cid:3) The important property of C Q is that for strongly regular right cells the corre-sponding cell 2-representations can be studied over C Q : Corollary 33.
Let Q be a strongly regular two-sided cell of C and R be a right cellof Q . Then the restriction of the cell -representation C R from C to C Q gives thecorresponding cell -representation for C Q .Proof. Let i ∈ C be such that R∩C i , i = ∅ . Denote by C QR the cell 2-representationof C Q associated to R . We will use the upper index Q for elements of this 2-representation. Consider C R as a 2-representation of C Q by restriction. By The-orem 24, we have the morphism of 2-representations Ψ := Ψ L G R : C QR → C R sending L Q G R to L G R .Let j ∈ C and F ∈ R ∩ C i , j . By Proposition 17(b), the morphism Ψ sends theindecomposable projective module F L Q G R of C QR ( j ) to the indecomposable projec-tive module F L G R in C R ( j ). As mentioned after Proposition 17, we have that2-morphisms in C Q surject onto the homomorphisms between indecomposable pro-jective modules both in C QR ( j ) and C R ( j ).To prove the claim it is left to show that Ψ is injective, when restricted to inde-composable projective modules in C QR ( j ). For this it is enough to show that theCartan matrices of C QR ( j ) and C R ( j ) coincide. For indecomposable F and H in R ∩ C i , j , using adjunction, we have(4) Hom C ( i , j ) (F L G R , H L G R ) = Hom C ( i , i ) (H ∗ ◦ F L G R , L G R ) . and similarly for C QR . The dimension of the right hand side of (4) equals themultiplicity of G R as a direct summand of H ∗ ◦ F. Since this multiplicity is thesame for C QR and C R , the claim follows. (cid:3) Cell -representations for strongly regular cells. In this section we fix astrongly regular two-sided cell Q in C . We would like to understand combinatoricsof the cell 2-representation C R for a right cell R ⊂ Q . By the previous subsection,for this it is enough to assume that C = C Q . We work under this assumption in the rest of this subsection and consider the direct sum C of all C R , where R runsthrough the set of all right cells in Q . To simplify our notation, by Hom C we denotethe homomorphism space in an appropriate module category C ( i ). Proposition 34.
Let Q be as above and F , H ∈ Q . ( a ) For some m F , H ∈ { , , , . . . } we have H ∗ ◦ F ∼ = m F , H G , where { G } = L H ∗ ∩R F ;moreover, m F , F = 0 . ( b ) If F ∼ R H , then m F , H = m H , F . ( c ) If H = H ∗ and F ∼ R H , then m F , F = dim End C (F L H ) . ( d ) If H = H ∗ and F ∼ R H , then F ◦ H ∼ = m H , H F and H ◦ F ∗ ∼ = m H , H F ∗ ( e ) If H = H ∗ and H ∼ L F , then m H , H = dim Hom C ( P F , H L F ) . ( f ) Assume G ∈ Q and H = H ∗ , G = G ∗ , H ∼ L F and G ∼ R F . Then m F , F m G , G = m F ∗ , F ∗ m H , H . Proof.
By our assumptions, every indecomposable direct summand of H ∗ ◦ F belongsto the right cell of F and the left cell of H ∗ , hence is isomorphic to G. Note thatF ∗ ◦ F is nonzero by adjunction since F L G R F = 0. This implies claim (a) and claim(c) follows from Proposition 17(b) using adjunction.If F ∼ R H, then H ∗ ◦ F ∼ = m F , H G R F by (a). By Proposition 30(a), the functor G R F is self-adjoint. Hence H ∗ ◦ F is self-adjoint, which implies claim (b).Set m = m H , H . Similarly to the proof of claim (a), we have F ◦ H ∼ = k F for some k ∈ { , , . . . } . Using associativity, we obtain k F = k (F ◦ H) = ( k F) ◦ H = (F ◦ H) ◦ H == F ◦ (H ◦ H) = F ◦ ( m H) = m (F ◦ H) = mk F . This implies claim (d) and claim (e) follows by adjunction.Claim (f) follows from the following computation: m F , F m G , G F (d) = m F , F (F ◦ G) = F ◦ ( m F , F G) (a) = F ◦ (F ∗ ◦ F) == (F ◦ F ∗ ) ◦ F (a) = ( m F ∗ , F ∗ H) ◦ F = m F ∗ , F ∗ (H ◦ F) (d) = m F ∗ , F ∗ m H , H F . This completes the proof. (cid:3)
As a corollary we obtain that the Cartan matrix of the cell 2-representation C R issymmetric. Corollary 35.
Assume that R is a strongly regular right cell. Then for any F , H ∈R we have [F L G R : L H ] = [H L G R : L F ] .Proof. We have [F L G R : L H ] = dim Hom C R (H L G R , F L G R ) . Using adjunction and Proposition 34(a), the latter equals m H , F . Now the claimfollows from Proposition 34(b). (cid:3) Corollary 36.
Let F , H ∈ Q be such that H = H ∗ and F ∼ L H . Then the module P F is a direct summand of H L F and m F , F ≤ m H , H .Proof. From Proposition 34 we have: m F , F = dim End C ( P F ) , m H , H = dim Hom C ( P F , H L F ) . Hence to prove the corollary we just need to show that P F is a direct summand ofH L F .By Proposition 34, the module F ◦ F ∗ L F decomposes into a direct sum of m F ∗ , F ∗ copies of the module H L F . Hence it is enough to show that P F is a direct summandof F ◦ F ∗ L F . ELL 2-REPRESENTATIONS 17
Let R be the right cell of F. We know that F ∗ L F = 0. Using adjunction andLemma 12, we obtain that every simple quotient of F ∗ L F = 0 is isomorphic to L G R . Hence F ∗ L F surjects onto L G R and, applying F, we have that F ◦ F ∗ L F surjects onto P F . Now the claim follows from projectivity of P F . (cid:3) Corollary 37.
For every F ∈ Q the projective module P F is injective.Proof. Let R be the right cell of F. Since the functorial actions of F and F ∗ on C R are biadjoint, they preserve both the additive category of projective modules andthe additive category of injective modules. Now take any injective module I andlet L H be some simple occurring in its top. Applying H ∗ we get an injective modulesuch that L G R occurs in its top. Applying now F we get an injective module inwhich the projective module P F ∼ = F L G R is a quotient. Hence P F splits off as adirect summand in this module and thus is injective. This completes the proof. (cid:3) Corollary 38.
Let F , H ∈ Q and R be the right cell of F . ( a ) We have F ∗ L F ∼ = P G R . ( b ) The module H L F is either zero or both projective and injective.Proof. Similarly to the proof of Corollary 36 one shows that the module P G R isa direct summand of F ∗ L F , so to prove claim (a) we have to show that F ∗ L F isindecomposable. We will show that F ∗ L F has simple socle. Since F annihilatesall simple modules in C R but L G R , using adjunction it follows that every simplesubmodule in the socle of F ∗ L F is isomorphic to L G R . On the other hand, usingadjunction and Proposition 17(b) we obtain that the homomorphism space from L G R to F ∗ L F is one-dimensional. This means that F ∗ L F has simple socle andproves claim (a).Assume that H L F = 0. Then, by Lemma 12, we have F ∗ ∼ R H (since Q is stronglyregular). Let G ∈ R be such that G ∼ L H. Then, by Proposition 34(a), we haveG ◦ F ∗ ∼ = m F ∗ , G ∗ H. So, to prove claim (b) it is enough to show that m F ∗ , G ∗ = 0 andthat G ◦ F ∗ L F is both projective and injective. By claim (a), we have F ∗ L F ∼ = P G R .Since G L G R = 0 by Proposition 17(b) and G is exact, it follows that G ◦ F ∗ L F = 0and hence m F ∗ , G ∗ = 0. Further, G P G R is projective as P G R is projective and G isbiadjoint to G ∗ . Finally, G P G R is injective by Corollary 37. Claim (b) follows andthe proof is complete. (cid:3) Corollary 39.
Let F , H ∈ Q be such that H = H ∗ and F ∼ L H . Then m F , F | m H , H .Proof. Let R be the right cell of F. By Lemma 12, H annihilates all simples of C R but L F . This and Corollary 38(b) imply that H L F = kP F for some k ∈ N . On theone hand, using Propositions 17 and 34 we have(5) (F ∗ ◦ H) L F = k F ∗ P F = k (F ∗ ◦ F) L G R = km F , F G R L G R = km F , F P G R . On the other hand, we have F ∗ ∼ R H and thus, using Proposition 34(d) andCorollary 38(a), we have:(6) (F ∗ ◦ H) L F = m H , H F ∗ L F = m H , H P G R . The claim follows comparing (5) and (6). (cid:3) Cyclic and simple -representations of fiat categories Cyclic -representations. Let C be a fiat category, M a 2-representationof C , i ∈ C and M ∈ M ( i ). We will say that M generates M if for any j ∈ C and X, Y ∈ M ( j ) there are F , G ∈ ˆ C ( i , j ) such that F M ∼ = X , G M ∼ = Y andthe evaluation map Hom ˆ C ( i , j ) (F , G) → Hom M ( j ) (F M, G M ) is surjective. The 2-representation M is called cyclic provided that there exists i ∈ C and M ∈ M ( i ) such that M generates M . Examples of cyclic 2-representations of C are given bythe following: Proposition 40. ( a ) For any i ∈ C the -representation P i is cyclic and gener-ated by P i . ( b ) For any right cell R of C the cell -representation C R is cyclic and generatedby L G R .Proof. Let j ∈ C , X, Y ∈ P i ( j ) and f : X → Y . Taking some projective presenta-tions of X and Y yields the following commutative diagram with exact rows:(7) X h / / f ′′ (cid:15) (cid:15) X / / / / f ′ (cid:15) (cid:15) X f (cid:15) (cid:15) Y g / / Y / / / / Y Now X , X , Y , Y are projective in P i ( j ) and we may assume that X = F P i , X = F P i , Y = G P i and Y = G P i for some F , F , G , G ∈ C ( i , j ). Fromthe definition of P i we then obtain that g , h , f ′ and f ′′ are given by 2-morphismsbetween the corresponding 1-morphisms (which we denote by the same symbols).It follows that X equals to the image of P i under H := Coker(F h → F ) ∈ ˆ C ( i , j ).Similarly, Y equals to the image of P i under H := Coker(G g → G ) ∈ ˆ C ( i , j ).Finally, f is induced by the diagramF h / / f ′′ (cid:15) (cid:15) F f ′ (cid:15) (cid:15) G g / / G . Claim (a) follows.To prove claim (b) we view every C R ( j ) as the corresponding full subcategory of P i ( j ). Let X, Y ∈ C R ( j ) and f : X → Y . From the proof of claim (a) we have thecommutative diagram (7) as described above. Our proof of claim (b) will proceedby certain manipulations of this diagram. Denote by I the ideal of C with respectto ≤ R generated by R and set I ′ := I \ R .To start with, we modify the left column of (7). Let X ′ and Y ′ denote the trace ofall projective modules of the form P G , G
6∈ I ′ , in X and Y , respectively. Considersome minimal projective covers ˆ X ։ X ′ and ˆ Y ։ X ′ of X ′ and Y ′ , respectively.Let can : ˆ X ։ X ′ ֒ → X and can ′ : ˆ Y ։ Y ′ ֒ → Y denote the correspondingcanonical maps and set ˆ h = h ◦ can and ˆ g = g ◦ can ′ . Then the cokernel of bothcan and can ′ has only composition factors of the form L F , F ∈ I ′ . By construction,the image of f ′′ ◦ can is contained in the image of can ′ . Hence, using projectivityof ˆ X , the map f ′′ lifts to a map ˆ f ′′ : ˆ X → ˆ Y such that the following diagramcommutes:(8) ˆ X h & & ˆ f ′′ (cid:15) (cid:15) can / / X f ′′ (cid:15) (cid:15) h / / X / / / / f ′ (cid:15) (cid:15) X f (cid:15) (cid:15) ˆ Y g can ′ / / Y g / / Y / / / / Y The difference between (7) and (8) is that the rows of the solid part of (8) are nolonger exact but might have homology in the middle. By construction, all simple
ELL 2-REPRESENTATIONS 19 subquotients of these homologies have the form L F , F ∈ I ′ . Further, all projectivedirect summands appearing in (8) have the form P F for F
6∈ I ′ .Denote by ˜ X , ˜ X , ˜ Y and ˜ Y the submodules of ˆ X , X , ˆ Y and Y , respectively,which are uniquely defined by the following construction: The corresponding sub-modules contain all direct summands of the form P F for F
6∈ R ; and for each directsummand of the form P F , F ∈ R , the corresponding submodules contain the sub-module Ker F of P F as defined in (3). By construction and Lemma 20, we haveˆ h : ˜ X → ˜ X , ˆ g : ˜ Y → ˜ Y , f ′ : ˜ X → ˜ Y and ˆ f ′′ : ˜ X → ˜ Y . Since X, Y ∈ C R , theimages (on diagram (8)) of ˜ X and ˜ Y in X and Y , respectively, are zero. Hence,taking quotients gives the following commutative diagram:(9) ˆ X / ˜ X h / / ˜ f ′′ (cid:15) (cid:15) X / ˜ X / / / / ˜ f ′ (cid:15) (cid:15) X f (cid:15) (cid:15) ˆ Y / ˜ Y g / / Y / ˜ Y / / / / Y, where ˜ h , ˜ g , ˜ f ′′ and ˜ f ′ denote the corresponding induced maps.By our construction of (9) and definition of C R , all indecomposable modules ap-pearing in the left square of (9) are projective in C R ( j ) (and hence, by definitionof C R , have the form F L G R for some F ∈ R ). Moreover, all simples of the form L F , F ∈ I ′ , become zero in C R ( j ) (since C R ( j ) is defined as a Serre subquotientand simples L F , F ∈ I ′ , belong to the kernel). This implies that both rows of (9)are exact in C R ( j ). As mentioned in the proof of claim (a), the maps g, h, f ′ and f ′′ on diagram (7) are given by 2-morphisms in C . Similarly, the maps ˆ g , ˆ h and ˆ f ′′ on diagram (8) are given by 2-morphisms in C as well. By construction of (9), themaps ˜ h , ˜ g , ˜ f ′′ and ˜ f ′ are induced by ˆ h , ˆ g , ˆ f ′′ and f ′ , respectively. Now the proofof claim (b) is completed similarly to the proof of claim (a). (cid:3) Simple -representations. A (nontrivial) 2-representation M of C is called quasi-simple provided that it is cyclic and generated by a simple module. FromProposition 40(b) it follows that every cell 2-representation is quasi-simple. A(nontrivial) 2-representation M of C is called strongly simple provided that it iscyclic and generated by any simple module. It turns out that for strongly regularright cells strong simplicity of cell 2-representations behaves well with respect torestrictions. Proposition 41.
Let Q be a strongly regular two-sided cell and R a right cell in Q . Then the cell -representation C R of C is strongly simple if and only if itsrestriction to C Q is strongly simple. To prove this we will need the following general lemma:
Lemma 42.
Let Q be two-sided cell and M a -representations of C . Let H ∈ ˆ C be such that for any F ∈ C the inequality Hom ˆ C (F , H) = 0 implies F < LR Q . Thenfor any G ∈ Q the functor M (G) annihilates the image of M (H) .Proof. Let H = Coker( α ), where α : H ′ → H ′′ is a 2-morphism in C . FromLemma 12, applied to an appropriate P i , it follows that G( α ) is surjective. Thisimplies that G ◦ H = 0 and yields the claim. (cid:3)
Proof of Proposition 41.
Let H ∈ C be such that H < LR Q . Then there exists a 1-morphism F in C and a 2-morphism α : F → H in C such that every indecomposabledirect summand of C R (F) has the form C R (G) for some G ∈ Q and the cokernelof C R ( α ) satisfies the condition that for any K ∈ C the existence of a nonzerohomomorphism from C R (K) to Coker( C R ( α )) implies K < LR Q . By Lemma 42, every 1-morphism in Q annihilates the image of Coker( C R ( α )). Since every simplein C R is not annihilated by some 1-morphism in Q , we have that the image ofCoker( C R ( α )) is zero and hence Coker( C R ( α )) is the zero functor. This meansthat C R (H) is a quotient of C R (F), which implies the claim. (cid:3) The following theorem is our main result (and a proper formulation of Theorem 1from Section 1).
Theorem 43 (Strong simplicity of cell 2-representations) . Let Q be a stronglyregular two-sided cell. Assume that (10) the function Q −→ N F m F , F is constant on left cells of Q . Then we have: ( a ) For any right cell R in Q the cell -representation C R is strongly simple. ( b ) If R and R ′ are two right cells in Q , then the cell -representations C R and C R ′ are equivalent.Proof. Let F , G ∈ R and H ∈ Q be such that H ∈ R F ∗ ∩ L G . The module H L F isnonzero by Lemma 12 and projective by Corollary 38(b). From Proposition 34 andCorollary 38(b) it follows that H L F ∼ = kP G for some k ∈ N . Hence, by adjunction, m H , H = dim End C R (H L F ) = dim End C R ( kP G ) = k dim End C R ( P G ) = k m G , G . On the other hand, H ∼ L G and thus m H , H = m G , G by our assumption (10), whichimplies k = 1. This means that every H ∈ R F ∗ maps L F to an indecomposableprojective module.To prove (a) it is left to show that 2-morphisms in C surject onto homomorphismsbetween indecomposable projective modules. By adjunction, it is enough to showthat for any H , J ∈ R F ∗ the space of 2-morphisms from H ∗ ◦ J to the identity surjectsonto homomorphisms from the projective module H ∗ ◦ J L F to L F . For the latterhomomorphism space to be nonzero, the functor H ∗ ◦ J should decompose into adirect sum of copies of K ∈ R F ∗ such that K ∼ = K ∗ (see Proposition 34(a)). Byadditivity, it is enough to show that there is a 2-morphism from K to the identitysuch that its evaluation at L F is nonzero. We have K L F = 0 by Lemma 12, whichimplies that the evaluation at L F of the adjunction morphism from K ◦ K to theidentity is nonzero. We have K ◦ K ∼ = m K , K K = 0 by Proposition 34(a). Byadditivity, the nonzero adjunction morphism restricts to a morphism from one ofthe summands such that the evaluation at L F remains nonzero. Claim (a) follows.To prove (b), consider the cell 2-representations C R and C R ′ . Without loss ofgenerality we may assume that Q is the unique maximal two-sided cell with respectto ≤ LR . Let G := G R and denote by F the unique element in R ′ ∩ L G . ThenG L F = 0 by Lemma 12. Moreover, from the proof of (a) we know that G L F is anindecomposable projective module and hence has simple top.Assume that i ∈ C is such that G ∈ C ( i , i ). Let K be a 1-morphism in C and α : K → G be a 2-morphism such that P i ( α ) is a projective presentation of L G .Denote by ˆG ∈ ˆ C the cokernel of α . Lemma 44.
The module ˆG L F surjects onto L F .Proof. It is enough to prove that ˆG L F = 0. Since G L F has simple top, it isenough to show that for any indecomposable 1-morphism M and any 2-morphism β : M → G which is not an isomorphism, the morphism β L F is not surjective.The statement is obvious if M L F = 0. If M L F = 0, we have M ≤ R F ∗ by Lemma 12.Hence either M ∼ R G or M < R G. If M = G, then β is a radical endomorphismof G, hence nilpotent (as C is a fiat category). This means that β L F is nilpotent ELL 2-REPRESENTATIONS 21 and thus is not surjective. If M ∈ R \
G, then M ∗
6∈ R and hence M ∗ L F = 0 byLemma 12. By adjunction this implies that L F does not occur in the top of M L F ,which means that β L F cannot be surjective. This implies the claim for all M ∈ Q .Consider now the remaining case M < R G and assume that β L F is surjective. LetM ′′ be a 1-morphism and γ : M ′′ → M be a 2-morphism such that γ gives the tracein M of all 1-morphisms J satisfying J < R R . Denote by M ′ and G ′ the cokernels of γ and β ◦ γ , respectively. Then both M ′ and G ′ are in ˆ C . Let β ′ : M ′ → G ′ be the2-morphism induced by β . Any direct summand of M ′′ which does not annihilate L F has the form ˆM for some ˆM ∈ Q because of our construction and maximality of Q . Hence from the previous paragraph it follows that the map β ′ L F is still surjective.On the other hand, because of our construction of M ′′ , an application of Lemma 42gives G ◦ M ′ = 0 in ˆ C . At the same time, the nonzero module G ′ L F is a quotientof G L F and hence has simple top L F . This implies G ◦ G ′ L F = 0. Therefore,applying G to the epimorphism β ′ L F : M ′ L F ։ G ′ L F annihilates the left hand side and does not annihilate the right hand side. Thiscontradicts the right exactness of G and the claim follows. (cid:3) By (a), any extension of L F by any other simple in C R ′ comes from some 2-morphism in C . Hence this extension cannot appear in ˆG L F by construction of ˆG.This and Lemma 44 imply ˆG L F ∼ = L F .Therefore, by Theorem 24, there is a unique homomorphism Ψ : C R → C R ′ of2-representations, which maps L G to L F . From claim (a) it follows that Ψ mapsindecomposable projectives to indecomposable projectives. Restrict Ψ to C Q . Thenfrom the proof of (a) we have that for any H , H ∈ R we havedim Hom C R ( P H , P H ) = dim Hom C R′ (Ψ P H , Ψ P H ) . Moreover, both spaces are isomorphic to C Q (H , H ). From (a) and constructionof Ψ it follows that Ψ induces an isomorphism between Hom C R ( P H , P H ) andHom C R′ (Ψ P H , Ψ P H ). This means that Ψ induces an equivalence between theadditive categories of projective modules in C R and C R ′ . Since Ψ is right exact,this implies that Ψ is an equivalence of categories and completes the proof. (cid:3) Examples
Projective functors on the regular block of the category O . Let g denote a semi-simple complex finite dimensional Lie algebra with a fixed triangulardecomposition g = n − ⊕ h ⊕ n + and O the principal block of the BGG-category O for g (see [Hu]). If W denotes the Weyl group of g , then simple objects in O are simple highest weight modules L ( w ), w ∈ W , of highest weight w · ∈ h ∗ .Denote by P ( w ) the indecomposable projective cover of L ( w ) and by ∆( w ) thecorresponding Verma module.Let S = S g denote the (strict) 2-category defined as follows: it has one object i (which we identify with O ); its 1-morphisms are projective functors on O , thatis functors isomorphic to direct summands of tensoring with finite dimensional g -modules (see [BG]); and its 2-morphisms are natural transformations of functors.For w ∈ W denote by θ w the unique (up to isomorphism) indecomposable projectivefunctor on O sending P ( e ) to P ( w ). Then { θ w : w ∈ W } is a complete and irre-dundant list of representatives of isomorphism classes of indecomposable projectivefunctors. Since O is equivalent to the category of modules over a finite-dimensionalassociative algebra, all spaces of 2-morphisms in S are finite dimensional. From[BG] we also have that S is stable under taking adjoint functors. It follows that S is a fiat category. The split Grothendieck ring [ S ] ⊕ of S is isomorphic to the integral group ring Z W such that the basis { [ θ w ] : w ∈ W } of [ S ] ⊕ corresponds tothe Kazhdan-Lusztig basis of Z W . We refer the reader to [Ma] for an overview andmore details on this category.Left and right cells of S are given by the Kazhdan-Lusztig combinatorics for W (see[KaLu]) and correspond to Kazhdan-Lusztig left and right cells in W , respectively.Namely, for x, y ∈ W the functors θ x and θ y belong to the same left (right or two-sided) cell as defined in Subsection 4.1 if and only if x and y belong to the sameKazhdan-Lusztig left (right or two-sided) cell, respectively. This is an immediateconsequence of the multiplication formula for elements of the Kazhdan-Lusztig basis(see [KaLu]). In particular, from [Lu] it follows that all cells for S are regular. If g ∼ = sl n , then W is isomorphic to the symmetric group S n . Robinson-Schenstedcorrespondence associates to every w ∈ S n a pair ( α ( w ) , β ( w )) of standard Youngtableaux of the same shape (see [Sa, Section 3.1]). Elements x, y ∈ S n belong to thesame Kazhdan-Lusztig right or left cell if and only if α ( x ) = α ( y ) and β ( x ) = β ( y ),respectively (see [KaLu]). It follows that in the case g ∼ = sl n all cells for S arestrongly regular.The 2-category S comes along with the defining -representation , that is the natu-ral action of S on O . Various 2-representations of S were constructed, as subquo-tients of the defining representation, in [KMS] and [MS] (see also [Ma] for a moredetailed overview). In particular, in [MS] for every Kazhdan-Lusztig right cell R there is a construction of the corresponding cell module . The later is obtained byrestricting the action of S to the full subcategory of O consisting of all modules M admitting a presentation X → X ։ M , where every indecomposable directsummand of both X and X is isomorphic to θ w L ( d ), where w ∈ R and d is theDuflo involution in R . Similarly to the proof of Theorem 43 one shows that thiscell module is equivalent to the cell 2-representation C R of S .Let Q be a strongly regular two-sided cell for S . In this case from [Ne, Theorem 5.3]it follows that the condition (10) is satisfied for Q . Hence from Theorem 43 weobtain that cell 2-representations of S for right cells inside a given two-sided cellare equivalent. This reproves, strengthens and extends the similar result [MS,Theorem 18], originally proved in the case g ∼ = sl n .7.2. Projective functors between singular blocks of O . The 2-category S g from the previous subsection admits the following natural generalization. For everyparabolic subalgebra p of g containing the Borel subalgebra b = h ⊕ n + let W p ⊂ W be the corresponding parabolic subgroup. Fix some dominant and integral weight λ p such that W p coincides with the stabilizer of λ p with respect to the dot action(to show the connection with the previous subsection we take λ b = 0). Let O λ p denote the corresponding block of the category O .Consider the 2-category S sing = S sing g defined as follows: its objects are the cat-egories O λ p , where p runs through the (finite!) set of parabolic subalgebras of g containing b , its 1-morphisms are all projective functors between these blocks, its2-morphisms are all natural transformations of functors. Similarly to the previoussubsection, the 2-category S sing is a fiat-category. The category S from the previ-ous subsection is just the full subcategory of S sing with the object O . A deformedversion of S sing (which has infinite-dimensional spaces of 2-morphisms and henceis not fiat) was considered in [Wi].Let us describe in more detail the structure of S sing in the smallest nontrivial caseof g = sl . In this case we have two parabolic subalgebras, namely b and g . Usingthe usual identification of h with C we set λ b = 0 and λ g = −
1. The objects of S sing are thus i = O and j = O − .The category S sing ( j , j ) contains a unique (up to isomorphism) indecomposableobject, namely j , the identity functor on j . The category S sing ( i , j ) contains ELL 2-REPRESENTATIONS 23 a unique (up to isomorphism) indecomposable object, namely the functor θ on oftranslation onto the wall. The category S sing ( j , i ) contains a unique (up to iso-morphism) indecomposable object, namely the functor θ out of translation out ofthe wall. The category S sing ( i , i ) contains exactly two (up to isomorphism) non-isomorphic indecomposable objects, namely the identity functor i and the functor θ := θ out ◦ θ on of translation through the wall.It is easy to see that there are exactly two two-sided cells: one containing only thefunctor i , and the other one containing all other functors. The right cells of thelatter two-sided cell are { j , θ out } and { θ, θ on } . The left cells of the latter two-sidedcell are { j , θ on } and { θ, θ out } . All cells are strongly regular. The values of thefunction m F , F from (10) are given by:F i j θ on θ out θm F , F { i } is given by the fol-lowing picture (with the obvious action of the identity 1-morphisms): C -mod θ =0 (cid:5) (cid:5) θ on =0 * * . θ out =0 k k By Theorem 43, the cell 2-representations for the right cells { j , θ out } and { θ, θ on } are equivalent and strongly simple. Consider the algebra D := C ( x ) / ( x ) of dualnumbers with the fixed subalgebra C consisting of scalars. The cell 2-representationfor the right cell { j , θ out } is given (up to isomorphism of functors) by the followingpicture: D -mod θ = D ⊗ − (cid:5) (cid:5) θ on =Res D C + + C -mod . θ out =Ind D C k k Projective functors for finite-dimensional algebras.
The last exampleadmits a straightforward abstract generalization outside category O . Let A = A ⊕ A ⊕ · · · ⊕ A k be a weakly symmetric self-injective finite-dimensional algebraover an algebraically closed field k with a fixed decomposition into a direct sum ofconnected components (here weakly symmetric means that the top and the socleof every projective module are isomorphic). Let C A denote the 2-category withobjects , , . . . , k , which we identify with the corresponding A i -mod. For i , j ∈{ , , . . . , k } define C A ( i , j ) as the full fully additive subcategory of the categoryof all functors from A i -mod to A j -mod, generated by all functors isomorphic totensoring with A i (in the case i = j ) and tensoring with all projective A j - A i bimodules (i.e. bimodules of the form A j e ⊗ k f A i for some idempotents e ∈ A j and f ∈ A i ) for all i and j . Functors, isomorphic to tensoring with projectivebimodules will be called projective functors . Lemma 45.
The category C A is a fiat category.Proof. The only nontrivial condition to check is that the left and the right adjointsof a projective functor are again projective and isomorphic. For any A -module M and idempotents e, f ∈ A , using adjunction and projectivity of f A we haveHom A ( Ae ⊗ k f A, M ) = Hom k ( f A, Hom A ( Ae, M ))= Hom k ( f A, eM )= Hom k ( f A, eA ⊗ A M )= Hom k ( f A, k ) ⊗ k eA ⊗ A M = ( f A ) ∗ ⊗ k eA ⊗ A M Since A is self-injective, ( f A ) ∗ is projective. Since A is weakly symmetric, ( f A ) ∗ ∼ = Af . This implies that tensoring with Af ⊗ k eA is right adjoint to tensoring with Ae ⊗ k f A . The claim follows. (cid:3) The category C A has a unique maximal two-sided cell Q consisting of all projectivefunctors. This cell is regular. Right and left cells inside Q are given by fixingprimitive idempotents occurring on the left and on the right in projective functors,respectively. In particular, they are in bijection with simple A - A -bimodules andhence Q is strongly regular. The value of the function m F , F on Ae ⊗ k f A is givenby the dimension of eA ⊗ A Ae ∼ = eAe , in particular, the function m F , F is constanton left cells. From Theorem 43 we thus again obtain that all cell 2-representationsof C corresponding to right cells in Q are strongly simple and isomorphic.The category S sing sl from the previous subsection is obtained by taking k = , A = C and A = D . In the general case we have the following: Proposition 46.
Let C be a fiat category, Q a strongly regular two-sided cell of Q and R a right cell in Q . For i ∈ C let A i be such that C R ( i ) ∼ = A i - mod and A = ⊕ i ∈ C A i . Assume that the condition (10) is satisfied. Then C R gives rise toa -functor from C Q to C A .Proof. We identify C R ( i ) with A i -mod. That A is self-injective follows from Corol-lary 38. That A is weakly symmetric follows by adjunction from Lemma 12 andstrong regularity of Q . Hence, to prove the claim we only need to show that forany F ∈ Q the functor C R (F) is a projective endofunctor of A -mod.As C R (F) is exact, it is given by tensoring with some bimodule, say B . Since C R (F)kills all simples but one, say L , and sends L to an indecomposable projective, say P (by Theorem 43), the bimodule B has simple top (as a bimodule) and hence isa quotient of some projective bimodule.By exactness of C R (F), the dimension of B equals the dimension of P times themultiplicity of L in A . This is exactly the dimension of the corresponding indecom-posable projective bimodule. The claim follows. (cid:3) References [AM] T. Agerholm, V. Mazorchuk; On selfadjoint functors satisfying polynomial relations,Preprint arXiv:1004.0094.[Au] M. Auslander; Representation theory of Artin algebras. I, II. Comm. Algebra (1974),177–268; ibid. (1974), 269–310.[BG] J. Bernstein, S. Gelfand; Tensor products of finite- and infinite-dimensional representa-tions of semisimple Lie algebras. Compositio Math. (1980), no. 2, 245–285.[Be] R. Bezrukavnikov; On tensor categories attached to cells in affine Weyl groups. Represen-tation theory of algebraic groups and quantum groups, 69–90, Adv. Stud. Pure Math., , Math. Soc. Japan, Tokyo, 2004.[CR] J. Chuang, R. Rouquier; Derived equivalences for symmetric groups and sl -categorifi-cation. Ann. of Math. (2) (2008), no. 1, 245–298.[Cr] L. Crane; Clock and category: is quantum gravity algebraic? J. Math. Phys. (1995),no. 11, 6180–6193.[CF] L. Crane, I. Frenkel; Four-dimensional topological quantum field theory, Hopf categories,and the canonical bases. Topology and physics. J. Math. Phys. (1994), no. 10, 5136–5154. ELL 2-REPRESENTATIONS 25 ∼ etingof/tenscat.pdf[EO] P. Etingof, V. Ostrik; Finite tensor categories. Mosc. Math. J. (2004), no. 3, 627–654,782–783.[Fr] P. Freyd; Representations in abelian categories. in: Proc. Conf. Categorical Algebra(1966), 95–120.[Hu] J. Humphreys; Representations of semisimple Lie algebras in the BGG category O .Graduate Studies in Mathematics, . American Mathematical Society, Providence, RI,2008.[KaLu] D. Kazhdan, G. Lusztig; Representations of Coxeter groups and Hecke algebras. Invent.Math. (1979), no. 2, 165–184.[Kh] O. Khomenko; Categories with projective functors. Proc. London Math. Soc. (3) (2005), no. 3, 711–737.[KM] O. Khomenko, V. Mazorchuk; On Arkhipov’s and Enright’s functors. Math. Z. (2005), no. 2, 357–386.[Kv] M. Khovanov; A functor-valued invariant of tangles. Algebr. Geom. Topol. (2002),665–741 (electronic).[KhLa] M. Khovanov, A. Lauda; A categorification of quantum sl n . Quantum Topol. (2010),1–92.[KMS] M. Khovanov, V. Mazorchuk, C. Stroppel; A categorification of integral Specht modules.Proc. Amer. Math. Soc. (2008), no. 4, 1163–1169.[Le] T. Leinster; Basic bicategories. Preprint arXiv:math/9810017.[Lu] G. Lusztig; Cells in affine Weyl groups. Algebraic groups and related topics (Ky-oto/Nagoya, 1983), 255–287, Adv. Stud. Pure Math., , North-Holland, Amsterdam,1985.[Ma] V. Mazorchuk; Lectures on algebraic categorification, Preprint arXiv:1011.0144.[MS] V. Mazorchuk, C. Stroppel; Categorification of (induced) cell modules and the roughstructure of generalised Verma modules. Adv. Math. (2008), no. 4, 1363–1426.[Ne] M. Neunh¨offer; Kazhdan-Lusztig basis, Wedderburn decomposition, and Lusztig’s ho-momorphism for Iwahori-Hecke algebras. J. Algebra (2006), no. 1, 430–446.[Os] V. Ostrik; Tensor ideals in the category of tilting modules. Transform. Groups (1997),no. 3, 279–287.[Ro1] R. Rouquier; Categorification of the braid groups, Preprint arXiv:math/0409593.[Ro2] R. Rouquier; 2-Kac-Moody algebras. Preprint arXiv:0812.5023.[Sa] B. Sagan; The symmetric group. Representations, combinatorial algorithms, and sym-metric functions. Second edition. Graduate Texts in Mathematics, . Springer-Verlag,New York, 2001.[St] C. Stroppel; Categorification of the Temperley-Lieb category, tangles, and cobordismsvia projective functors. Duke Math. J. (2005), no. 3, 547–596.[Wi] G. Williamson; Singular Soergel bimodules. Preprint arXiv:1010.1283. Volodymyr Mazorchuk, Department of Mathematics, Uppsala University, Box 480,751 06, Uppsala, SWEDEN, [email protected] [email protected]@uea.ac.uk