LECTURES ON ALGEBRAIC CATEGORIFICATION

Transcription

1 LECTURES ON ALGEBRAIC CATEGORIFICATION VOLODYMYR MAZORCHUK Content Introduction 3 1. Baic: decategorification and categorification The idea of categorification Grothendieck group Decategorification (Pre)categorification of an F-module Graded etup Some contruction 8 2. Baic: from categorification of linear map to 2-categorie Categorification of linear map Naïve categorification Weak categorification categorie (Genuine) categorification Baic: 2-repreentation of finitary 2-categorie repreentation of 2-categorie Fiat-categorie Principal 2-repreentation of fiat-categorie Cell Cell module Homomorphim from a cell module Serre ubcategorie and quotient Naturally commuting functor Category O: definition Definition of category O Verma module Block decompoition BGG reciprocity and quai-hereditary tructure Tilting module and Ringel elf-duality Parabolic category O gl 2 -example Category O: projective and huffling functor Projective functor Tranlation through wall Decription via Harih-Chandra bimodule Shuffling functor Singular braid monoid gl 2 -example Category O: twiting and completion Zuckerman functor 28 Date: September 7,

2 2 VOLODYMYR MAZORCHUK 6.2. Twiting functor Completion functor Alternative decription via (co)approximation Serre functor Category O: grading and combinatoric Double centralizer property Endomorphim of the antidominant projective Grading on B λ Hecke algebra Categorification of the right regular H-module Combinatoric of O λ S n -categorification: Soergel bimodule, cell and Specht module Soergel bimodule Kazhdan-Luztig cell Cell module Categorification of the induced ign module Categorification of Specht module S n -categorification: (induced) cell module Categorie O ˆR Categorification of cell module Categorification of permutation module Parabolic analogue of O Categorification of induced cell module Category O: Kozul duality Quadratic dual of a poitively graded algebra Linear complexe of projective Kozul duality Kozul dual functor Alternative categorification of the permutation module l 2 -categorification: imple finite dimenional module The algebra U v (l 2 ) Finite dimenional repreentation of U v (l 2 ) Categorification of V1 n Categorification of V n Kozul dual picture Application: categorification of the Jone polynomial Kauffman bracket and Jone polynomial Khovanov idea for categorification of J(L) Quantum l 2 -link invariant Functorial quantum l 2 -link invariant l 2 -categorification: of Chuang and Rouquier Genuine l 2 -categorification Affine Hecke algebra Morphim of l 2 -categorification Minimal l 2 -categorification of imple finite dimenional module l 2 -categorification on category O Categorification of the imple reflection Application: block of F[S n ] and Broué conjecture Jucy-Murphy element and formal character Induction and retriction Categorification of the baic repreentation of an affine Kac-Moody algebra 65

3 LECTURES ON ALGEBRAIC CATEGORIFICATION Broué conjecture Broué conjecture for S n Divided power Application: of S n -categorification Wedderburn bai for C[S n ] Kotant problem Structure of induced module Exercie 71 Reference 73 Index 78 Introduction The mot traditional approach to olving mathematical problem i: tart with a difficult problem and implify it until it become eay enough to be olved. However, developing mathematical theorie quite often goe in the oppoite direction: tarting with an eay theory one trie to generalize it to omething more complicated which could hopefully decribe a much bigger cla of phenomena. An example of the latter i the development of what i now known a Khovanov homology. The Jone polynomial i a very elementary claical combinatorial invariant appearing in the low dimenional topology. However, a mot of known topological invariant, it i not an abolute one (it doe not ditinguih all knot). Some twelve year ago Mikhail Khovanov ha developed a very advanced refinement of Jone polynomial which eriouly increaed the level of theoretical ophitication neceary to be able to define and work with it. Intead of elementary combinatoric and baic algebra, Khovanov definition wa baed on category theory and homological algebra and very oon led to the tudy of higher categorical tructure. Thi categorification of the Jone polynomial created a new direction in topology and attracted a lot of attention from ome other part of mathematic, notably algebra and category theory. Within a few year categorification became an intenively tudied ubject in everal mathematical area. It completely changed the viewpoint on many long tanding problem and led to everal pectacular reult and application. Thi text i a write-up of the lecture given by the author during the Mater Cla Categorification at Århu Univerity, Denmark in October It motly concentrate on algebraical apect of the theory, preented in the hitorical perpective, but alo contain everal topological application, in particular, an algebraic (or, more preciely, repreentation theoretical) approach to categorification of the Jone polynomial mentioned above. The text conit of fifteen ection correponding to fifteen one hour lecture given during the Mater Cla and fairly decribe the content of thee lecture. There are ome exercie (which were propoed to participant of the Mater Cla) collected at the end of the text and a rather extenive lit of reference. Unfortunately, the time contrain on the Mater Cla reulted in the fact that everal recent development related to categorification did not make it into the text. The text i aimed to be and introductory overview of the ubject rather than a fully detailed monograph. The emphai i made on definition, example and formulation of the reult. Mot proof are either hortly outlined or omitted, however, complete proof could be found by tracking reference. It i aumed that the reader i familiar with baic of category theory, repreentation theory, topology and Lie algebra.

4 4 VOLODYMYR MAZORCHUK 1. Baic: decategorification and categorification 1.1. The idea of categorification. The term categorification wa introduced by Loui Crane in [Cr] and the idea originate from the earlier joint work [CF] with Igor Frenkel. The term refer to the proce of replacing et-theoretic notion by the correponding category-theoretic analogue a hown in the following table: Set Theory et element relation between element function relation between function Category Theory category object morphim of object functor natural tranformation of functor The general idea (or hope) i that, replacing a impler object by omething more complicated, one get a bonu in the form of ome extra tructure which may be ued to tudy the original object. A priori there are no explicit rule how to categorify ome object and the anwer might depend on what kind of extra tructure and propertie one expect. Example 1.1. The category FS of finite et may be conidered a a categorification of the emi-ring (N 0, +, ) of non-negative integer. In thi picture addition i categorified via the dijoint union and multiplication via the Carteian product. Note that the categorified operation atify commutativity, aociativity and ditributivity law only up to a natural iomorphim. In thee lecture we will deal with ome rather pecial categorification of algebraic object (which are quite different from the above example). There exit many other, even for the ame object. Our categorification are uually motivated by the naturality of their contruction and variou application. It i alway eaier to forget information than to make it up. Therefore it i much more natural to tart the tudy of categorification with the tudy of the oppoite proce of forgetting information, called decategorification. One of the mot natural claical way to forget the categorical information encoded in a category i to conider the correponding Grothendieck group Grothendieck group. Originally, the Grothendieck group i defined for a commutative monoid and provide the univeral way of making that monoid into an abelian group. Let M = (M, +, 0) be a commutative monoid. The Grothendieck group of M i a pair (G, ϕ), where G i a commutative group and ϕ : M G i a homomorphim of monoid, uch that for every monoid homomorphim ψ : M A, where A i a commutative group, there i a unique group homomorphim Ψ : G A making the following diagram commutative: M ϕ G ψ Ψ A In the language of category theory, the functor that end a commutative monoid M to it Grothendieck group G i left adjoint to the forgetful functor from the category of abelian group to the category of commutative monoid. A uual, uniquene of the Grothendieck group (up to iomorphim) follow directly from the univeral property. Exitence i guaranteed by the following contruction: Conider the et G = M M/, where (m, n) (x, y) if and only if m+y + = n + x + for ome M.

5 LECTURES ON ALGEBRAIC CATEGORIFICATION 5 Lemma 1.2. (a) The relation i a congruence on the monoid M M (i.e. a b implie ac bc and ca cb for all a, b, c M M) and the quotient G i a commutative group (the identity element of G i (0, 0); and the invere of (m, n) i (n, m)). (b) The map ϕ : M G defined via ϕ(m) = (m, 0) i a homomorphim of monoid. (c) The pair (G, ϕ) i a Grothendieck group of M. Thi idea of the Grothendieck group can be eaily generalized to the ituation of eentially mall categorie with ome additional tructure. Recall that a category i called eentially mall if it keleton i mall. Categorie which will normally appear in our context are module categorie, additive ubcategorie of module categorie, and derived categorie of module categorie. All uch categorie are eaily een to be eentially mall. The mot claical example i the Grothendieck group of an abelian category. Let A be an eentially mall abelian category with a fixed keleton A. Then the Grothendieck group [A] = K 0 (A) of A i defined a the quotient of the free abelian group generated by [X], where X A, modulo the relation [Y ] = [X] + [Z] for every exact equence (1.1) 0 X Y Z 0 in A. Thi come together with the natural map [ ] : A [A] which map M A to the cla [M ] in [A], where M A i the unique object atifying M = M. The group [A] ha the following natural univeral property: for every abelian group A and for every additive function χ : A A (i.e. a function uch that χ(y ) = χ(x) + χ(z) for any exact equence (1.1)) there i a unique group homomorphim χ : [A] A making the following diagram commutative: A [ ] [A] χ χ A The Grothendieck group of A i the eaiet way to make A into jut an abelian group. In ome claical cae the group [A] admit a very natural decription: Example 1.3. Let k be a field and A = A-mod the category of finite-dimenional (left) module over ome finite dimenional k-algebra A. A every A-module ha a compoition erie, the group [A] i iomorphic to the free abelian group with the bai given by clae of imple A-module. Similarly one define the notion of a Grothendieck group for additive and triangulated categorie. Let A be an eentially mall additive category with biproduct and a fixed keleton A. Then the plit Grothendieck group [A] of A i defined a the quotient of the free abelian group generated by [X], where X A, modulo the relation [Y ] = [X] + [Z] whenever Y = X Z. Note that any abelian category i additive, however, if A i abelian, then the group [A] can be bigger than [A] if there are exact equence of the form (1.1) which do not plit. Let C be an eentially mall triangulated category with a fixed keleton C. Then the Grothendieck group [C] of C i defined a the quotient of the free abelian group generated by [X], where X C, modulo the relation [Y ] = [X] + [Z] for every ditinguihed triangle X Y Z X[1]. Again, a triangulated category i alway additive, but [C] i uually bigger than [C] by the ame argument a for abelian categorie.

6 6 VOLODYMYR MAZORCHUK Let k be a field and A a finite dimenional k-algebra. Then we have two naturally defined triangulated categorie aociated with A-mod: the bounded derived category D b (A) and it ubcategory P(A) of perfect complexe (i.e. complexe, quai-iomorphic to finite complexe of A-projective). For projective P the map i a group homomorphim. ϕ [P(A)] [A-mod] [P ] [P ] Example 1.4. If A ha finite global dimenion, then D b (A) = P(A) and the map ϕ i an iomorphim. Thi mean that in thi cae the group [A-mod] ha another ditinguihed bai, namely the one correponding to iomorphim clae of indecompoable projective A-module. Example 1.5. Conider the algebra D = C[x]/(x 2 ) of dual number. Thi algebra ha a unique imple module L := C (which i annihilated by x). The projective cover of L i iomorphic to the left regular module P := D D. The module P ha length 2. Therefore the group [D-mod] i the free abelian group with bai [L]. The group [P(D)] i the free abelian group with bai [P ] and ϕ([p ]) = 2[L]. Thi mean that [P(D)] i a proper ubgroup of [D-mod] Decategorification. From now on all abelian, triangulated and additive categorie are aumed to be eentially mall. Definition 1.6. Let C be an abelian or triangulated, repectively additive, category. Then the decategorification of C i the abelian group [C], rep. [C]. In what follow, object which we would like to categorify will uually be algebra over ome bae ring (field). Hence we now have to extend the notion of decategorification to allow bae ring. Thi i done in the uual way (ee [MS4, Section 2]) a follow: Let F be a commutative ring with 1. Definition 1.7. Let C be an abelian or triangulated, repectively additive, category. Then the F-decategorification of C i the F-module [C] F := F Z [C] (rep. [C] F := F Z [C] ). The element 1 [M] of ome F-decategorification will be denoted by [M] for implicity. We have [C] = [C] Z and [C] = [C] Z (Pre)categorification of an F-module. Definition 1.8. Let V be an F-module. An F-precategorification (C, ϕ) of V i an abelian (rep. triangulated or additive) category C with a fixed monomorphim ϕ from V to the F-decategorification of C. If ϕ i an iomorphim, then (C, ϕ) i called an F-categorification of V. Wherea the decategorification of a category i uniquely defined, there are uually many different categorification of an F-module V. For example, in cae F = Z we can conider the category A-mod for any k-algebra A having exactly n imple module and realize A-mod a a categorification of the free module V = Z n. In particular, V ha the trivial categorification given by a emiimple category of the appropriate ize, for example by C n -mod. Definition 1.9. Let V be an F-module. Let further (C, ϕ) and (A, ψ) be two F- (pre)categorification of V via abelian (rep. triangulated or additive) categorie. An exact (rep. triangular or additive) functor Φ : C A i called a morphim of

7 LECTURES ON ALGEBRAIC CATEGORIFICATION 7 categorification provided that the following diagram commute: [Φ] [C] F [A] F ϕ ψ V where [Φ] denote the F-linear tranformation induced by Φ. Definition 1.9 turn all F-(pre)categorification of V into a category. In what follow we will uually categorify F-module uing module categorie for finite dimenional k-algebra. We note that extending calar without changing the category C may turn an F-precategorification of V into an F -categorification of F F V. Example Conider the algebra D of dual number (ee Example 1.5). Then there i a unique monomorphim ϕ : Z [D-mod] uch that ϕ(1) = [D]. The homomorphim ϕ i not urjective, however, it induce an iomorphim after tenoring over Q, that i ϕ : Q [D-mod] Q. Hence (D-mod, ϕ) i a pre-categorification of Z while (D-mod, ϕ) i a Q-categorification of Q = Q Z Z. The lat example to ome extend explain the neceity of the notion of precategorification. We will uually tudy categorification of variou module. Module tructure will be categorified uing functorial action, ay by exact functor. Such functor are completely determined by their action on the additive category of projective module. Thi mean that in mot cae the natural bai for categorification in the one given by indecompoable projective. A we aw in Example 1.5, iomorphim clae of indecompoable projective do not have to form a bai of the decategorification Graded etup. By graded we will alway mean Z-graded. Let R be a graded ring. Conider the category R-gMod of all graded R-module and denote by 1 the hift of grading autoequivalence of R-gMod normalized a follow: for a graded module M = i Z M i we have (M 1 ) j = M j+1. Aume that C i a category of graded R-module cloed under ±1 (for example the abelian category R-gMod or the additive category of graded projective module or the triangulated derived category of graded module). Then the group [C] (rep. [C] ) become a Z[v, v 1 ]- module via v i [M] = [M i ] for any M C, i Z. To extend the notion of decategorification to a category of graded module (or complexe of graded module), let F be a unitary commutative ring and ι : Z[v, v 1 ] F be a fixed homomorphim of unitary ring. Then ι define on F the tructure of a (right) Z[v, v 1 ]-module. Definition The ι-decategorification of C i the F-module [C] (F,ι) := F Z[v,v 1 ] [C] (rep. [C] (F,ι) := F Z[v,v 1 ] [C] ). In mot of our example the homomorphim ι : Z[v, v 1 ] F will be the obviou canonical incluion. In uch cae we will omit ι in the notation. We have [C] = [C] (Z[v,v 1 ],id), [C] = [C] (Z[v,v 1 ],id). Definition Let V be an F-module. A ι-precategorification (C, ϕ) of V i an abelian or triangulated, repectively additive, category C with a fixed free action of Z and a fixed monomorphim ϕ from V to the (F, ι)-decategorification of C. If ϕ i an iomorphim, (C, ϕ) i called a ι-categorification of V. Example The algebra D of dual number i naturally graded with x being of degree 2 (thi i motivated by the realization of D a the cohomology ring of a

8 8 VOLODYMYR MAZORCHUK flag variety). Let C = D-gmod. Then [C] = Z[v, v 1 ] a a Z[v, v 1 ]-module, hence the graded category C i a (Z[v, v 1 ], id)-categorification of Z[v, v 1 ] Some contruction. A already mentioned before, for any F we have the trivial categorification of the free module F n given by k n -mod and the iomorphim ϕ : F n [k n -mod] F which map the uual bai of F n to the bai of [k n -mod] F given by iomorphim clae of imple module. If A-mod categorifie ome F k and B-mod categorifie F n for ome finite dimenional k-algebra A and B, then A B-mod categorifie F k F n. Thi follow from the fact that every imple A B-module i either a imple A-module or a imple B-module. If A-mod categorifie F k and B-mod categorifie F n for ome finite dimenional k-algebra A and B, then A k B-mod categorifie F k F F n. Thi follow from the fact that imple A k B-module are of the form L k N, where L i a imple A-module and N i a imple B-module. Let A = A-mod categorify F k uch that the natural bai of F k i given by the iomorphim clae of imple A-module. Let C be a Serre ubcategory of A (i.e. for any exact equence (1.1) in A we have Y C if and only if X, Z C). Then there i an idempotent e A uch that C i the category of all module annihilated by e. Thu C i equivalent to B-mod, where B = A/AeA. The group [C] i a ubgroup of [A] panned by all imple A-module belonging to C. Hence C categorifie the correponding direct ummand V of F k. We alo have the aociated abelian quotient category A/C which ha the ame object a A and morphim given by A/C(X, Y ) := lim A(X, Y/Y ), where the limit i taken over all X X and Y Y uch that X/X, Y C. The category A/C i equivalent to C-mod, where C = eae (ee for example [AM, Section 9] for detail). It follow that A/C categorifie the quotient F k /V, which i alo iomorphic to the direct complement of V in F k. 2. Baic: from categorification of linear map to 2-categorie The aim of the ection i to dicu variou approache to categorification of algebra and module. An important common feature i that any uch approach categorifie linear map a a pecial cae Categorification of linear map. Let V, W be F-module and f : V W be a homomorphim. Aume that (A, ϕ) and (C, ψ) are abelian (rep. additive, triangulated) F-categorification of V and W, repectively. Definition 2.1. An F-categorification of f i an exact (rep. additive, triangular) functor F : A C uch that [F] ϕ = ψ f, where [F] : [A] F ( ) [C]F ( ) denote the induced homomorphim. In other word, the following diagram commute: V ϕ [A] F ( ) f [F] W ψ [C] F ( ) For example, the identity functor i a categorification of the identity morphim; the zero functor i the (unique) categorification of the zero morphim.

9 LECTURES ON ALGEBRAIC CATEGORIFICATION Naïve categorification. Let A be an (aociative) F-algebra with a fixed generating ytem A = {a i : i I}. Given an A-module M, every a i define a linear tranformation a M i of M. Definition 2.2. A naïve F-categorification of M i a tuple (A, ϕ, {F i : i I}), where (A, ϕ) i an F-categorification of M and for every i I the functor F i i an F-categorification of a M i. There are everal natural way to define morphim of naïve categorification. Let A and A be a above, M and N be two A-module and (A, ϕ, {F i : i I}), (C, ψ, {G i : i I}) be naïve categorification of M and N, repectively. In what follow dealing with different categorification we alway aume that they have the ame type (i.e. either they all are abelian or additive or triangulated). By a tructural functor we will mean an exact functor between abelian categorie, an additive functor between additive categorie and a triangular functor between triangulated categorie. Definition 2.3. A naïve morphim of categorification from (A, ϕ, {F i : i I}) to (C, ψ, {G i : i I}) i a tructural functor Φ : A C uch that for every i I the following digram commute: [A] F [F i] [A] F [Φ] [C] F [G i ] where [Φ] denote the morphim, induced by Φ. [Φ] [C] F, Definition 2.4. A weak morphim from (A, ϕ, {F i : i I}) to (C, ψ, {G i : i I}) i a tructural functor Φ : A C uch that for every i I the following digram commute (up to iomorphim of functor): A F i A Φ C G i C Definition 2.5. A trict morphim from (A, ϕ, {F i : i I}) to (C, ψ, {G i : i I}) i a tructural functor Φ : A C uch that for every i I the following digram commute trictly: F i A A Φ C G i C Definition give rie to the naïve (rep. weak or trict) category of naïve categorification of A-module (with repect to the bai A). Thee categorie are in the natural way (not full) ubcategorie of each other. Example 2.6. Conider the complex group algebra C[S n ] of the ymmetric group S n. For λ n let S λ be the correponding Specht module and d λ be it dimenion (the number of tandard Young tableaux of hape λ). Chooe in C[S n ] the tandard generating ytem A coniting of tranpoition i = (i, i+1), i = 1, 2,..., n 1. In S λ chooe the bai coniting of tandard polytabloid (ee e.g. [Sa, Chapter 2]). Then the action of every i in thi bai i given by ome matrix M i = (m i t) d λ,t=1 with integral coefficient. Categorify S λ via D b (C d λ ), uch that the bai of tandard Φ Φ

10 10 VOLODYMYR MAZORCHUK polytabloid correpond to the uual bai of [D b (C d λ )] given by imple module. Categorify the action of every i uing the appropriate direct um, given by the correponding coefficient in M i, of the identity functor (hifted by 1 in homological poition in the cae of negative coefficient). We obtain a (trivial) naïve categorification of the Specht module S λ. Similarly to Example 2.6 one can contruct trivial naïve categorification for any module with a fixed bai in which the action of generator ha integral coefficient. Intead of C-mod one can alo ue the category of module over any local algebra. We refer the reader to [Ma7, Chapter 7] for more detail Weak categorification. To define the naïve categorification of an A-module M we imply required that the functor F i categorifie the action of a M i only numerically, that i only on the level of the Grothendieck group. Thi i an extremely weak requirement o it i natural to expect that there hould exit lot of different categorification of M and that it hould be almot impoible to claify, tudy and compare them in the general cae. To make the claification problem more realitic we hould impoe ome extra condition. To ee what kind of condition we may conider, we have to analyze what kind of tructure we (uually) have. The mot important piece of information which we (intentionally) neglected up to thi point i that A i an algebra and hence element of A can be multiplied. A we categorify the action of the element of A via functor, it i natural to expect that the multiplication in A hould be categorified a the compoition of functor. A we have already fixed a generating ytem A in A, we can conider ome preentation of A (or the correponding image with repect to the action on M) relative to thi generating ytem. In other word, the generator a i could atify ome relation. So, we can try to look for functorial interpretation of uch relation. Here i a lit of ome natural way to do thi: equalitie can be interpreted a iomorphim of functor; addition in A can be interpreted a direct um of functor; for triangulated categorification one could interpret the negative coefficient 1 a the hift by 1 in homological poition (ee Example 2.6), in particular, ubtraction in A can be ometime interpreted via taking cone in the derived category; negative coefficient could be made poitive by moving the correponding term to the other ide of an equality. Another quite common feature i that the algebra A we are working with uually come equipped with an anti-involution. The mot natural way for the functorial interpretation of an anti-involution i via (bi)adjoint functor. Of coure one ha to emphaize that none of the above interpretation i abolutely canonical. Still, we might give the following looe definition from [MS4] (from now on, if F i fixed, we will omit it in our notation for implicity): Definition 2.7. A naïve categorification (A, ϕ, {F i : i I}) of an A-module M i called a weak categorification if it atifie the condition given by ome choen interpretation of defining relation and eventual anti-involution for A. In what follow for a functor F we will denote by F the biadjoint of F (if it exit). Example 2.8. Let A = C[a]/(a 2 2a) and A = {a}. Let further M = C be the A-module with the action a 1 = 0 and N = C the A-module with the action a 1 = 2. Conider C = C-mod, F = 0 and G = Id C Id C. Define ϕ : M [C] and ψ : N [C] by ending 1 to [C] (the cla of the imple C-module). The algebra A ha the C-linear involution defined via a = a. We have both F = F

11 LECTURES ON ALGEBRAIC CATEGORIFICATION 11 and G = G. We interpret a 2 2a = 0 a a a = a + a. We have F F = F F and G G = G G. Hence (C, ϕ, F) and (C, ψ, G) are weak categorification of M and N, repectively. Note that A = C[S 2 ] and that under thi identification the module M and N become the ign and the trivial C[S 2 ]-module, repectively categorie. Following our decription above we now can ummarize that to categorify the action of ome algebra A on ome module M we would like to lift thi action to a functorial action of A on ome category C. So, the image of our lift hould be a nice ubcategory of the category of endofunctor on C. Thi latter category ha an extra tructure, which we already tried to take into account in the previou ubection, namely, we can compoe endofunctor. Thi i a pecial cae of the tructure known a a 2-category. Definition 2.9. A 2-category i a category enriched over the category of categorie. Thi mean that if C i a 2-category, then for any i, j C the morphim C(i, j) form a category, it object are called 1-morphim and it morphim are called 2-morphim. Compoition 0 : C(j, k) C(i, j) C(i, k) i called horizontal compoition and i trictly aociative and unital. Compoition 1 of 2-morphim inide C(i, j) i called vertical compoition and i trictly aociative and unital. We have the following interchange law for any compoable 2-morphim α, β, γ, δ: Thi i uually depicted a follow: 1 (α 0 β) 1 (γ 0 δ) = (α 1 γ) 0 (β 1 δ) 1 = = 0 There i a weaker notion of a bicategory, in which, in particular, compoition of 1-morphim i only required to be aociative up to a 2-iomorphim. There i a natural extenion of the notion of categorical equivalence to bicategorie, called biequivalence. We will, however, alway work with 2-categorie, which i poible thank to the following tatement: Theorem 2.10 ([MP, Le]). Every bicategory i biequivalent to a 2-category. A typical example of a 2-category i the category of functor on ome category. It ha one object, it 1-morphim are functor, and it 2-morphim are natural tranformation of functor. One can alternatively decribe thi uing the notion of a (trict) tenor category, which i equivalent to the notion of a 2-category with one object. A 2-functor F : A C between two 2-categorie i a triple of function ending object, 1-morphim and 2-morphim of A to item of the ame type in C uch that it preerve (trictly) all the categorical tructure. If G : A C i another 2-functor, then a 2-natural tranformation ζ from F to G i a function ending i A to a 1-morphim ζ i C uch that for every 2-morphim α : f g, where 1 Thank to Hnaef on Wikimedia for the L ATEX-ource file!

12 12 VOLODYMYR MAZORCHUK f, g A(i, j), we have F(f) G(f) F(i) F(α) F(g) F(j) ζ j G(j) = F(i) ζ i G(i) G(α) G(g) G(j) In particular, applied to the identity 2-morphim we get that ζ i an ordinary natural tranformation between the aociated ordinary functor F and G. Note that 2-categorie with 2-functor and 2-natural tranformation form a 2-category. A 2-category i called additive if it i enriched over the category of additive categorie. If k i a fixed field, a 2-category i called k-linear if it i enriched over the category of k-linear categorie. Now ome notation: for a 2-category C, object of C will be denoted by i, j and o on, object of C(i, j) (that i 1-morphim) will be called f, g and o on, and 2-morphim from f to g will be written α, β and o on. The identity 1-morphim in C(i, i) will be denoted 1 i and the identity 2-morphim from f to f will be denoted id f. Compoition of 1-morphim will be denoted by, horizontal compoition of 2-morphim will be denoted by 0 and vertical compoition of 2-morphim will be denoted by (Genuine) categorification. Let C be an additive 2-category. Definition The Grothendieck category [C] of C i the category defined a follow: [C] ha the ame object a C, for i, j [C] we have [C](i, j) = [C(i, j)] and the multiplication of morphim in [C] i given by [M] [N] := [M 0 N]. Note that [C] i a preadditive category (i.e. it i enriched over the category of abelian group). A before, let F be a commutative ring with 1. Definition The F-decategorification [C] F of C i the category F Z [C]. The category [C] F i F-linear (i.e. enriched over F-Mod) by definition. Now we are ready to define our central notion of (genuine) categorification. Definition Let A be an F-linear category with at mot countably many object. A categorification of A i a pair (A, ϕ), where A i an additive 2-category and ϕ : A [A] F i an iomorphim. In the pecial cae when A ha only one object, ay i, the morphim et A(i, i) i an F-algebra. Therefore Definition 2.13 contain, a a pecial cae, the definition of categorification for arbitrary F-algebra. Example Let A = C[a]/(a 2 2a). For the algebra D of dual number conider the bimodule X = D C D and denote by C the 2-category with one object i = D-mod uch that C(i, i) i the full additive ubcategory of the category of endofunctor of i, coniting of all functor iomorphic to direct um of copie of Id = Id D-mod and F = X D. It i eay to check that X D X = X X, which implie that C(i, i) i cloed under compoition of functor. The clae [Id] and [F] form a bai of [C(i, i)]. Since F F = F F, it follow that the map ϕ : A [C(i, i)] C uch that 1 [Id] and a [F] i an iomorphim. Hence (C, ϕ) i a C-categorification of A. The above lead to the following major problem: Problem Given an F-linear category A (atifying ome reaonable integrality condition), contruct a categorification of A.

13 LECTURES ON ALGEBRAIC CATEGORIFICATION 13 Among variou olution to thi problem in ome pecial cae one could mention [Ro1, Ro2, La, KhLa]. Wherea [Ro1, Ro2] propagate algebraic approach (uing generator and relation), the approach of [La, KhLa] ue diagrammatic calculu and i motivated and influenced by topological method. The idea to ue 2-categorie for a proper definition of algebraic categorification eem to go back at leat to [Ro1] and i baed on the reult of [CR] which will be mentioned later on. One of the main advantage of thi approach when compared with weak categorification i that now all extra propertie (e.g. relation or involution) for the generator of our algebra can be encoded into the internal tructure of the 2- category. Thu relation between generator now can be interpreted a invertability of ome 2-morphim and biadjointne of element connected by an anti-involution can be interpreted in term of exitence of adjunction morphim, etc. 3. Baic: 2-repreentation of finitary 2-categorie repreentation of 2-categorie. Let C be a 2-category and k a field. A uual, a 2-repreentation of a C i a 2-functor to ome other 2-category. We will deal with k-linear repreentation. Denote by A k, R k and D k the 2-categorie whoe object are fully additive k-linear categorie with finitely many iomorphim clae of indecompoable object, categorie equivalent to module categorie of finite-dimenional k-algebra, and their (bounded) derived categorie, repectively; 1-morphim are functor between object; and 2-morphim are natural tranformation of functor. Define the 2-categorie C-amod, C-mod and C-dmod of k-linear additive 2-repreentation, k-linear 2-repreentation and k-linear triangulated 2-repreentation of C a follow: Object of C-amod (rep. C-mod and C-dmod) are 2-functor from C to A k (rep. R k and D k ); 1-morphim are 2-natural tranformation (thee are given by a collection of tructural functor); 2-morphim are the o-called modification, defined a follow: Let M, N C-mod (or C-amod or C-dmod) and ζ, ξ : M N be 2-natural tranformation. A modification θ : ζ ξ i a function, which aign to every i C a 2-morphim θ i : ζ i ξ i uch that for every 1-morphim f, g C(i, j) and any 2-morphim α : f g we have F(f) ζ j ζ i G(f) F(i) F(α) F(g) F(j) θ j ξ j G(j) = F(i) θ i ξ i G(i) G(α) G(g) G(j) For implicity we will identify object in C(i, k) with their image under a 2- repreentation (i.e. we will ue the module notation). We will alo ue 2-action and 2-module a a ynonym for 2-repreentation. Define an equivalence relation on 2-repreentation of C a the minimal equivalence relation uch that two 2-repreentation of C are equivalent if there i a morphim between thee 2-repreentation, uch that the retriction of it to every object of C i an equivalence of categorie. Now we can define genuine categorification for A-module. Definition 3.1. Let k be a field, A a k-linear category with at mot countably many object and M an A-module. A (pre)categorification of M i a tuple (A, M, ϕ, ψ), where (A, ϕ) i a categorification of A;

14 14 VOLODYMYR MAZORCHUK M A-amod or M A-mod or M A-dmod i uch that for every i, j A and a A(i, j) the functor M(a) i additive, exact or triangulated, repectively; ψ = (ψ i ) i A, where every ψ i : M(i) [M(i)] i a monomorphim uch that for every i, j A and a A(i, j) the following diagram commute: M(i) M(ϕ 1 ([a])) M(j) ψ i [M(i)] k ( ) [M(a)] ψ j [M(j)] k ( ). If every ψ i i an iomorphim, (A, M, ϕ, ψ) i called a categorification of M. It i worth to note the following: if (A, M, ϕ, ψ) and (A, M, ϕ, ψ ) are categorification of M and M, repectively, and, moreover, the 2-repreentation M and M are equivalent, then the module M and M are iomorphic. An important feature of the above definition i that categorification of all relation and other tructural propertie of A i encoded into the internal tructure of the 2-category A. In particular, the requirement for M(a) to be tructural i often automatically atified becaue of the exitence of adjunction in A (ee the next ubection). Later on we will ee many example of categorification of module over variou algebra. For the moment we would like to look at the other direction: to be able to categorify module one hould develop ome abtract 2-repreentation theory of 2-categorie. Some overview of thi (baed on [MM2]) i the aim of thi ection Fiat-categorie. An additive 2-category C with a weak involution i called a fiat-category (over k) provided that (I) C ha finitely many object; (II) for every i, j C the category C(i, j) i fully additive with finitely many iomorphim clae of indecompoable object; (III) for every i, j C the category C(i, j) i enriched over k-mod (in particular, all pace of 2-morphim are finite dimenional) and all compoition are k-bilinear; (IV) for every i C the identity object in C(i, i) i indecompoable; (V) for any i, j C and any 1-morphim f C(i, j) there exit 2-morphim α : f f 1 j and β : 1 i f f uch that (α 0 id f ) 1 (id f 0 β) = id f and (id f 0 α) 1 (β 0 id f ) = id f. If C i a fiat category and M a 2-repreentation of C, then for every 1-morphim f the functor M(f) i both left and right adjoint to M(f ), in particular, both are exact if M C-mod. Example 3.2. The category C from Example 2.14 i a fiat-category. Let M C-mod, i C and M M(i). For j C denote by M (j) the additive cloure in M(j) of all object of the form f M, where f C(i, j). Then M inherit from M the tructure of an additive 2-repreentation of C, moreover, M C-amod if C i fiat Principal 2-repreentation of fiat-categorie. Let C be a fiat category and i C. Conider the 2-functor P i : C A k defined a follow: for all j C the additive category P i (j) i defined to be C(i, j); for all j, k C and any 1-morphim f C(j, k) the functor P i (f) i defined to be the functor f 0 : C(i, j) C(i, k);

15 LECTURES ON ALGEBRAIC CATEGORIFICATION 15 for all j, k C, all 1-morphim f, g C(j, k) and any 2-morphim α : f g, the natural tranformation P i (α) i defined to be α 0 : f 0 g 0. The 2-repreentation P i i called the i-th principal additive 2-repreentation of C. Define alo the abelianization 2-functor : C-amod C-mod a follow: Let M C-amod. For i C define M(i) a the category with object X a Y, where X, Y M(i) and a Hom M(i) (X, Y ). Morphim in M(i) are equivalence clae of diagram a given by the olid part of the following picture: X a Y c b X a Y where X, X, Y, Y M(i), a M(i)(X, Y ) a M(i)(X, Y ), b M(i)(X, X ) and b M(i)(Y, Y ), modulo the ideal generated by all morphim for which there exit c a hown by the dotted arrow above uch that a c = b. In particular, the category M(i) i equivalent to the category of module over the oppoite category of a keleton of M(i). The 2-action of C on M(i) i induced from that on M(i) by applying element of C component-wie. The ret of the 2-tructure of i alo defined via component-wie application. The 2-repreentation P i i called the i-th principal 2-repreentation of C. Propoition 3.3 (The univeral property of P i ). Let M C-amod, M M(i). (a) For j C define Φ M j : P i (j) M(j) a follow: Φ M j (f) := M(f) M for every object f P i (j); Φ M j (α) := M(α) M for every object f, g P i (j) and every morphim α : f g. Then Φ M = (Φ M j ) j C i the unique morphim from P i to M ending 1 i to M. (b) The correpondence M Φ M i functorial. Idea of the proof. Thi follow from the 2-functoriality of M Cell. Let C be a fiat-category. For i, j C denote by C i,j the et of iomorphim clae of indecompoable 1-morphim in C(i, j). 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 exit h C(j, l) uch that g occur a a direct ummand of h f (note that thi i poible only if i = k). Similarly, we will write f L g provided that there exit h C(k, i) uch that g occur a a direct ummand of f h (note that thi i poible only if j = l). Finally, we will write f LR g provided that there exit h 1 C(k, i) and h 2 C(j, l) uch that g occur a a direct ummand of h 2 f h 1. The relation L, R and LR are partial preorder on C. The map f f preerve LR and wap L and R. For f C the et of all g C uch 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-ided cell LR f are defined analogouly. Example 3.4. The 2-category C from Example 2.14 ha two right cell, namely {Id} and {F}, which are alo left cell and thu two-ided cell a well Cell module. Let C be a fiat-category. For i, j C indecompoable projective module in P i (j) are indexed by object of C i,j. For f C i,j we denote by L f the unique imple quotient of the indecompoable projective object P f := (0 f) P i (j). Propoition 3.5. (a) For f, g C the inequality f L g 0 i equivalent to f L g. (b) For f, g, h C the inequality [f L g : L h ] 0 implie h R g. b,

16 16 VOLODYMYR MAZORCHUK (c) For g, h C uch that h R g there exit f C uch that [f L g : L h ] 0. (d) Let f, g, h C. If L f occur in the top or in the ocle of h L g, then f R g. (e) For any f C i,j there i a unique (up to calar) nontrivial homomorphim from P 1i to f L f. In particular, f L f 0. Idea of the proof. To prove (a), without lo of generality we may aume g C i,j and f C j,k. Then f L g 0 if and only if there i h C i,k uch that Hom Pi (k) (P h, f L g ) 0. Uing P h = h P 1i and adjunction we obtain 0 Hom Pi (k) (P h, f L g ) = Hom Pi (j) (f h P 1i, L g ). Thi inequality i equivalent to the claim that P g = g P 1i i a direct ummand of f h P 1i, that i g i a direct ummand of f h. Claim (a) follow. Other claim are proved imilarly Fix i C. Let R be a right cell in C uch that R C i,j for ome j C. Propoition 3.6. (a) There i a unique ubmodule K = K R of P 1i which ha the following propertie: (i) Every imple ubquotient of P 1i /K i annihilated by any f R. (ii) The module K ha imple top, which we denote by L gr, and f L gr 0 for any f R. (b) For any f R the module f L gr ha imple top L f. (c) We have g R R. (d) For any f R we have f L g R and f R gr. (e) We have gr R. Idea of the proof. The module K i defined a the minimal module with property (ai). Then for every f R we have f K = f P 1i = P f. The latter module ha imple top, which implie that K ha imple top. The ret follow from Propoition 3.5. Set L = L gr. For j C denote by D R,j the full ubcategory of P i (j) with object g L, g R C i,j. Note that 2-morphim in C urject onto homomorphim between thee g L. Theorem 3.7 (Contruction of cell module.). (a) For every f C and g R, the module f g L i iomorphic to a direct um of module of the form h L, h R. (b) For every f, h R C i,j we have dim Hom Pi (j) (f L, h L) = [h L : L f ]. (c) For f R let Ker f be the kernel of P f f L. Then the module f R Ker f i table under any endomorphim of f R P f. (d) The full ubcategory C R (j) of P i (j) coniting of all object M which admit a two tep reolution X 1 X 0 M, X 1, X 0 add( f R Ci,j f L), i equivalent to D op R,j -mod. (e) Retriction from P i define the tructure of a 2-repreentation of C on C R, which i called the cell module correponding to R. Example 3.8. Conider the category C from Example For the cell repreentation C {1i} we have G {1i} = 1 i, which implie that C {1i}(i) = C-mod; C {1i}(F) = 0 and C {1i}(α) = 0 for all radical 2-morphim α. For the cell repreentation C {F} we have G {F} = F, which implie that C {F} (i) = D-mod, C {F} (F) = F and C {F} (α) = α for all radical 2-morphim α.

17 LECTURES ON ALGEBRAIC CATEGORIFICATION Homomorphim from a cell module. Let C be a fiat category, R a right cell in C and i C be uch that g R C i,i. Let further f C(i, i) and α : f g R be uch that P i (α) : f P 1i g R P 1i give a projective preentation for L gr. Theorem 3.9. Let M be a 2-repreentation of C. Denote by Θ = Θ M R the full ubcategory of M(i) coniting of all object iomorphic to thoe which appear a cokernel of M(α). (a) For every morphim Ψ from C R to M we have Ψ(L gr ) Θ. (b) For every M Θ there i a morphim Ψ M from C R to M ending L gr to M. Idea of the proof. Uing the univeral property of P i, thi follow from the 2- functoriality of M Serre ubcategorie and quotient. Let C be a fiat category and M C-mod. In every M(i) chooe a et of imple module and denote by N(i) the Serre ubcategory of M(i) which thee module generate. If N turn out to be table with repect to the action of C (retricted from M), then N become a 2- repreentation of C (a Serre ubmodule). Moreover, the quotient Q, defined via Q(i) := M(i)/N(i), alo carrie the natural tructure of a 2-repreentation of C. Example The cell module C {1i } in Example 3.8 i a Serre ubmodule of P i and the cell module C {F} i equivalent to the correponding quotient. Serre ubmodule of M can be organized into a Serre filtration of M Naturally commuting functor. A 2-morphim between two 2- repreentation of ome 2-category C can be undertood via functor naturally commuting with the functor defining the action of C in the terminology of [Kh]. Note that thi notion i not ymmetric, that i if ome functor F naturally commute with the action of C it doe not follow that an element of thi action naturally commute with F (in fact, the latter doe not really make ene a F i not pecified a an object of ome 2-action of a 2-category). Another intereting notion from [Kh] i that of a category with full projective functor, which jut mean that we have a 2-repreentation of a 2-category C uch that 2-morphim of C urject onto homomorphim between projective module of thi repreentation. 4. Category O: definition 4.1. Definition of category O. One of the main ource for categorification model i the Berntein-Gelfand-Gelfand (BGG) category O aociated to a fixed triangular decompoition of a emi-imple complex finite dimenional Lie algebra g. Thi category appear in [BGG2] a a non-emi-imple extenion of the emi-imple category of finite-dimenional g-module, which, on the one hand, contain a lot of new object, notably all imple highet weight module, but, on the other hand, ha everal very nice propertie, notably the celebrated BGG reciprocity, ee Theorem 4.5. The tudy of category O that followed revealed a number of different kind of ymmetrie (e.g. Ringel elf-duality and Kozul elf-duality) and pectacular connection to, in particular, combinatoric and geometry. Mot importantly for u, the Lie-theoretic nature of O lead to a variety of naturally defined functorial action on thi category which, a we will ee, are very ueful and play an important role in variou categorification. Let u tart with a hort recap of baic propertie of O (ee alo [Di, Hu]). For n N conider the reductive complex Lie algebra g = g n = gl n = gl n (C). Let h denote the commutative ubalgebra of all diagonal matrice (the Cartan

18 18 VOLODYMYR MAZORCHUK ubalgebra of g). Denote by n + and n the Lie ubalgebra of upper and lower triangular matrice, repectively. Then we have the tandard triangular decompoition g = n h n +. The ubalgebra b = h n + i the Borel ubalgebra of g. For a Lie algebra a we denote by U(a) the univeral enveloping algebra of a. Definition 4.1. The category O = O(g) i the full ubcategory of the category of g-module which conit of all module M atifying the following condition: M i finitely generated; the action of h on M i diagonalizable; the action of U(n + ) on M i locally finite, that i dim U(n + )v < for all v M. For example, all emi-imple finite-dimenional g-module are object of O. Element of h are called weight. For a g-module M and λ h define the correponding weight pace M λ := {v M : hv = λ(h)v for all h h}. Then the condition of h-diagonalizability can be written in the following form: M = λ h M λ. For i, j {1, 2,..., n} denote by e ij the correponding matrix unit. Then {e ii } form a tandard bai of h. For i < j et α ij = e ii e jj. Then R := {±α ij} i a root ytem of type A n (in it linear hull). Let W = S n be the correponding Weyl group. It act on h (and h) by permuting indexe of element of the tandard bai. For a root α we denote by α the correponding reflection in W. For i = 1, 2,..., n 1 we alo denote by i the imple reflection αii+1. We denote by S the et of all imple reflection. The correponding poitive root form a bai of R. In term of the generator 1, 2,..., n 1 the et of defining relation for W conit of the relation 2 i = e for all i (meaning that every i i an involution) together with the following relation, called braid relation: i j = j i for all i, j uch that i j > 1; i j i = j i j for all i, j uch that i j = 1. A preentation of the braid group B n i given by the ame generator uing only braid relation (o, the generator of B n are no longer involutive). In thi way B n appear with the natural epimorphim onto W given by the identity map on the generator. Let ρ be the half of the um of all poitive root. Define the dot-action of W on h a follow: w λ = w(λ + ρ) ρ. For any M, λ and i < j we have e ij M λ M λ+αij and e ji M λ M λ αij. For i < j we have poitive root {α ij } and negative root { α ij }, moreover, the matrix unit e ij and e ji are root element for root α ij and α ij, repectively. In particular, n + and n are the linear pan of all poitive and negative root pace, repectively. With repect to the ytem S we have the length function l : W {0, 1, 2,... }. We denote by w o the longet element of W and by the Bruhat order on W. Define the tandard partial order on h a follow: λ µ if and only if µ λ i a linear combination of poitive root with non-negative integral coefficient. Let h dom denote the et of all element in h dominant with repect to the dot-action Verma module. For λ h let C λ be the one-dimenional h-module on which element of h act via λ. Setting n + C λ = 0 define on C λ the tructure of a b-module. The induced module M(λ) := U(g) U(b) C λ