Three lectures on categorification of Verma modules

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1 Three lectures on categorfcaton of Verma modules Pedro Vaz Insttut de Recherche en Mathématque et Physque Unversté Catholque de Louvan Chemn du Cyclotron Louvan-la-Neuve Belgum ABSTRACT. We provde an ntroducton to the hgher representaton theory of Kac Moody algebras and categorfcaton of Verma modules. CONTENTS Lecture 1: background 5 Lecture 2: Categorfcaton of Verma modules: sl 2 19 Lecture 3: Algebrac categorfcaton of Verma modules for symmetrzable g 30 Hgher Representaton Theory studes actons of groups, algebras,..., on categores. In HRT the usual basc structures of representaton theory, lke vector spaces and lnear maps, are replaced by category theory analogs, lke categores and functors. Opposte to vector spaces and lnear maps, the world of categores s tremendously bg, offerng enough room for fndng rcher structures: for example, replacng lnear maps by a functors always comes accompaned by a hgher structure whch s assocated to natural transformatons between them. Ths hgher structure s nvsble to tradtonal representaton theory. Categorcal actons of Le algebras were frst developed by Chuang and Rouquer [2] to solve a conjecture on modular representaton theory of the symmetrc group called the Broué conjecture. There were parallel deas beng developed at that tme by Frenkel, Khovanov and Stroppel [4] based on earler work of Khovanov and collaborators. All these deas were boosted by the E-mal address: pedro.vaz@uclouvan.be. Ths survey s an extended verson of the materal covered n a three 90 mnute lecture seres on categorfcaton of Verma modules, held as part of the Junor Hausdorff Trmester Program Symplectc Geometry and Representaton Theory of the Hausdorff Research Insttute n Bonn n November I d lke to aknowledge the organzers of the program. Specal thanks to Danel Tubbenhauer and Grégore Nasse for comments on a prelmnary verson of these notes. 1

2 categorfcaton of quantum groups by Lauda [12], Khovanov Lauda [9, 10, 11] and Rouquer [21] and converged to what s called nowadays Hgher Representaton Theory. Besdes t relatons wth representaton theory, HRT has shown to share nterestng connectons wth other subjects, lke for example topology [14, 18, 20, 22]. A popular example s the constructon of Khovanov s lnk homology n [22] (see also [23]), gvng t a conceptual context n terms of HRT of sl 2. Overvew of the lectures: In ths seres of lectures I gve an ntroducton to my jont work wth Grégore Nasse on categorfcaton of Verma modules for quantum Kac Moody algebras [16, 17, 19]. In Lecture 1, we frst gve the necessary background on representaton theory of (quantum) sl 2 adjustng the exposton n [15] to the quantum case. We the gve a somehow detaled overvew on the categorfcaton of the fnte-dmensonal rreducble representatons of quantum sl 2 usng categores of modules for cohomologes of fnte-dmensonal Grassmannans and partal flag varetes. Ths s due to Frenkel Khovanov Stroppel [4] and ndependently to Chuang Rouquer [2], and s an example of how such categorfcatons arse naturally. In Lecture 2, by workng wth nfnte Grassmannans and addng a bt more structure we are able to categorfy Verma modules for quantum sl 2. One natural sub-product of the geometrc approach to categorfcaton of Verma modules s a certan superalgebra extendng the wellknown nlhecke algebra, one of the fundamental ngredents n the categorfcaton of quantum sl 2. In Lecture 3 we explan the case of categorfcaton of Verma modules for Kac Moody algebras. Ths requres a generalzaton of KLR algebras, the latter beng the man ngredent n the categorfcaton of quantum groups by Khovanov Lauda Rouquer. CONTENTS (DETAILED) Contents 1 1. Lecture I: Background sl 2 -actons Quantum sl sl 2 -modules Integrable modules Weght modules Dense modules Verma modules The Shapovalov form 7 2

3 Fnte-dmensonal modules Categorcal sl 2 -actons What should a categorcal sl 2 -acton be? Integrable categorcal sl 2 -actons Categorfcaton of the fnte-dmensonal rreducbles: the CR FKS approach Categorfcaton of the weght spaces: the cohomology of Grassmannans Movng between the weght spaces: the categorcal sl 2 -acton Categorfcaton of V (n) Categorfyng the dual canoncal bass Unravelng the hgher structure: the nlhecke algebra Axomatc defnton of ntegrable 2-representatons (optonal) Integrable 2-representatons Strong ntegrable 2-representatons CR FKS gves a strong ntegrable 2-representaton An even stronger noton of 2-representaton: proper 2-representatons References for Lecture I Lecture II: Categorfcaton of Verma modules: sl (Unversal) Verma modules revsted Towards a categorfcaton of a Verma module: ntal constrants Topologcal Grothendeck groups Extendng CR FKS to Verma modules Categorfcaton of the weght spaces of M(λq 1 ): H (G(n)) and H (G(n))! Movng between the weght spaces: the categorcal sl 2 -acton The categorfcaton theorem DGAs and the recoverng of CK FKS s categorfcaton of V (n) DG rngs The snake lemma and consequences Derved equvalences nlhecke acton References for Lecture II Appendx to Lecture II: remnders on super structures Superrngs Supermodules and superbmodules Supercategores, superfunctors and supernatural transformatons Lecture III: Algebrac categorfcaton of Vermas for symmetrzable g Towards 2-Verma modules for sl Cyclotomc nlhecke algebra: categorfcaton of V (n) usng NH The superalgebras A n 31 3

4 Bases for A n (optonal) The algebra A n s somorphc to a matrx algebra Categorcal sl 2 -acton and a new categorfcaton of M(λq 1 ) The categorfcaton theorem Derved equvalences Verma modules for sl 2 (optonal) k-lnear 2-Verma modules, and abelan 2-Verma modules Unqueness of strong 2-representatons: sl (Quantum, symmetrzable) Kac Moody algebras and Verma modules Quantum Kac Moody algebras Unversal Verma modules (Parabolc) Verma modules and fnte-dmensonal rreducbles The Shapovalov form Bases for M p (β, N ) p-klr algebras KLR algebras and ther cyclotomc quotents Cyclotomc KLR algebras Extended KLR superalgebras: p-klr superalgebras Tght floatng dots and a bass for R p Sem-cyclotomc p-klr algebras Categorcal g-acton R b and Rp N as a DGAs New bases for cyclotomc KLR algebras Categorfcaton of Verma modules An axomatc defnton of (strong) 2-Verma modules and unqueness References for Lecture III 51 References 51 4

5 1. LECTURE I: BACKGROUND 1.1. sl 2 -actons Quantum sl 2. Let k = C(q). Quantum sl 2 s the assocatve k-algebra U generated by e, f and k ±1, modulo the relatons k f = q 2 f k, ke = q 2 ek, kk 1 = k 1 k = 1, e f f e = k k 1 q q 1. It s a Hopf algebra wth comultplcaton (e) = 1 e +e k 1, (f ) = k f + f 1, (k ±1 ) k ±1, antpode S(k ±1 ) = k 1, S(e) = ek and S(f ) = k 1 f. Ths s a quantzaton of the unversal envelopng algebra of sl sl 2 -modules. We say that (quantum) sl 2 acts on the k-vector space M f we have operators such that, for every m M we have and E,F,K ±1 End k (M) K F (m) = q 2 F K (m), K E(m) = q 2 EK (m), K K 1 (m) = m = K 1 K (m), EF (m) = F E(m) + K (m) K 1 (m) q q 1. We say that sl 2 acts on M through the applcaton f F, e E, k ±1 K ±1 or that M s an sl 2 -module, or even that M s a representaton of sl 2. Sometmes we denote F (m) = F.m, E(m) = E.m, etc... A subspace N M whch s closed under the sl 2 -acton (U.N N ) s called a submodule. An sl 2 -module s rreducble f t does not contan any proper submodule (.e. dfferent from {0} and M) Integrable modules. We say that an sl 2 -acton on M s ntegrable (or that M s ntegrable) f for every m M we have E r 1 (m) = 0 and F r 2 (m) = 0 for r 1,r 2 0. Note that r 1 and r 2 depend on m. In the case of weght modules (see below) and q = 1, the name comes from the fact that one can ntegrate these up to the group Weght modules. Fx a complex number ξ and for α ξ + Z C put q α = λq α ξ k[λ ±1 ]. Suppose that M has an egenvector for K that s, M has a vector m µ for some ξ C and some µ ξ + Z, such that K (m µ ) = q µ m µ. We say that m µ s a weght vector of weght µ. Note that M becomes a k[λ ±1 ]-vector space. It s easy to show that n ths case F (m µ ) and E(m µ ) are also weght vectors of weghts µ 2 and µ+2 respectvely. We see that the subvector space U.m µ M s a submodule whch conssts 5

6 only of weght vectors. Ths s an example of a type of modules called weght modules. The weght space M µ M s the subspace consstng of weght vectors of weght µ: M µ = { m M K (m) = q µ m }. Defne the support supp(m) as the set of all ts weghts: supp(m) = { µ C M µ 0 }. Then, we have that M µ M µ supp(m) s a submodule and, n general M s a weght module f as a vector space, t s the drect-sum of all ts weght spaces: M = M µ. µ supp(m) From now on we wll only consder weght modules. There are also non-weght modules, and these are necessarly nfnte-dmensonal Dense modules. In the specal case M = U.m µ all weght spaces are 1-dmensonal, and t s useful to depct M as n the dagram below: F F F F F F... m µ 4 m µ 2 m µ m µ+2 m µ+4... E K E K E K E K E K E Ths s a collecton of 1-dmensonal k-vector spaces fxed by K ±1, whle F and E allow movng between them. Ths s an example of a class of modules called dense modules. In ths case supp(m) = µ + 2Z Verma modules. Suppose that n a dense module as the one above we have E(m β ) = 0 for some β supp(m). Then U.m β M s a submodule wth support β 2N 0 whch s called a Verma module and denoted M(β). F F F F... m β 6 m β 4 m β 2 m β 0 E K E K E K E K E The vector m β for whch E(m β ) = 0 s a hghest weght vector (of hghest weght β) and M(β) s sad to be a hghest weght module 1 (of hghest weght β) (the termnology should be clear from the dagrams). 1 There s also the correspondng noton of lowest weght module. 6

7 Verma modules are also called standard modules. They are defned as nduced modules. Let U (b) U be the (Hopf) subalgebra generated by k ±1 and e. It s an example of Borel subalgebra and ths one s the standard Borel subalgebra. Let k β = kv β be a 1-dmensonal representaton of U (b) generated by a weght vector v β of weght β: k.v β = q β v β, e.v β = 0. The Verma module M(β) s the nduced module M(β) = U U (b) k β. It s easy to see that t s a hghest weght module of hghest weght β wth the vector 1 v β the hghest weght vector, and that all weght spaces are 1-dmensonal, and therefore t agrees wth the descrpton above as a submodule of a dense module. Physcsts lke lowest-weght modules, as do Rouquer [21]. The descrpton of M(β) as an nduced module has the advantage of gvng mmedately a bass, the F bass. From now on we fnd convenent to label the bass vectors m 0,m 1,...: m +1 m m 2 m 1 m 0 0 [k + 2][β 1] [k + 1][β ] q β 2 [2][β 1] [β] q β Here, for α ξ + Z we put [α] := qα q α q q 1 k(q)[λ±1 ]. There are other nterestng bases: the canoncal bass {m 0,m 1,...}: [ + 2] [ + 1] [2] [1] m +1 m m 2 m 1 m 0 0 [β 1] [β ] q β 2 [β 1] [β] q β The Shapovalov form. Verma modules come equpped wth a blnear form, called the Shapovalov form (, ) β. It s the blnear form on M(β) unquely defned, for m,m M(β), u U, and f k, by (m 0,m 0 ) β = 1, (um,m ) β = (m,ρ(u)m ) β, where ρ s the q-lnear antautomorphsm U U defned by ρ(e) = q 1 k 1 f, ρ(f ) = q 1 ke and ρ(k) = k, f (m,m ) β = (f m,m ) β = (m, f m ) β. 7

8 For example, (β ) [β + 1][β + 2] [β] (m,m ) β = q. [ ]! Note that the canoncal bass and the F bass are both orthogonal w.r.t. the Shapovalov form and ths wll be mportant later When M(β) s rreducble, the Shapovalov form s nondegenerate (ths s n fact a ff condton, snce the radcal of, s a submodule). Ths allows defnng a dual canoncal bass of M(β), denoted {m 0,m 1,...}, as (m,m j ) = δ,j. Ths gves m = q (β ) [ ]! [β + 1][β + 2] [β] m. We need the [β + ] s to be nvertble n k (we can work for example n k(q β )). In ths bass the sl 2 -acton s descrbed n the dagram below. q β 2 1 [β ] q β 3 [β 1] q β 1 [β] m +1 m m 2 m 1 m 0 0 q β+2+1 [ + 1] q β 2 q β+3 [2] q β+1 [1] q β The Verma module M(β) s rreducble unless β N 0. M( n 2) as a submodule. If β = n N 0 then M(n) contans Fnte-dmensonal modules. We denote V (n) the quotent M(n)/M( n 2). It has a canoncal bass {v 0, v 1,...}: [n] [n 1] [2] [1] 0 v n v n 1 v v 2 v 1 v 0 0 q n [1] [2] q n 2 [n 1] [n] q n The bass {v 0, v 1, } s a partcular case of Lusztg-Kashwara s canoncal bases for fnte-dmensonal rreducble representatons of quantum groups. Note that V (n) has several symmetres: t s nvarant under the operaton that swtches v v n 2 for all {0,...,n} and E F. Note also that n ths case k k 1 q q 1 v = [n 2 ]v = (q n q n q n q n+2+1 )v f n 2 0 (and ts negatve f n 2 0) s a fnte sum, and therefore, the man sl 2 -relaton can be wrtten EF (v) F E(v) = [µ]v 8

9 for v n the weght space V µ. Note that, as defned above, [µ] s a polynomal n q. As long as we have the weght space decomposton and the representaton s fnte dmensonal we don t really need K ±1. actually ths s more general: we can get rd of K ±1 actng on a weght module M whenever the support supp(m) (Z). The Shapovalov form descends to an nondegenerate blnear form (, ) n on V (n) (snce we have modded out by ts radcal. Ths allows defnng the dual canoncal bass of V (n) n the same way as before yeldng 2 : (1) v = q ( n+1) [ ]! [n + 1]! v. Ether the canoncal bass and the dual canoncal bass are orthogonal bases w.r.t. to (, ) n. For each n N 0 there s a unque somorphsm class of n + 1-dmensonal rreducble representaton for sl 2. Moreover, every fnte-dmensonal representaton of sl 2 decomposes nto a drect sum of V (n) s for varous n s Categorcal sl 2 -actons What should a categorcal sl 2 -acton be? Roughly speakng, a categorcal sl 2 -acton on a category C conssts of functors F, E, K ±1 on C that satsfy the sl 2 -relatons. There are several ways of defnng what s to satsfy the sl 2 -relatons, and apparently we have to make a choce. The Grothendeck group of a category C endowed wth a class of dstngushed trples (e.g. exact sequences n abelan categores, trangles n trangulated categores, drect-sum decompostons A = B C n addtve categores) s the abelan group K 0 (C ), freely generated by symbols [A] for objects A of C, subjected to relatons [B] = [A] + [C ] for each dstngushed trple (A, B,C ). Sometmes we can take dfferent Grothendeck groups for the same category. We wll then use the notaton G 0 ( ) n the case we take the Grothendeck group w.r.t. exact sequences. Snce we are assumng almost nothng about C, at the tme beng t seems reasonable to ask that the functors F, E, K ±1 nduce an sl 2 -acton on the Grothendeck group 3 of C. Ths means the assgnment f [F], e [E], k ±1 [K ±1 ] defnes an sl 2 -acton on K 0 (C ). Ths seems to be the smplest defnton, but t doesn t say much about the functors, nor about C. The only nformaton we can extract at ths moment s that C s a graded category (n whch q corresponds to the gradng shft {1} va q[a] = [A{1}]). We also know that the functors {E,F,K ±1 } nduce operators on the Grothendeck group. Assumng t exsts, we call t a naïve categorcal 2 Ths s a conventon whch s dfferent from [4]. The dual canoncal bass elements there were defned as (v, v ) = q j (n j ) δ j. 3 In other words, we well be only nterested n exact functors, where exact means they preserve the trples (.e. addtve, exact or trangulated). 9

10 sl 2 -acton. Ths can be slghtly mproved by demandng that the functor F s somorphc to a left adjont of E: sl 2 acts weakly on C f the functors F, E, K ±1 nduce an sl 2 -acton on the Grothendeck group of C, and F s somorphc to a left adjont of E. We say that (C,F,E,K ±1 ) s a weak categorfcaton of the sl 2 -module K 0 (C ). Note that (naïve, weak) categorcal actons on C gves blnear forms on K 0 (C ) (dependng on the type of category C s). For example, one can have [X ],[Y ] = gdm ( Hom C (X,Y ) ). To go any further we have to restrct the class of categores C on whch we act Integrable categorcal sl 2 -actons. Suppose that sl 2 acts weakly on C. Suppose also that C has a zero object and f for every object X of C we have E r 1 (X ) = 0 and F r 2 (X ) = 0 for r 1,r 2 0, where E r 1 (resp F r 2 ) s the composte of E (resp. F) wth tself r 1 (resp. r 2 ) tmes. In ths case, the Grothendeck group of C s a drect sum of fnte-dmensonal rreducble representatons. Moreover, all weghts occurrng n K 0 (C ) are ntegers: supp(k 0 (C )) Z. It seems reasonable to assume that C has fnte coproducts (drect sums) and moreover to ask that t has a block decomposton (recall the orthogonalty of the several bases w.r.t. the Shapovalov form) C = µ ZC µ, where C µ C s the full subcategory generated by objects M such that [M] K 0 (C ) µ. In ths case we can gve a step further and ask the functors {E,F} to satsfy the followng somorphsms (recall we don t need K ±1 anymore): EF(X ) = FE(X ) Id [µ] (X ), for µ 0, FE(X ) = EF(X ) Id [ µ] (X ), for µ 0, for every object X C. Here Id [µ] (X ) = X {µ 1} X {µ 3} X { µ + 3} X { µ + 1}. Lookng further at the form of the drect sum decompostons above we see that we d better work wth categores that are at least addtve. We have a bt more nformaton about the functors that realze the acton (we have a bt more of knowledge about the hgher structure). We have made a crucal choce: we have mposed that the functors F and E satsfy a drectsum decomposton realzng the sl 2 -commutator. Later we wll fnd mportant to reformulate ths condton. We can now summarze. 10

11 Defnton 1.1. An ntegrable categorcal sl 2 -acton on an addtve category C = µ ZC µ conssts of functors F and E such that F s somorphc to a left adjont of E up to degree shft, and they satsfy and F r 1 (X ) = 0, E r 2 (X ) = 0, for r = r (µ) 0 ( = 1,2), for every object X C µ. EF(X ) = FE(X ) Id [µ] (X ), for µ 0, FE(X ) = EF(X ) Id [ µ] (X ), for µ 0, Note we haven t gven a noton of sl 2 -acton that s not naïve, nor weak, nor ntegrable. Ths wll be done later n the context of categorfcaton of Verma modules Categorfcaton of the fnte-dmensonal rreducbles: the CR FKS approach. The man dea s to replace the weght spaces wth categores, on whch f and e act va (exact) functors F and E: F F F V n V n+2 V n 4 V n 2 V n E and ask these functors to be an adjont par ((F,E) as always) and to satsfy the sl 2 -relatons: EF(V n 2k ) = FE(V n 2k ) Id [n 2k] V n 2k, n 2k 0 FE(V n 2k ) = EF(V n 2k ) Id [ n+2k] V n 2k, n 2k 0 E E Categorfcaton of the weght spaces: the cohomology of Grassmannans. For 0 k n, let G k (n) denote the varety of complex k-planes n C N. The cohomology rng of G k (n) has a natural structure of a Z-graded Q-algebra, We wrte H k := H (G k (n),q). H (G k (n),q) = 0 k k(n k) H k (G k,q). The graded rng H k can be gven an explct descrpton n terms of Chern classes. We have H k = Q[c 1,...,c k, c 1,..., c n k ]/I k,n 11

12 where degc j = 2j = deg c j, and I k,n s the deal generated by equatng the terms n the equaton ( 1 + c1 t + c 2 t 2 + c k t k)( 1 + c 1 t + c 2 t c n k t n k) = 1 that are homogeneous n t. Ths s a neat way of encodng a large number of relatons at once. Example 1.2. As a smple example, take k = 1. Then G 1 (n) s the complex projectve space, and H 1 = Q[x]/(x n ) wth deg(x) = 2. Let and set H k -gmod: the category of graded, fntely generated, projectve H k -modules, wth degreepreservng maps, V n 2k = H k -gmod. The rngs H k beng graded local rngs (they have an unque maxmal left/rght deal) mples that ther Grothendeck group s a free Z[q, q 1 ]-module, generated by a unque ndecomposable projectve module, snce objects satsfy the Krull-Schmdt property. Hence, so that the category K 0 (V n ) Z[q,q 1 ] Q(q) = Q(q) V = n V n 2k k=0 categorfes the rreducble representaton V (n) n the sense that as Q(q)-vector spaces. ( ) n K 0 V (n) = K 0 (V n 2k ) Z[q ±1 ] Q(q) = V (n) k=0 At ths pont we do not have any categorcal sl 2 -acton Movng between the weght spaces: the categorcal sl 2 -acton. Consder the partal flag varety G k,k+1 (n) = { (W k,w k+1 ) dm C W k = k, dm C W k+1 = (k + 1), 0 W k W k+1 C n}. We wrte H k,k+1 := H ( G k,k+1 (n) ) for the cohomology rng of ths varety. Agan, ths rng s smple to descrbe explctly n terms of Chern classes: polynomal rng: (2) H k,k+1 := Q[c 1,c 2,...c k ;ξ; c 1, c 2,..., c n k 1 ]/I k,k+1,n, where I k,k+1,n s the deal generated by equatng the homogeneous terms n the equaton ( 1 + c1 + c 2 t c k t k)( 1 + ξt )( 1 + c 1 t + c 2 t c n k 1 t n k 1) = 1. 12

13 As before, everythng s completely explct. Here the generator ξ has degree 2 and corresponds to the Chern class of the natural lne bundle over G k,k+1 (n)) whose fbre over a pont 0 W k W k+1 C n n G k,k+1 (n) s the lne W k+1 /W k. Ths varety has natural forgetful maps G k,k+1 (n) G k+1 (n) G k (n) nducng ncluson maps H k,k+1 H k+1 H k on cohomology. These nclusons make H k,k+1 an (H k+1, H k )-bmodule. Snce these rngs are commutatve we can also thnk of H k,k+1 as an (H k, H k+1 )-bmodule whch we wll denote by H k+1,k. Recall that we get functors between categores of modules by tensorng wth a bmodule. We compose these functors by tensorng the correspondng bmodules. The acton of e and f s gven by tensorng wth the bmodules H k+1,k and H k,k+1, respectvely. V k,k V k+1 V k Defne the functors F k : V n 2k V n 2k 2 E k : V n 2kk+2 V n 2k Res k,k+1 ( Hk,k+1 k+1 k ( ) ) {1 n + k} Res k,k+1 ( Hk,k+1 k k+1 ( ) ) { k} The gradng shfts n the defnton of E and F are necessary to ensure that these functors satsfy the sl 2 -relatons n Proposton 1.4 below. Proposton 1.3. The functors F k, E k have both left and rght adjonts and commute wth the gradng shft functor on graded modules. Ths mples that they are exact, take projectves to projectves and therefore nduce maps on the Grothendeck groups. As a matter of fact, F k (H k ) = [k] H k+1 {1 n + k}, 13 E k (H k+1 ) = [n k] H k { k}.

14 Proposton 1.4. The functors F k, E k satsfy the sl 2 -relatons Categorfcaton of V (n). Put E k F k = Fk E k Id [n 2k] V k, n 2k 0 F k E k = Ek F k Id [n 2k] V k, n 2k 0 F = F k and E = k. k 0 k 0E Theorem 1.5 (Frenkel-Khovanov-Stroppel, Chuang-Rouquer). (1) Functors E and F nduce an acton on the Grothendeck group K 0 (V (n)). (2) Wth ths acton K 0 (V (n)) s somorphc wth V (n), as sl 2 -modules. (3) The somorphsm sends classes of projectve ndecomposables to canoncal bass elements. (4) ([M],[N ]) n = gdmhom V (n) (M, N ). Due to results of Chuang-Rouquer and Rouquer we know that V (n) s essentally unque. Ths wll be our typcal example of a strong categorcal acton, where the somorphsms are fxed by the 2-morphsms. An mportant ngredent s an acton of the nlhecke algebra Categorfyng the dual canoncal bass. In order to categorfy the dual canoncal bass we have to work wth a bgger category. We consder and set H k -fmod: the category of graded, fntely generated, H k -modules, wth degree-preservng maps, V n 2k = H k -fmod, V = n k=1 V n 2k = H k -fmod. The Grothendeck group G 0 (V n 2k ) s a free Z[q, q 1 ]-module generated by the unque smple module snce objects n V n 2k ether have fnte length (and the unque ndecomposable projectve module after tensorng wth Q(q) over Z[q ±1 ]). Example 1.6. For example, for H 1 we have that the smple S of Q[x]/x n s the quotent of Q[x]/x n by the (maxmal) deal generated by x, and ts projectve cover s the ndecomposable Q[x]/x n. The projectve ndecomposable Q[x]/x n has a composton seres 0 x n 1 Q[x]/x n x 2 Q[x]/x n xq[x]/x n Q[x]/x n, where x m Q[x]/x n x m 1 Q[x]/x n s the submodule generated by x m. We have x Q[x]/x n x +1 Q[x]/x n = S{2 }, 14

15 and so, n the Grothendeck group we have [Q[x]/x n n 1 ] = q 2 [S] = q n 1 [n] q [S], =0 where we have wrtten [ ] q for quantum numbers to avod confuson wth the notaton for the classes on the Grothendeck group (cf. (1) whch gves v 1 = q n 1 [n]v 1 ). For the Grothendeck group of V (n) we have an somorphsm as Q(q)-vector spaces. ( G 0 V (n) ) = n k=0 G 0 ( V n 2k ) Z[q ±1 ] Q(q) = V (n) The categorcal acton s constructed as as before, and we have the followng. Theorem 1.7 (Frenkel-Khovanov-Stroppel, Chuang-Rouquer). (1) Functors E and F nduce an acton on the Grothendeck group G 0 (V (n)). (2) Wth ths acton G 0 (V (n)) s somorphc wth V (n), as sl 2 -modules. (3) The somorphsm sends classes of projectve ndecomposables to canoncal bass elements, and classes of rreducbles to dual canoncal bass elements. The dualty between a canoncal bass vector and a dual canoncal bass vector s categorfed by a Hom-lke form. Wthn ths constructon t s not possble to categorfy the change-of bass, snce the formulas expressng dual canoncal bass vectors n terms of canoncal bass vectors (and vce versa) nvolve denomnators. One way to go around s to work wth completed Grothendeck groups à la Achar-Stroppel. A dfferent approach s to consder slghtly dfferent categores and work wth topologcal Grothendeck groups, as we wll see n 2.3. Agan n ths case we have an acton of the nlhecke algebra Unravelng the hgher structure: the nlhecke algebra. We are nterested n studyng the natural transformatons between varous compostes of the functors F k s and E k s. The presentaton of H k,k+1 we have makes t easy to explctly construct bmodule homomorphsms and determne relatons between them. Up to a shft, the functor F m decomposes nto a drect sum of functors, each one nvolvng tensorng (at the left) wth a bmodule lke H r +m,r +m 1 r +m 1 H r +m 1,r +m 2 r +m 2 r +2 H r +2,r +1 r +1 H r +1,r. Ths bmodule s somorphc to the bmodule H r +m,...,r = H ( G r,r +1,...,r +m (n),q ), 15

16 where G r,r +1,...,r +m (n) = { (W r,w r +1,...,W r +m ) dmc W j = j, 0 W r W r +m C n}. Once agan, we can gve an explct descrpton of ths cohomology rng usng Chern classes: H r,,r +m = Q[c 1,...,c r,ξ 1,ξ 2,...,ξ m, c 1,..., c n m ]/I r,...,r +m, where I r,...,r +m s the deal generated by the homogeneous terms n the equaton (1 + x 1 t + x 2 t x r t r )(1 + ξ 1 t)(1 + ξ 2 t) (1 + ξ m t)(1 + y 1 t + + y n m t n m ). The degree two generators ξ j arse from the Chern classes of the lne bundles W r +j /W r +j 1. Lemma 1.8. The operators ξ and j : H r,,r +m H r,,r +m (1 n, 1 j n 1), defned on f H r,,r +m by are (H r, H r +m )-bmodule maps. ξ(f ) = ξ f and j (f ) = f s j (f ) ξ j ξ j +1, The Q-algebra generated by the operators ξ (1 n) and (1 n 1) s called the nlhecke algebra and wll be denoted NH m. It can be defned over any assocatve untal rng k and has a presentaton by the generators above and relatons ξ ξ j = ξ j ξ, (3) ξ j = ξ j f j > 1, j = j f j > 1, ξ = ξ , 2 = 0, ξ = ξ , +1 = An mmedate consequence of Lemma 1.8 s the followng. Proposton 1.9. The composte functors F m carry an acton of the nlhecke algebra NH m. By adjuncton, the E s also carry an acton of the nlhecke algebra. The nlhecke algebra s Z-graded wth deg(ξ) = 2 and deg( ) = 2. Later we wll call ths gradng the q-gradng. Note also that the category V (n) s abelan. In order for t to have a nlhecke acton s enough t s k-lnear. For example we can work wth projectve objects and K 0. 16

17 1.4. Axomatc defnton of ntegrable 2-representatons (optonal) Integrable 2-representatons. The data of an ntegrable categorcal sl 2 - acton allows to sketch a noton of ntegrable 2-representaton. If (F,E} defne an ntegrable sl 2 -acton on C we can form the 2-category C whose objects are the categores {C µ } µ Z, and for where Hom C (C µ,c ν ) s the full subcategory f Fun(C µ,c ν ) generated by drect sums, compostons and gradng shfts of functors F and E. It s mplct from the start that C has a Grothendeck group. In order to have a meanngful noton of a 2-representaton we have to mpose the same s true for C: n ths case K 0 (C) wll result n a representaton of U gven n the form of a category, lke the dagrams n the form of a quver as we have seen before. The acton of U conssts of sendng e and f to the arrows there Strong ntegrable 2-representatons. Usng the example of the categorfcaton of V (n) usng the cohomology of Grassmannans as an nspraton we can defne a noton of (strong) ntegrable 2-representaton of U as follows (see [1]). Defnton 1.10 (Cauts-Lauda). A strong ntegrable 2-representaton of U conssts of a graded, addtve k-lnear dempotent complete 2-category K where: The objects of K are ndexed by the weghts µ Z. There are dentty 1-morphsms Id µ for each µ, as well as 1-morphsms F µ : µ µ 2 n K. We also assume that F µ has a rght adjont and defne the 1-morphsm E µ : µ 2 µ as E µ := (F µ ) R {µ + 1}, On ths data we mpose the followng condtons: () (Integrablty) The dentty 1-morphsm Id µ+2r of the object µ + 2r s somorphc to the zero 1-morphsm for r 0 or r 0. () Hom K (1 µ,1 µ {l}) s zero f l < 0 and one-dmensonal f l = 0. Moreover, the space of 2-morphsms between any two 1-morphsms s fnte dmensonal. () We have the followng somorphsms n K: F µ E µ = Eµ F µ Id [µ] µ, f µ 0, E µ F µ = Fµ E µ Id [ µ] µ, f µ < 0, (v) The F s carry an acton of the nlhecke algebra. We denote K(n) the strong ntegrable 2-representaton where µ s somorphc to the zero object unless µ { n, n + 2,...,n 2,n}. Remark (1) Note that we do not requre an acton of the nlhecke algebra on the E s. The exstence of such an acton wll follow formally from the acton on the F s. (2) The ntegrablty condton mples that 17

18 (a) F µ also has (a shft of) E µ as left adjont (the par (F µ,e µ ) s a then called a badjont par), (b) the 1-morphsms E µ have no negatve degree endomorphsms. (3) We can replace the condton (ntegrablty) by the two condtons above. Note that n some cases badjontness can be hard to check CR FKS gves a strong ntegrable 2-representaton. Usng the CR FKS constructon we defne a 2-category V = V(n) as follows. The objects of V are the blocks V µ for µ supp(v (n)) (recall these are H (n µ)/2 -gmod for µ {n,n 2,..., n + 2, n}), For each par of objects V µ, V ν, the morphsms are formed by the full subcategory of Fun(V µ,v ν ) and natural transformatons, enrched n Q-lnear categores, generated by gradng shfts of drect sums of compostons of the varous F α s and E α s wth the correct doman and target objects. By the results of CR ths 2-representaton s essentally unque. Ths s false n the naïve or weak level An even stronger noton of 2-representaton: proper 2-representatons. Khovanov-Lauda and Rouquer defned the noton of a categorfed quantum group, sometmes referred to as a 2-Kac-Moody algebra. It s a certan monodal 2-category whose Grothendeck group s somorphc to a quantum group (seen as a category). We say that a 2-representaton C of sl 2 s proper f s consstent wth U n the followng sense: there s a 2-functor R : U C such that we have a commutatve dagram U R C U K 0 R K 0 K 0 (C) Cauts and Lauda proved n [1] that strong ntegrable 2-representatons are proper References for Lecture I. [2] J. Chuang and R. Rouquer, Derved equvalences for symmetrc groups and sl 2 - categorfcaton, Ann. of Math. (2) 167 (2008), no. 1, [4] I. Frenkel, M. Khovanov and C. Stroppel, A categorfcaton of fnte-dmensonal rreducble representatons of quantum sl 2 and ther tensor products. Selecta Math. (N.S.) 12 (2006), no. 3-4, [13] A. Lauda, An ntroducton to dagrammatc algebra and categorfed quantum sl 2, Bull. Inst. Math. Acad. Sn. (N.S.) 7 (2012), no. 2, [15] V. Mazorchuk, Lectures on sl(2, C)-modules, Imperal College Press, London, x+263 pp. 18

19 2. LECTURE II: CATEGORIFICATION OF VERMA MODULES: sl (Unversal) Verma modules revsted. Recall that we put λ = q β and treat λ as a formal parameter. Ths gves the unversal Verma module whch, abusng notaton, we wrte M(λ). It s unversal n the sense that any Verma module can be obtaned from ths one by specalzng λ: there s an evaluaton map M(λ) M(β) (λ q β ). Note that our ground feld now contans C(q,λ). We prefer to work over C((q,λ)), the feld of formal Laurent seres for techncal reasons related to the fact that we nterpret denomnators n C(q,λ) as Laurent seres. In terms of the canoncal bass, the unversal Verma module M(λ) has the form: [ + 2] [ + 1] [2] [1] m +1 m m 2 m 1 m 0 0 [λ, 1] [λ, ] λq 2 [λ, 1] [λ,0] λ where (recall ths s [β + r ]). [λ,r ] = λqr λ 1 q r q q Towards a categorfcaton of a Verma module: ntal constrants. Just by lookng at the dagram above for M(λ) one sees that one needs to categorfy multplcaton by [λ,k]. We see that we need (at least) a bgraded category. Note also that we cannot wrte [λ,k] as a fnte sum. One way forward s to nterpret the denomnator as a power sum 1 q q 1 = q 1( 1 + q + q 2 + ), and ask our categores to have (controlled) nfnte coproducts. But we also have the mnus sgns!. One possblty to deal wth them s to work wth supercategores 4. Notaton: For an object X n such a category we denote by X r, s ts shft up by r unts n the frst gradng (the q ) and by s unts n the second (the λ ) and by ΠX ts shft n the Z/2Z-degree, called the party. 1 To categorfy multplcaton by one can (and we wll!) consder the nfnte coproduct q q 1 Π( ) 2 1,0. N 0 4 See 2.7 for defntons and conventons on super structures. 19

20 We have made some choces, and they seem reasonable at ths pont! Altogether, f we are to categorfy Verma modules we can work wth (at least) addtve, bgraded, supercategores, whch have nfnte coproducts. In ths context a weak categorfcaton of the Verma module M(λ) should consst of an addtve, bgraded, supercategory, wth nfnte coproducts M (λ) = µ supp(m(λ)) M (λ) µ, and functors F, E, K ±1 Fun ( M (λ) ) such that F s a left adjont of E, and they satsfy a categorcal verson of the sl 2 -relatons. Let s look at what we mean by sl 2 -relatons n ths context. There s a pont whose mportance s fundamental: F s a left adjont but not a rght adjont of E (the par (F,E) s not an adjont par), otherwse, there would be an r N such that F r = 0 (see the remarks rght after the defnton of strong ntegrable 2-representaton). Ths means that F, E and K ±1 cannot be connected through a drect-sum decomposton as n the case of categorfcaton of ntegrable representatons. Otherwse, that maps realzng the decomposton could be used to mply that (F,E) s a badjont par. Recall that we want ths EF-relaton to mply the sl 2 -commutator on the Grothendeck group. Therefore, the next type of relaton one can thnk of s to ask that M (λ) admts exact sequences and the composte functors EF and FE be related through an exact sequence... One of the prncpal features of the unversal Verma module s that t projects to the rreducble representaton V (n) (for any n). It seems reasonable to mpose that a weak categorfcaton of M(λ) comes equpped wth a categorcal projecton onto a categorfcaton of V (n). Let s sketch a provsonal defnton. Defnton 2.1 (Provsonal defnton). A weak categorfcaton of the Verma module M(λ) s an addtve, bgraded, supercategory M (λ), wth nfnte coproducts and admttng exact sequences, M (λ) = µ supp(m(λ)) M (λ) µ, and functors F, E, K ±1 Fun(M (λ)), whch commute wth gradng shfts, and descend to operators on a Grothendeck group of M (λ) and satsfy: (1) F ( M (λ) µ ) M (λ)µ 2, E ( M (λ) µ ) M (λ)µ+2, K ( M (λ) µ ) M (λ)µ, for all µ supp(m(λ)), (2) F s somorphc to a left adjont of E (up to a shft), (3) KF = FK 2,0, KE = EK 2,0, KK ±1 = K ±1 K = Id M (λ). (4) F, E K ±1 and commute wth gradng shfts and wth the party change Π, 20

21 (5) there s a (non-splt) exact sequence 0 EF FE QK ΠQK 1, where Q s the nfnte coproduct Q( ) = N 0 (Π( )) 2 1,0, (6) For any n N 0 there s a projecton from M (λ) to a categorfcaton of V (n) Topologcal Grothendeck groups. The fact that we are to work wth nfnte coproducts mpose severe restrctons on the categores we wll work wth. Recall that we n order to categorfy Verma modules we need the Grothendeck group of each block M (λ) µ (µ supp(m(λ))) to be fnte dmensonal (and non-zero). We wll work wth (bgraded, super, locally addtve 5 ) categores C whose enrched Hom spaces HOM(M, N ) = l,r Z HOM l,r (M, N ) = l,r Z are fnte-dmensonal n each degree. Moreover we demand that Hom(M, N l,r ) pars (l,r ) Z 2 for whch HOM l,r (M, N ) 0, le nsde a cone C Z 2 compatble wth an order n Z 2, HOM l,r (M, N ) = 0 for r 0 or r 0, We also requre that C has: or or both. local Krull-Schmdt property: every object decomposes nto a locally fnte drect sum of small 6 objects havng local endomorphsms rngs (see [16, 5.1]), local Jordan-Hölder property (f C s abelan: every object has locally fntely many composton factors, plus a stablty condton) (see [16, 5.2]), Remark 2.2. A locally Krull-Schmdt category s dempotent complete. Moreover, an object wth local endomorphsm rng s ndecomposable, and have only 0 and 1 as dempotents. For these type of categores we can defne the Topologcal Grothendeck group G 0 (C ): ths s the free Z((q,λ))-module generated by the classes of smple objects (up to shfts). Topologcal splt Grothendeck group K 0 (C ): ths s the free Z((q,λ))-module generated by the classes of ndecomposables (up to shfts). 5 We say that an addtve, strctly Z-graded category C (A{r } = A for all A C ) s locally addtve f all ts locally fnte coproducts are bproducts. 6 An object A n a category C s small f every map f : A 21 I B factors through B j for a fnte subset J I. j J

22 Both Grothendeck groups above are modules over Z π = Z[π]/π 2 1. When specalzng the parameter π = 1 and extendng the scalars to Q, we wrte G 0 (C ) = G 0 (C )) Zπ Q[π]/(π + 1), and the same for K 0 (C ). Remark 2.3. The condtons above are manly techncal and are necessary to be able to defne Grothendeck groups wth the correct propertes for the sake of categorfcaton. A complete descrpton of these categores and the detals ts Grothendeck groups can be found n [16, 5] Extendng CR FKS to Verma modules. In the followng t seems more natural to categorfy M(λq 1 ) and ths wll be clear very soon. We want to fnd nce bgraded (super)rngs Ω k wth 1-dmensonal Grothendeck groups and (Ω k,ω k+1 )-bmodules (denoted Ω k,k+1 ), such that Ω k,k+1 s a free Ω k+1 -left module of graded rank [k + 1], Ω k,k+1 s a free Ω k -left module of graded rank [λ; k], The (Ω k,ω k )-bmodules Ω k(k+k)k := Ω k,k+1 k+1 Ω k+1,k, Ω k(k k)k := Ω k,k 1 k 1 Ω k 1,k are related through a short exact sequence 0 Ω k(k 1)k Ω k(k+1)k Ω k [ξ] 2k + 2, 1 ΠΩ k [ξ] 2k,1 [k + 1] [k] Ω k,k+1 m k+1 m k m k 1 [λ; k] [λ; k + 1] λq 2(k 1) Ω k+1 Ω k Categorfcaton of the weght spaces of M(λq 1 ): H (G(n)) and H (G(n))!. Let G k be the Grassmannan varety of k-planes n C. Its (ratonal) cohomology rng s just a polynomal rng generated by the Chern classes H(G k ) = Q[x 1,..., x k ], deg(x k ) = 2k, The Ext-algebra Ext H(Gk )(Q,Q) s an exteror algebra Ext H(Gk )(Q,Q) = (ω 1,...,ω k ), wth deg(ω k ) = ( ) qdeg(ω k ),λdeg(ω = ( 2k,2). We form Ω k = H(G k ) Ext H(Gk )(Q,Q) whch we regard as a bgraded superrng. Here the x s are even whle the ω s are odd. 22

23 Put M λq 1 2k = Ω k -mod lf The latter beng bgraded, left (super) Ω k -modules that are fnte dmensonal on each degree and cone bounded Movng between the weght spaces: the categorcal sl 2 -acton. Let G k,k+1 be the nfnte 1-step partal flag varety {0 W k W k+1 C dm C (W j ) = j }. We have Put H(G k,k+1 ) = Q[y 1,..., y k,ξ], deg(y j ) = 2j,deg(ξ) = 2. ( ) Ω k,k+1 = H(G k,k+1 ) Ext H(Gk+1 ) Q,Q We consder the maps Ω k,k+1 H(G k,k+1 ) Ext H(Gk+1 )(Q,Q) H(G k+1 ) Ext H(Gk+1 )(Q,Q) Ω k+1 Explctly, these maps are φ k : Ω k Ω k,k+1, H(G k ) Ext H(Gk )(Q,Q) Ω k x j y j, ω j ω j + ξω j +1, and ψ k+1 : Ω k+1 Ω k,k+1, x j y j + ξj j 1, ω j ω j. wth y 0 = 1 and y +1 = 0. Defne the functors F k : M λq 1 2k M λq 1 2(k+1) E k : M λq 1 2(k+1) M λq 1 2k K k : M λq 1 2k M λq 1 2k Res k,k+1 ( Ωk,k+1 k+1 k ( ) ) k,0 Res k,k+1 ( Ωk,k+1 k k+1 ( ) ) k + 2, 1 ( ) 2k,1 23

24 and Q k : M λq 1 2k M λq 1 2k, defned for all k 0 by Q( ) = Π( ) Q[ξ] 1,0 and put and M (λq 1 ) = k 0M λq 1 2k F = F k, k 0 E = E k, K = k. k 0 k 0K Proposton 2.4. Functors F, E are exact and E s somorphc to a rght adjont of F (they are not badjont!). Moreover, there s an acton of the nlhecke algebra on F m (and on E m ). Actually, there s a bgger (super)algebra actng on F m and E m. It can be computed through bmodule homomorphsms the same way we dd NH m. We wll fnd ths algebra agan n Theorem 2.5. We have natural somorphsms KK 1 = Id = K 1 K, KF = FK 2,0, KE = EK 2,0, and a natural exact sequence The categorfcaton theorem. Theorem FE EF KQ ΠK 1 Q 0. (1) The functors F, E and K nduce an acton of quantum sl 2 on the Grothendeck group of M (λq 1 ). Wth ths acton K 0 (M ) s somorphc wth the Verma module M(λq 1 ) after specalzng the acton of [Π] to 1. (2) The somorphsm from K 0 (M (λq 1 )) sends classes of projectve ndecomposable objects to canoncal bass elements, and classes of rreducbles to dual canoncal bass elements DGAs and the recoverng of CK FKS s categorfcaton of V (n) DG rngs. For n N 0 and for each k we turn Ω k and Ω k,k+1 nto DG rngs by ntroducng dfferentals dn k k,k+1 and dn, both wth degrees n, 2 and Z/2Z-degree 1. These act trvally on H(G k ) and H(G k,k+1 ) and send the generators of Ext H(G )(Q,Q) to elements of H(G k ) (and H(G k,k+1 ) respectvely). Moreover, these dfferentals commute wth the canoncal maps that gve Ω k,k+1 the structure of a (Ω k,ω k+1 )-bmodule, so that t becomes a DG bmodule. Proposton 2.7. The DG rngs (Ω k,dn k) and (Ω k,k+1,dn k ) are formal. Moreover we have quassomorphsms (Ω k,d k n ) = q.. (H(G k (n)),0) (Ω k,k+1,d k n ) = q.. (H(G k,k+1 (n)),0). 24

25 The snake lemma and consequences. Recall that n the case of V (n) we had a drect sum decomposton for the functors EF and FE. We wll now see that ths comes naturally as a consequence of our exact sequence and dfferentals. We can equp Ω k [ξ] ΠΩ k [ξ] 2k 2 wth a dfferental d n such that t becomes a DG bmodule over (Ω k,dn k). Ths s not a drect sum of two DG bmodules, snce d n mxes terms of both summands. Ths dfferental s compatble wth the maps n the SES and we have a short exact sequence of DG ((Ω k,d n ),(Ω k,d n ))-bmodules 0 ( ) ( ) Ω k(k 1)k,d n Ωk(k+1)k,d n (Ωk [ξ] 2k, 1 ΠΩ k [ξ] 2k 2,1,d n ) 0. By the snake lemma, t descends to a long exact sequence of H(Ω k,d n ) = H(G k;n )-bmodules... H 1 ( ) Ω k(k+1)k,d n H 1 (Ω k [ξ] 2k, 1 ΠΩ k [ξ] 2k 2,1,d n ) H 0 ( Ω k(k 1)k,d n ) H 0 ( Ω k(k+1)k,d n ) H 0 (Ω k [ξ] 2k, 1 ΠΩ k [ξ] 2k 2,1,d n ) H 1 ( Ω k(k 1)k,d n )... We know that the homology of ( Ω k(k+1)k,d n ) s concentrated n party 0 and thus we have a long exact sequence 0 H 1 (Ω k [ξ] 2k, 1 ΠΩ k [ξ] 2k 2,1,d n ) H 0 ( Ω k(k 1)k,d n ) H 0 ( Ω k(k+1)k,d n ) H 0 (Ω k [ξ] 2k, 1 ΠΩ k [ξ] 2k 2,1,d n ) 0. Snce we have explct formulas and nce decompostons we can easly compute the homologes: (1) For n 2k 0 the homology of ( ) Ω k [ξ] 2k, 1 ΠΩ k [ξ] 2k 2,1,d n s concentrated n party 0 and gven by q 2k H(G k;n ). {n 2k} Therefore we get the followng short exact sequence 0 H(G k,k 1;n ) H(Gk 1;n ) H(G k 1,k;n ) H(G k,k+1;n ) H(Gk+1;n ) H(G k+1,k;n ) q 2k H(G k;n ) 0. {n 2k} 25

26 (2) For n 2k 0 the homology s concentrated n party 1 and t s somorphc to q 2k 2 λ 2 ΠH(G k;n ). {2k n} After shftng by the degree of the connectng homomorphsm, t yelds the short exact sequence 0 {2k n} q 2k H(G k;n ) H(G k,k 1;n ) H(Gk 1;n ) H(G k 1,k;n ) H(G k,k+1;n ) H(Gk+1;n ) H(G k+1,k;n ) 0. Proposton 2.8. Both exact sequences splt, and recover the well-known sl 2 categorcal acton of CR FKS [2, 4] usng cohomology of the fnte Grassmannans and 1-step flag varetes. Defne the DG ((Ω k+1,d n ),(Ω k,d n ))-bmodule and the DG ((Ω k,d n ),(Ω k+1,d n ))-bmodule ( Ω k+1,k,d n ) = (Ω k+1,k,d n ) 0,0, ( Ω k,k+1,d n ) = (Ω k,k+1,d n ) n,1. Proposton 2.9. We have quas-somorphsms of bgraded DG ((Ω k,d n ),(Ω k,d n ))-bmodules ( Ω k,k+1 k+1 Ω k+1,k,d n ) = ( Ω k,k 1 k 1 Ω k 1,k,d n ) [n 2k] (Ω k,d n ), f n 2k 0, ( Ω k,k 1 k 1 Ω k 1,k,d n ) = ( Ω k,k+1 k+1 Ω k+1,k,d n ) [2k n] (Ω k,d n ), f n 2k Derved equvalences. Let D c ( Ω k,d n ) and D c ( Ω k,k+1,d n ) be respectvely the derved category of bgraded, left, compact ( Ω k,d n ) modules and the derved category of bgraded, left, compact ( Ω k+1,k,d n )-modules. Proposton There are equvalences of trangulated categores between D c ( Ω k,d n ) = D b (H(G k (n))-gmod) = D b (V k+1,k ) D c ( Ω k+1,1,d n ) = D b (H(G k,k+1 (n))-gmod) = D b (V k+1,k ), where D b ( ) s the bounded derved category. Recall that V k and V k+1,k are the categores used n CR FKS. The nducton (derved) functor Ind k+1,k k bmodule ( Ω k+1,k,d n ): Ind k+1,k k and the restrcton functor Res k+1,k k Res k+1,k k s the derved tensor functor assocated wth the DG = ( Ω k+1,k,d n ) L k ( ): Dc ( Ω k,d n ) D c ( Ω k+1,k,d n ) concdes wth the (derved) functor = RHom (Ωk,d n )( ( Ω k+1,k,d n ), ) : D c ( Ω k+1,k,d n ) D c ( Ω k,d n ). 26

27 Analogously, we defne and Ind k+1,k k+1 : D c ( Ω k+1,d n ) D c ( Ω k+1,k,d n ), Res k+1,k : D c ( Ω k+1 k+1,k,d n ) D c ( Ω k+1,d n ). For each k 0 defne the functors F k ( ) = Res k,k+1 ( ( Ω k+1 k+1,k,d n ) L k ( )), E k ( ) = Res k,k+1 ( ( Ω k k,k+1,d n ) L k+1 ( )), where ( Ω k+1,k,d n ) s seen as a DG ((Ω k+1,k,d n ),(Ω k,d n ))-bmodule and ( Ω k,k+1,d n ) as a DG ((Ω k,d n ),(Ω k,k+1,d n ))-bmodule. Corollary The functors F k and E k are badjont up to a shft. Moreover we have natural somorphsms and E k F k = Fk 1 E k 1 [n 2k] Id k, f n 2k 0, F k 1 E k 1 = Ek F k [2k n] Id k, f n 2k 0. Corollary Defne the category W (n) = k 0D c ( Ω k,d n ). We have a Z[q, q 1 ]-lnear somorphsm of U q (sl 2 )-modules, K 0 (W (n)) = V (n), for all n nlhecke acton. By takng tensor products we can form the DG ((Ω k,d n ),(Ω k+m,d n )- bmodule ( Ω k,...,k+m,d n ) and the DG (Ω k+m,d n ),(Ω k,d n )) n -bmodule ( Ω k+m,...,k,d n ). Proposton The nlhecke algebra NH m acts as endomorphsms of the DG bmodules ( Ω k,...,k+m,d n ) and ( Ω k+m,...,k,d n ). Corollary The nlhecke algebra NH s acts as endomorphsms of E s and of F s. Ths acton concdes wth the one from Lauda and Chuang-Rouquer References for Lecture II. [16] G. Nasse and P. Vaz, An approach to categorfcaton of Verma modules, arxv: v2 [math.rt] (2016). 27

28 2.7. Appendx to Lecture II: remnders on super structures. The followng materal s standard and can be found for example n [3, 7] or [8] Superrngs. A superrng s a Z/2Z-graded rng. Let R = R 0 R 1 be a superrng. We wll call the Z/2Z-gradng the party and use the notaton p(r ) to ndcate the party of a homogeneous element r R. Elements wth party 0 are called even, elements of party 1 are called odd. Whenever we refer to an element of R as even or odd, we wll always be assumng that t s homogeneous. A subsuperrng of R s a subrng whch s tself a superrng, that s, the canoncal ncluson preserves the party. The supercenter Z s (R) of R s the set of all elements of R whch supercommute wth all elements of R, that s Z s (R) = {x R xr = ( 1) p(x)p(r ) r x for all r R} Supermodules and superbmodules. A left R-supermodule M s a Z/2Z-graded module over R such that R M j M + (, j Z/2Z). A left supermodule map f : M N s a homogeneous group homomorphsm that supercommutes wth the acton of R, f (r m) = ( 1) p(f )p(r ) r f (m), for all r R and m M. A rght supermodule map s a homogeneous rght module homomorphsm. An (R,R )-superbmodule s both a left R-supermodule and a rght R -supermodule, wth compatble actons. A superbmodule map s both a left supermodule map and a rght supermodule map. Then, f R has a supercommutatve rng structure and f we vew t as an (R,R)- superbmodule, multplyng at the left by an element of R gves rse to a superbmodule endomorphsm. Let M and N be respectvely an (R,R) and an (R,R )-superbmodules. One form ther tensor product over R n the usual way for bmodules, gvng a superbmodule. Gven two superbmodule maps f : M M and g : N N, we can form the tensor product f g : M N M N, whch s defned by and gves a superbmodule map. (f g )(b m) = ( 1) p(g )p(b) f (b) g (m), Now defne the party shft of a supermodule M, denoted ΠM = {π(m) m M}, where π(m) s the element m wth the party nversed, and f M s a left supermodule (or superbmodule) wth left acton gven by r π(m) = ( 1) p(r ) π(r m), for r R and m M. The acton on the rght remans the same. In ths context, the map R ΠR defned by r π(ar ) for some odd element a R s a Z/2Zgradng preservng homomorphsm of (R,R)-superbmodules. 28

29 Let π : M ΠM denote the change of party map x π(x). It s a supermodule map wth party 1 and satsfes π 2 = Id. The map π π : ΠM N M ΠN s Z/2Z-gradng preservng and such that (π π) 2 = Id, thus Π(M N ) = ΠM N = M ΠN are somorphsms of supermodules. Of course, all the above extends to the case when R has addtonal gradngs, makng t a multgraded superrng. In ths context we can speak of (mult)graded modules, (mult)graded bmodules, etc Supercategores, superfunctors and supernatural transformatons. A supercategory s a category C equpped wth a strong categorcal acton of Z/2Z. A superfunctor s a morphsm of strong categorcal Z/2Z-actons. More precsely, Defnton A supercategory s the followng data: a category C, a functor Ψ C : C C, and a natural somorphsm ξ C : Ψ 2 C 1 C, such that ξ C 1 ΨC = 1Ψ C ξ C as natural somorphsms from Ψ 3 C to Ψ C. The data of a superfunctor (F,α F ) : (C,Ψ C ) (D,Ψ D ) s: a functor F : C D and a natural somorphsm α F : F Ψ C Ψ D F, and the only condton s that 1 F ξ C = (ξ D 1 F )(1 ΨD α F )(α F 1 ΨC ). Left (resp. rght) supermodules and left (resp. rght) supermodule maps form supercategores, as well as superbmodules and superbmodule maps. Defnton Let (F,α F ),(G,α G ) : (C,Ψ C ) (D,Ψ D ) be superfunctors. A supernatural transformaton between (F,α F ) and (G,α G ) s a natural transformaton ϕ : F G whch commutes wth the natural somorphsms α F,α G n the sense that the dagram ϕ1 Ψ F Ψ C GΨ C commutes. α F Ψ D F 1 Ψ ϕ α G Ψ D G 29