Séminaire BOURBAKI Juin ème année, , n o CATEGORIFICATION OF LIE ALGEBRAS [d après Rouquier, Khovanov-Lauda,...

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1 Sémnare BOURBAKI Jun ème année, , n o 1072 CATEGORIFICATION OF LIE ALGEBRAS [d après Rouquer, Khovanov-Lauda,...] by Joel KAMNITZER INTRODUCTION Categorfcaton s the process of fndng hdden hgher level structure. To categorfy a natural number, we look for a vector space whose dmenson s that number. For example, the passage from Bett numbers to homology groups was an mportant advance n algebrac topology. To categorfy a vector space V, we look for a category C whose Grothendeck group s that vector space, K(C) = V. If V carres an acton of a Le algebra g, then t s natural to look for functors F a : C C for each generator a of g, such that F a gves the acton of a on the Grothendeck group level. In ths case, we say that we have categorfed the representaton V. There are two general motvatons for tryng to categorfy representatons. Frst, by studyng the category C, we hope to learn more about the vector space V. For example, we get a specal bass for V comng from classes of ndecomposable obects of C. Second, we may use the acton of g on C to learn more about C. For example, Chuang-Rouquer used categorfcaton to prove Broué s abelan defect group conecture for symmetrc groups. Recently, there has been amazng progress towards constructng categorfcatons of representatons of semsmple (or more generally Kac-Moody) Le algebras. In ths report, we am to gve an ntroducton to ths theory. We start wth the categorfcaton of sl 2 and ts representatons. We explan the nave defnton and then the true defnton, due to Chuang-Rouquer [CR]. We also explan how ths defnton leads to nterestng equvalences of categores. We then address general Kac-Moody Le algebras, reachng the defnton of the Khovanov-Lauda-Rouquer 2-category [R2, KL3]. We explan the relatonshp to Lusztg s categores of perverse sheaves, due to Varagnolo-Vasserot [VV] and Rouquer [R3]. We close by dscussng three fundamental examples of categorcal representatons: modular representaton theory of symmetrc groups (due to Lascoux-Leclerc-Thbon [LLT], Gronowsk [Gr], and Chuang-Rouquer [CR]), cyclotomc quotents of KLR algebras (due to Kang-Kashwara [KK] and Webster [W1]), and quantzed quver varetes (due to Zheng [Z] and Rouquer [R3]). In order to keep the exposton readable, we have made a number of smplfcatons and glossed over many detals. In partcular, we only address smply-laced Kac-Moody

2 Le algebras (and when t comes to the geometry, only fnte-type). We suggest that nterested readers consult the lterature for more detals. Throughout ths paper, we work over C; all vector spaces are C-vector spaces (sometmes they are actually C(q)-vector spaces) and all addtve categores are C-lnear. I would lke to thank R. Rouquer, M. Khovanov, and A. Lauda for developng the beautful mathematcs whch s presented here and for ther many patent explanatons (an extra thank you to A. Lauda for allowng me to use hs dagrams). I also thank D. Ben-Zv, R. Bezrukavnkov, A. Braverman, J. Brundan, C. Dodd, D. Gatsgory, H. Nakama, A. Kleshchev, A. Lcata, D. Nadler, B. Webster, G. Wllamson, and O. Yacob for nterestng dscussons about categorfcaton over many years and a specal thank you to S. Cauts for our long and frutful collaboraton. Fnally, I thnk S. Cauts, M. Khovanov, A. Lauda, C. Lu, S. Morgan, R. Rouquer, B. Webster and O. Yacob for ther helpful comments on a frst draft of ths paper. 1. CATEGORIFICATION OF sl 2 REPRESENTATIONS 1.1. The structure of fnte-dmensonal representatons The Le algebra sl 2 (C) has the bass e = [ ], h = [ ], f = [ ]. Consder a fnte-dmensonal representaton V of sl 2. A basc theorem of representaton theory states that h acts semsmply on V wth nteger egenvalues. Thus we may wrte V = r Z V r as the drect sum of the egenspaces for h. Moreover the commutaton relatons between the generators e, f, h mply the followng. 1. For each r, e restrcts to a lnear map e : V r V r Smlarly, f restrcts to a lnear map f : V r V r These restrctons obey the commutaton relaton (1) ef fe Vr = ri Vr. Conversely, a graded vector space V = V r, along wth rasng and lowerng operators e, f as above, defnes a representaton of sl 2 f these operators satsfy the relaton (1). The followng example wll be very nstructve. Example 1.1. Let X be a fnte set of sze n. Let V = C P (X) be a vector space whose bass conssts of the subsets of X. For r = n, n + 2,..., n, defne V r to be the span of subsets of sze k, where r = 2k n. Defne lnear maps e : V r V r+2, f : V r V r 2 by the formulas (2) e(s) = T, f(s) = T T S, T = S +1 T S, T = S 1

3 It s easy to check that (ef fe)(s) = (2k n)s, f S has sze k. (The basc reason s that there are n k ways to add somethng to S and k ways to take somethng away from S.) Thus ths defnes a representaton of sl 2. In fact, ths representaton s somorphc to an n-fold tensor product (C 2 ) n of the standard representaton of sl 2. We wll also need the concept of a representaton of the quantum group U q sl 2, though we wll nether need nor gve an explct defnton of U q sl 2. For each nteger r, let [r] := qr q r q q 1 = qr 1 + q r q r+1 denote the quantum nteger (the second expresson s only vald f r 0). A representaton of U q sl 2 s a graded C(q) vector space V = V r along wth rasng e : V r V r+2 and lowerng f : V r V r 2 operators such that ef fe Vr = [r]i Vr Nave categorcal acton Once we thnk of an sl 2 representaton n terms of a sequence of vector spaces together wth rasng and lowerng operators, we are led to the noton of an acton of sl 2 on a category. Defnton 1.2. A nave categorcal sl 2 acton conssts of a sequence D r of addtve categores along wth addtve functors E : D r D r+2, F : D r D r 2, for each r, such that there exst somorphsms of functors (3) (4) EF Dr F E Dr = F E Dr I r D r, f r 0 = EF Dr I r D r, f r 0. Suppose that the categores D r carry a nave categorcal sl 2 acton. Then we can construct a usual sl 2 representaton as follows. We set V r = K(D r ), the complexfed splt Grothendeck group. The functors E, F gve rse to lnear maps e : V r V r+2, f : V r V r 2 and we can easly see that (3) and (4) gve the commutaton relaton (1). Thus we get a representaton of sl 2 on V = V r. We say that the categores D r categorfy the representaton V = V r. It s also useful to consder a graded verson of the above defnton. A graded addtve category s a category C along wth an addtve functor 1 : C C. We defne a graded nave categorcal sl 2 acton as above but wth (3), (4) replaced by EF Dr = F E Dr I Dr r 1 I Dr r + 1, f r 0 F E Dr = EF Dr I Dr r 1 I Dr r + 1, f r 0. The Grothendeck groups K(D r ) wll then carry an acton of U q sl 2. We wll now gve an example of a nave categorcal acton whch wll buld on Example 1.1.

4 In Example 1.1, we studed subsets of a fnte set. There s a well-known analogy between subsets of an n-element set and subspaces of an n-dmensonal vector space over a fnte feld F q, where q s a power of a prme. Ths analogy suggests that we try to construct a representaton of sl 2 on V r, where V r = C G(k,Fn q ) s a C-vector space whose bass s G(k, F n q ), the set of k-dmensonal subspaces of F n q (where r = 2k n as before). If we defne e, f as n (2), then we get a representaton of the quantum group U qsl 2 (after a slght modfcaton). The fnte set G(k, F n q ) s the set of F q -ponts of a proectve varety, called the Grassmannan. By Grothendeck s fonctons-fasceaux correspondence, we can categorfy C G(k,Fn q ) usng an approprate category of sheaves on G(k, F n q ). For smplcty, we swtch to characterstc 0 and consder sheaves on G(k, C n ), the Grassmannan of k-dmensonal subspaces of C n. For each r = n, n+2,..., n, we let D r = Dc(G(k, b C n )) denote the bounded derved category of constructble sheaves (agan here r = 2k n). These are graded categores, where the gradng comes from homologcal shft. Wth the above motvatons, we wll defne a categorcal sl 2 acton usng these categores. For each k, we defne the 3-step partal flag varety F l(k, k + 1, C n ) = {0 V V C n : dm V = k, dm V = k + 1}. F l(k, k +1, C n ) serves as a correspondence between G(k, C n ) and G(k +1, C n ) and thus t can be used to defne functors between categores of sheaves on these varetes. Let p : F l(k, k + 1, C n ) G(k, C n ) and q : F l(k, k + 1, C n ) G(k + 1, C n ) denote the two proectons. We defne E : D r = D b c(g(k, C n )) D r+2 = D b c(g(k + 1, C n )) A q (p A) F : D r D r 2 A p (q A). The above defnton of E, F parallels the defnton (2). The followng result was proven n an algebrac context (.e. after applyng the Belnson-Bernsten correspondence) by Bernsten-Frenkel-Khovanov [BFK]. Theorem 1.3. Ths defnes a graded nave categorcal sl 2 acton. The proof of ths theorem s relatvely straghtforward. To llustrate the dea, let us fx V G(k, C n ) and consder A 1 = {V : V V, dm V = k + 1} and A 2 = {V : V V, dm V = k 1}; these are the varetes of ways to ncrease or decrease V. Note that A 1 s a proectve space of dmenson n k 1 and A 2 s a proectve space of dmenson k 1. Thus dm H (A 2 ) dm H (A 1 ) = 2k n. Ths observaton combned wth the decomposton theorem proves the above result.

5 Remark 1.4. The Grothendeck group of these categores D r s actually nfntedmensonal. To cut down to a fnte dmensonal stuaton, we can consder the full subcategores D r = P Sch (G(k, C n )) consstng of drect sums of homologcal shfts of IC-sheaves on Schubert varetes. The subcategores D r carry a nave categorcal sl 2 acton and by consderng dmensons of weght spaces, we can see that they categorfy the representaton (C 2 ) n Categorcal sl 2 -acton In the defnton of nave categorcal sl 2 acton, we only demanded that there exst somorphsms of functors n (3) and (4). We dd not specfy the data of these somorphsms. Ths s very unnatural from the pont of vew of category theory. However, t s not mmedately obvous how to specfy these somorphsms nor what relatons these somorphsms should satsfy. In ther breakthrough paper, Chuang-Rouquer [CR] solved ths problem. Frst, t s natural to assume that the functors E, F be adont (ths s a categorfcaton of the fact that e, f are adont wth respect to the Shapovalov form on any fnte-dmensonal representaton of sl 2 ). Now (assume r 0), we desre to specfy a somorphsm of functors (φ, ψ 0,..., ψ r 1 ) : EF Dr F E Dr I r D r so φ Hom(EF, F E) = Hom(EE, EE) (usng the aduncton) and ψ s Hom(EF, I) = Hom(E, E) (agan usng the aduncton). Thus t s natural to choose two elements T Hom(EE, EE) and X Hom(E, E) such that φ corresponds to T and ψ s corresponds to X s for s = 0,..., r 1. Ths leads us to the followng defnton, essentally due to Chuang-Rouquer [CR]. Defnton 1.5. A categorcal sl 2 acton conssts of 1. a sequence D r of addtve categores, wth D r = 0 for r 0, 2. functors E : D r D r+2, F : D r D r 2, for each r, 3. natural transformatons ε : EF I, η : I F E, X : E E, T : E 2 E 2 such that the followng hold (5) 1. The morphsms ε, η are the unts and counts of adunctons. 2. If r 0, the morphsm (σ, ε, ε XI F..., ε X r 1 I F ) : EF Dr F E Dr I r D r s an somorphsm, where σ : EF F E s defned as the composton EF ηi EF F EEF I F T 1 F I F EEF F E ε F E. (And we mpose a smlar somorphsm condton f r 0.) 3. The morphsms X, T obey the followng relatons (a) In Hom(E 2, E 2 ), we have XI E T T I E X = I E 2 = T XI E I E X T.

6 (b) In Hom(E 2, E 2 ), we have T 2 = 0. (c) In Hom(E 3, E 3 ), we have T I E I E T T I E = I E T T I E 1 E T. Remark 1.6. If we work n the graded settng, then t s natural to ask that X have degree 2,.e. that t be a morphsm X : E E 2. Lkewse, we gve T degree 2. The degrees of ε and η depend on r. At frst glance, t s not apparent where the relatons among the X, T come from. To motvate them, we ntroduce the nl affne Hecke algebra. Defnton 1.7. The nl affne Hecke algebra H n s the algebra wth generators x 1,..., x n, t 1,..., t n 1 and relatons t 2 = 0, t t +1 t = t +1 t t +1, t t = t t f > 1, x x = x x, t x x +1 t = 1 = x t t x +1. Suppose that we have a categorcal sl 2 -acton. Then the morphsms X, T generate an acton of H n on E n. More precsely, we have an algebra morphsm H n Hom(E n, E n ) by sendng x to I E 1XI E n and t to I E 1T I E n 1. The above relatons among X, T ensure that the relatons of H n hold. Remark 1.8. In ther orgnal paper, Chuang-Rouquer [CR] used relatons among X, T modelled after the affne Hecke algebra or degenerate affne Hecke algebra, rather than the nl affne Hecke algebra. The nl affne Hecke relatons were frst ntroduced by Lauda [La]. The nl affne Hecke algebra arses qute naturally n the study of the topology of the flag varety. Let F l(c n ) denote the varety of complete flags n C n. The followng result appears to be due to Araba [A] (see also [G, Prop. 12.8]). Proposton 1.9. There s an somorphsm of algebras H n = H GL n (F l(c n ) F l(c n )) where the rght hand sde carres an algebra structure by convoluton Categorcal sl 2 actons comng from Grassmannans Let us return to constructble sheaves on Grassmannans. Consder the functor E p : Dc(G(k, b C n )) Dc(G(k b + p, C n )). It s gven by the correspondence wth the partal flag varety F l(k, k + 1,..., k + p, C n ) = {0 V 0 V 1 V p C n : dm V = k + }. The map F l(k, k + 1,..., k + p, C n ) G(k, C n ) G(k + p, C n ) s a fbre bundle onto ts mage F l(k, k + p, C n ) wth fbre F l(c p ). By Proposton 1.9 ths provdes an acton of the algebra H p on the functor E p. Ths can be used to upgrade Theorem 1.3 to the followng result.

7 Theorem The nave graded categorcal sl 2 acton on D r = Dc(G(k, b C n )) extends to a graded categorcal sl 2 acton. The above result s well-known but does not appear explctly n the lterature. It s a specal case of the man result of [W2]. It s worth mentonng a more elementary verson of ths categorcal sl 2 acton. For each k = 0,..., n, let D r be the category of fnte-dmensonal H (G(k, n))-modules (wth r = 2k n). We have a functor D r D r gven by global sectons. The followng result was sketched by Chuang-Rouquer [CR, secton 7.7.2] and a complete proof was gven by Lauda [La, Theorem 7.12]. Theorem There exsts a categorcal sl 2 acton on D r compatble wth the functor D r D r. Ths categorfes the n + 1-dmensonal rreducble representaton of sl 2. Moreover, ths categorcal sl 2 representaton s the smplest possble categorfcaton of ths rreducble representaton; more precsely, t s a mnmal categorfcaton, accordng to the results of Chuang-Rouquer [CR]. A related constructon was gven by Cauts, Lcata, and the author n [CKL]. We consdered derved categores of coherent sheaves on cotangent bundles to Grassmannans D r := D b Coh(T G(k, C n )), where agan r = 2k n. We proved the followng result. Theorem There s a graded categorcal sl 2 acton on D r where the functors E, F come from the conormal bundles to the correspondences F l(k, k + 1, C n ). Ths categorfes the representaton (C 2 ) n Equvalences We wll now see how a categorcal sl 2 acton can be used to produce nterestng equvalences of categores, followng Chuang-Rouquer [CR]. To motvate the constructon, suppose that V = V r s a fnte-dmensonal representaton of sl 2. Then the group SL 2 acts on V r. In partcular the matrx s = [ ] acts on V. Snce s s a lft of the non-trval element n the Weyl group of SL 2, t gves an somorphsm of vector spaces s : V r V r for all r. We would lke to do somethng smlar for categorcal sl 2 actons. To do ths, let us fx r 0 and note that the acton of s on V r s gven by s Vr = F (r) EF (r+1) + E (2) F (r+2) where E (n) = 1 n! En. (Note that ths sum s fnte snce for large enough p, V r 2p = 0.) The alternatng sgns n ths expresson suggest that we try to categorfy s usng a complex. Ths complex was ntroduced by Chuang-Rouquer [CR], nspred by certan complexes of Rckard. The followng result s due to Chuang-Rouquer [CR] n the abelan case and Cauts-Kamntzer-Lcata [CKL] n the trangulated case (whch s the one we state below).

8 Theorem Suppose that D r s a sequence of trangulated categores carryng a graded categorcal sl 2 acton such that all functors E, F are exact. Then the complex S = [F (r) EF (r+1) 1 E (2) F (r+2) 2 ] provdes an equvalence S : D r D r. Here E (n) s defned usng a splttng E n = E (n) n! whch s acheved usng the acton of H n on E n (see Secton of [R2] or Secton 9.2 of [La]). The maps n ths complex come from the adunctons. See Secton 6.1 of [CR] for more detals. Example Suppose that we have a categorcal sl 2 acton wth ust D 2, D 0, D 2 non-zero. Then choosng r = 0, the above complex has two terms S = [I EF 1 ]. In ths case, the equvalence S s actually a Sedel-Thomas [ST] sphercal twst wth respect to the functor E : D 2 D 0. Thus we see that the equvalences comng from categorcal sl 2 actons generalze the theory of sphercal twsts. Chuang-Rouquer appled Theorem 1.13 to prove that certan blocks of modular representatons of symmetrc groups were derved equvalent. Ths proved Broué s abelan defect group conecture for symmetrc groups. See Theorem 4.1 for the constructon of the relevant categorcal acton. Another very nterestng applcaton of Theorem 1.13 concerns constructble sheaves on Grassmannans, as n Theorem In ths case, t can be shown that the resultng equvalence D b c(g(k, C n )) D b c(g(n k, C n )) s gven by the Radon transform. More precsely, S s gven by the ntegral transform wth respect to the kernel C U, where U G(k, C n ) G(n k, C n ) s the open GL n -orbt consstng pars of transverse subspaces (1). Yet another applcaton of Theorem 1.13 nvolves coherent sheaves on cotangent bundles of Grassmannans. In [CKL], by combnng Theorem 1.13 wth Theorem 1.12, we were able to construct an equvalence D b Coh(T G(k, C n )) D b Coh(T G(n k, C n )), thus answerng an open problem posed by Kawamata and Namkawa. (Ths approach was prevously suggested by Rouquer n [R1].) The exact descrpton of the equvalence n ths case was gven by Cauts [C]. 2. THE KHOVANOV-LAUDA-ROUQUIER CATEGORIFICATION We wll now rephrase the noton of categorcal sl 2 acton (Defnton 1.5) from a more general vewpont. We wll then proceed to defne the categorfcaton of any smply-laced Kac-Moody Le algebra. 1. Ths result wll appear n a forthcomng paper by Cauts, Dodd, and the author.

9 Generaltes on categorfcaton Let C be an addtve category. Let K(C) denote the (complexfed) splt Grothendeck group of C; ths s the vector space spanned by somorphsm classes [A] of obects of C modulo the relaton [A B] = [A] + [B]. If C s a graded addtve category, then K(C) s a C[q, q 1 ]-module, where we defne q[a] = [A 1 ]. We can then tensor to obtan a C(q)-vector space, whch we wll also denote by K(C). Let V be a vector space. A categorfcaton of V s an addtve category C, along wth an somorphsm of vector spaces K(C) = V. If V s a C(q)-vector space, then a categorfcaton of V s a graded addtve category C, along wth an somorphsm of C(q)-vector spaces K(C) = V. We wll also need the noton of categorfcaton of algebras. A monodal category s an addtve category C, along wth an addtve bfunctor : C C C, such that A (B C) = (A B) C (2). If C s a monodal category, then K(C) acqures the structure of an algebra where the multplcaton s defned by [A][B] = [A B]. Let A be an algebra. A categorfcaton of A s a monodal category C, along wth an somorphsm of algebras K(C) = A. (Ths generalzes n an obvous way to C(q)-algebras and graded monodal categores.) Example 2.1. The smplest algebra s A = C. Ths algebra s categorfed by Vect, the category of fnte-dmensonal vector spaces. Smlarly, C(q) s categorfed by the category of graded vector spaces. More generally, f G s a fnte group, then the category Rep(G) of fnte-dmensonal representatons of G categorfes the algebra C c (G) of class functons on G. The somorphsm K(Rep(G)) C c (G) s provded by the character map. An algebra A can be regarded as a lnear category wth one obect whose set of endomorphsms s A and where the composton of morphsms s the multplcaton n A. From ths perspectve, t s natural to try to categorfy more general categores, especally those wth very few obects. To ths end, we wll need to look at 2-categores. A 2-category C (for our purposes) s a category enrched over the category of addtve categores. That means we have a set of obects C and for any two obects A, B C, a category Hom(A, B). We also have assocatve composton functors Hom(B, C) Hom(A, B) Hom(A, C). Note that a monodal category s the same as a 2-category wth one obect. The smplest example of a 2-category s Cat, the 2-category of addtve categores. The obects of Cat are addtve categores and for any two addtve categores A, B, we defne Hom(A, B) to be the category of functors from A to B (the morphsms n Hom(A, B) are natural transformatons of functors). If C s a 2-category, then we wll defne K(C) to be the category whose obects are the same as C and whose morphsm sets are defned by Hom K(C) (A, B) = K(Hom(A, B)). 2. Actually, ths defnes the noton of strct monodal category.

10 Let A be a lnear category. A categorfcaton of A s an addtve 2-category C along wth an somorphsm K(C) = A. We wll also need the noton of dempotent completon (or Karoub envelope). Recall that f C s an addtve category, an dempotent n C s a morphsm T : A A n C such that T 2 = T. We say that T splts f we can wrte A as a drect sum A = A 0 A 1, such that T acts by 0 on A 0 and by 1 on A 1. The dempotent completon (C) of C s the smallest enlargement of C such that all dempotents splt n (C). If C s a 2-category, then (C) wll denote the 2-category wth the same obects, but where we perform dempotent completon on every Hom-category categorcal rephrasng for sl 2 Let us apply ths setup to A = Usl 2, the unversal envelopng algebra. Actually we wll need Lusztg s dempotent form Usl 2. Snce Usl 2 carres a system of dempotents, we can regard t as a category. Defnton 2.2. The category Usl 2 has obects r Z. It s the C-lnear category wth generatng morphsms e Hom(r, r + 2) and f Hom(r, r 2), for all r, subect to the relaton ef fe = ri r for all r (ths s an equaton n Hom(r, r)). A representaton of an algebra A s the same thng as a lnear functor A Vect, where A s the category wth one obect constructed usng A. Thus we can speak more generally of a representaton of a lnear category C as a lnear functor C Vect. In partcular, we can consder lnear functors Usl 2 Vect. From our dscusson n Secton 1.1, we can see that a fnte-dmensonal representaton V = V r of sl 2 s the same thng as a lnear functor Usl 2 Vect whch takes the obect r to the vector space V r. We also have U q sl 2, whch s defned n the same fashon, except that t s C(q)-lnear and the relaton s ef fe = [r]i r. Now we proceed to the queston of tryng to categorfy Usl 2. Snce t s a category wth obects r Z, t wll be categorfed by a 2-category wth the same set of obects. In the prevous secton we explaned Chuang-Rouquer s defnton (Defnton 1.5) of a categorcal sl 2 acton. By thnkng about ths defnton, we reach the defnton of a 2-category whch categorfes Usl 2. Defnton 2.3. Let Usl 2 denote the addtve 2-category wth 1. obects r Z, 2. 1-morphsms generated under drect sum and composton by E Hom(r, r + 2) and F Hom(r, r 2) for all r, 3. 2-morphsms generated by subect to the relatons X : E E, T : E 2 E 2, η : I F E, ε : EF I

11 (6) 1. n Hom(E, E), we have εi E I E η = I E, 2. n Hom(E 2, E 2 ), we have XI E T T I E X = I E 2 = T XI E I E X T, 3. n Hom(E 2, E 2 ), we have T 2 = 0, 4. n Hom(E 3, E 3 ), we have T I E I E T T I E = I E T T I E 1 E T, 5. f r 0, the followng 2-morphsm (σ, ε, ε XI F..., ε X r 1 I F ) : EF F E I r r s an somorphsm, where σ s defned as n Defnton 1.5 (plus a smlar condton f r 0). More precsely, the last condton means that for each r, n the category Hom(r, r) we adon the nverse of (σ, ε, ε XI F..., ε X r 1 I F ). Now that we have defned the 2-category Usl 2, t s natural to consder 2-functors Usl 2 Cat (these are 2-representatons of Usl 2 ). Wth the above defnton, t s easy to see that a categorcal sl 2 acton on some categores D r (Defnton 1.5) s the same thng as a 2-functor Usl 2 Cat whch takes r to D r for all r. Remark 2.4. In ths defnton, we are followng Rouquer s defnton [R2] of the 2-category. In the Lauda [La] verson, whch we denote by U L sl 2, we do not nvert (σ, ε,..., ε X r 1 I F ), but rather add extra relatons to ensure that ths map s nvertble. In a recent paper, Cauts-Lauda [CL] proved that under some mld assumptons a 2-functor from Usl 2 to Cat gves rse to a 2-functor from U L sl 2 to Cat (the converse s automatcally true). The followng result s due to Lauda [La]. Theorem 2.5. The 2-category U L sl 2 categorfes Usl 2. Remark 2.6. The graded verson of Usl 2 categorfes Lusztg s U q sl 2. There s also a more precse verson of Theorem 2.5, whch states that the dempotent completon (U L sl 2 ) categorfes Lusztg s Z[q, q 1 ]-form of U q sl 2 (f we look at the Z[q, q 1 ] verson of the Grothendeck group) The 2-category for general g Suppose that g s an arbtrary Kac-Moody Le algebra. It s natural to try to extend the above constructon from sl 2 to g, n partcular to construct a 2-category Ug whch categorfes Ug. Roughly equvalent constructons of ths 2-category were acheved ndependently and smultaneously by Khovanov-Lauda [KL1, KL2, KL3] and by Rouquer [R2]. For smplcty, we wll assume that g s smply-laced. Let us fx notaton as follows. Let X denote the weght lattce of g. Let I denote the ndexng set for the smple roots and let α for I denote the smple roots. Let ZI X be the root lattce and let NI denote the postve root cone. Let, denote the symmetrc blnear form on X. Then α, α are the entres of the Cartan matrx of g (these le n the set {2, 1, 0} by

12 assumpton). We choose an orentaton of the Dynkn dagram of g n order to produce a drected graph, called a quver and denoted by Q. We wrte f there s an orented edge from to n Q. The category Ug s constructed from Lusztg s dempotent form of the unversal envelopng algebra Ug and ts defnton parallels Usl 2 (Defnton 2.2). In partcular, t has obects λ X and generatng morphsms e Hom(λ, λ + α ) and f Hom(λ, λ α ) for I and λ X (for reasons of brevty, we do not gve a complete lst of the relatons n Ug). As before, there s a quantum verson U q g whch s obtaned by replacng all ntegers n the defnton of U g by quantum ntegers. We wll descrbe the 2-category Ug usng graphcal notaton due to Khovanov and Lauda. In ths graphcal notaton, 2-morphsms are vewed as strng dagrams n the plane, wth strngs orented and labelled from I. The orentatons and labels on the strands tell you the source and target of the 2-morphsm. An arrow labelled pontng up (resp. down) denotes E (resp. F ). For more nformaton on ths graphcal notaton see [La, Secton 4]. Defnton 2.7. The 2-category Ug s defned as follows The obects are λ for λ X. The 1-morphsms are generated by E Hom(λ, λ + α ), F Hom(λ, λ α ) for I and λ X. The 2-morphsms are generated by X = : E E, X = : F F, T = : E E E E, T = : F F F F : E F I, : F E I, : I F E, : I E F for I and λ X. (We have suppressed λ n the above notaton t should label a regon n each elementary strng dagram. Ths label tells you the source and target of the E, F.) The 2-morphsms are subect to the followng relatons The KLR algebra relatons among upward pontng strng dagrams (7) 1. If all strands are labeled by the same I, the nl affne Hecke algebra relatons = 0, =, = =

13 For (8) = f α, α ) = 0, f, (9) 3. For the dot sldng relatons = = 4. Unless = k and α, α < 0 the relaton (10) k Otherwse, α, α < 0 and = f. k (11) = The cap and cup morphsms are badunctons f, f. (12) = = = = Moreover the dots and crossng are compatble wth these badunctons. For each, we have (13) (14) where we defne = := = =

14 (The equalty comes from the badontness of the crossng.) For each and each λ such that r = λ, α 0, the followng 2-morphsm s nvertble, ( ) 1 r 1 (15),,,..., : E F F E I r λ Here a dot wth a postve nteger k ndcates that we put k dots on that strand (n other words, t means X k ). We also mpose a smlar condton f r 0. Ths defnton s qute complcated, so let us see where these relatons come from. When g = sl 2, ths defnton gves the 2-category from Defnton 2.3. In fact, (7) s relatons 2, 3, 4 from Defnton 2.3 wrtten n dagrammatc form and (12) and (15) correspond to relatons 1 and 5 from Defnton 2.3. (Actually there s a slght dfference, n that the above defnton mposes badontness, whereas Defnton 2.3 only nvolves one-sded adontness. For more dscusson on ths, see [R2, Theorem 5.16].) Khovanov-Lauda and Rouquer dscovered the relatons (8), (9), (10) and (11) based on computatons nvolvng cohomology of partal flag varetes and quver varetes (essentally to get Theorem 3.4 to hold). Remark 2.8. As n Remark 2.4, ths the Rouquer verson of the 2-category, because of (15). Khovanov-Lauda s verson, denoted U KL g, bears the same relatonshp to Ug as Lauda s verson, U L sl 2, dd n the sl 2 case. Consder the Grothendeck group K(Ug) as a 1-category. The generatng morphsms are e = [E ], f = [F ] as above. From (13), we see that we have e f = f e, and from (15), we see that e f f e = λ, α I λ n Hom(λ, λ). As these are most of the relatons of U g (there reman the Serre relatons), ths suggests the followng result, whch was proven by Khovanov-Lauda [KL3] n the case of sl n and for general g by Webster [W1]. Theorem 2.9. There s an somorphsm of categores K(U KL g) = Ug. In other words, the 2-category U KL g s a categorfcaton of Ug. Remark There s a graded verson of Ug wth the degree of X equal to 2 and the degree of T equal to α, α. Ths graded verson categorfes U q g. Agan, there s a more precse form relatng the dempotent completon of U KL g and Lusztg s Z[q, q 1 ]-form of U q g Categorfcaton of the upper half It s mportant to solate the categorfcaton of the upper half of the enveloppng algebra U + g, where U + g Ug s the subalgebra generated by all E (or equvalently, t s the enveloppng algebra of n). Note that U + g has no dempotents, so we regard t as an algebra, not as a category. We have the usual gradng U + g = ν NI (U + g) ν.

15 Defnton Let U + g denote the monodal category whose obects are generated (under drect sum and tensor product) by E, for I, and whose morphsms are upward pontng strng dagrams as n Defnton 2.7 (so the morphsms are generated by the upward pontng dot and crossng wth the KLR algebra relatons). Remark In the above defnton of U + g, the E do not have a source and target obect as they do n Ug. Thus U + g does not st nsde Ug n any way. Ths s not that surprsng, as U + g s not a subalgebra of Ug. Let ν NI. Let Seq ν = { = ( 1,..., m ) : α α m = ν}. Ths s the set of all ways to wrte ν as an ordered sum of smple roots. For Seq ν, we let E = E 1 E m (here the -operaton n U + g s wrtten as concatenaton). Let (U + g) ν denote the full subcategory of U + g whose obects are drects sums of the E for Seq ν. We defne algebras R ν :=, Seq ν Hom U + g(e, E ). These algebras have become known as Khovanov-Lauda-Rouquer (KLR) algebras, though the term quver Hecke algebras has also been used. See [B] for a survey paper on these algebras. Example Suppose that g = sl 2. Then ν = nα for some n, Seq ν has only one element and R ν = H n, the nl affne Hecke algebra. By general prncples, we have an equvalence of categores (U + g) ν = R ν -pmod between the dempotent completon of (U + g) ν and the category of proectve modules over the KLR algebra R ν. In partcular, K((U + g) ν ) acqures a bass of ndecomposable proectve R ν modules (these are the same as the ndecomposable obects of (U + g) ν under the above equvalence). Note that under the above equvalence, the monodal structure on (U + g) comes from the ncluson R µ R ν R µ+ν gven by horzontal concatenaton of strng dagrams. The followng result s due to Khovanov-Lauda [KL1, Theorem 1.1] (n the smplylaced case). Theorem (U + g) = ν R ν -pmod s a categorfcaton of U + g. In fact, K((U + g) ) can be gven the structure of a balgebra and then the above result can be strengthened to an somorphsm of balgebras. Remark There s a graded verson of U + g whch categorfes U + q g. 3. LUSZTIG S PERVERSE SHEAVES AND KLR ALGEBRAS We wll now explan a geometrc ncarnaton of the KLR algebras and of the category U + g. For smplcty, let us assume that g s of fnte type.

16 Lusztg s perverse sheaves Recall that we chose an orentaton of the Dynkn dagram of g to produce the quver Q. Defnton 3.1. A representaton of Q s a graded vector space V = I V along wth lnear maps A : V V for every drected edge n Q. The dmenson-vector of a representaton V s defned by dm V = I dm V α NI. Let M ν denote the modul stack of representatons of Q of dmenson-vector ν. More explctly, we can present M ν as a global quotent M ν = Hom(C ν, C ν )/ GL(C ν ) where ν = ν α. Example 3.2. When g = sl 2, then the quver Q conssts of ust one vertex wth no arrows. Thus a representaton of Q s ust a vector space. So we see that M nα = pt/gl n. When g = sl 3, then the quver Q conssts of two vertces wth an arrow between them. Thus a representaton of Q s a par of vertces and a lnear map between them. Thus, we see that M nα1 +mα 2 = Hom(C n, C m )/GL n GL m. We let D(M) := ν D(M ν ) denote the derved category of constructble sheaves on the stack M = M ν. Note that we may consder D(M ν ) as the GL(Cν )-equvarant derved category of (,) Q Hom(C ν, C ν ). Followng Lusztg [Lu1, Lu2], we defne a monodal structure on the category D(M). We consder the modul stack of short exact sequences S = {0 V 1 V 3 V 2 0} of representatons of Q. We have three proecton morphsms π 1, π 2, π 3 : S M and thus for A, B D(M), we can defne A B = π 3 (π 1A π 2B). If A D(M ν ) and B D(M µ ), then A B D(M ν+µ ). The smple perverse sheaves n D(M ν ) are precsely the IC-sheaves of the GL(Cν )- orbts n Hom(C ν, C ν ) and thus are n becton wth the somorphsm classes of representatons of Q of dmenson-vector ν. Rngel s theorem tells us that the ndecomposable representatons of Q have the postve roots as ther dmenson-vectors. Thus, the number of somorphsm classes of representatons of Q of dmenson-vector ν equals the dmenson of (U + g) ν. Let P (M ν ) be the subcategory of D(M ν ) consstng of drect sums of homologcal shfts of smple perverse sheaves n D(M ν ). By the decomposton theorem, P (M) = P (M ν ) s a monodal subcategory. Note that P (M) has a graded structure gven by homologcal shft. Lusztg [Lu1, Lu2] proved the followng theorem concernng P (M)

17 Theorem 3.3. The Grothendeck rng of P (M) s somorphc to U q + g. words, P (M) s a categorfcaton of U q + g. In other By ths theorem, U + q g acqures a bass comng from the classes of the IC-sheaves n P (M). Ths bass s called Lusztg s canoncal bass Relatonshp to KLR algebras It s natural to expect that Lusztg s categorfcaton of U + q g s related to the categorfcaton of U + q g defned by generators and relatons n Secton 2.4. Ths result was proven ndependently by Varagnolo-Vasserot [VV] and by Rouquer [R3]. Theorem 3.4. There s an equvalence of addtve monodal categores (U q + ) P (M) defned on generators as follows E C Mα X x Ext (C Mα, C Mα ) = H C (pt) = C[x ] T t. Here we use that M α = pt/c. The defnton of t s a bt nvolved and depends on cases, so we skp the defnton. We can reformulate ths theorem usng a convoluton algebra defned usng M ν. We defne M ν to be the modul stack of complete flags 0 V 1 V m of representatons of Q wth dm V m = ν. Then we can form the stack Z ν := M ν Mν Mν. Then H (Z ν ) s an algebra under convoluton. By Gnzburg [G, Prop 5.1], H (Z ν ) s an Ext-algebra n P (M). Wth ths setup, Theorem 3.4 s equvalent to the exstence of compatble somorphsms R ν = H (Z ν ) for all ν. Example 3.5. If we take g = sl 2 and ν = nα, then M ν = F l(c n )/GL n and Z ν = (F l(c n ) F l(c n ))/GL n. In ths case the somorphsm R ν = H (Z ν ) s precsely the statement of Proposton 1.9. As a corollary of Theorem 3.4, we obtan the followng result: Corollary 3.6. The bass of U + q g provded by ndecomposable graded proectve R ν -modules under Theorem 2.14 s Lusztg s canoncal bass. 4. EXAMPLES OF CATEGORICAL g-representations 4.1. Defnton of categorcal g-representatons Usng the 2-category Ug, we can now defne a categorcal g-representaton to be an addtve lnear 2-functor Ug Cat. In partcular, a categorcal g-representaton conssts of a collecton of categores D µ for µ X, badont functors E, F : D µ D µ±α and natural transformatons X : E E, T : E E E E satsfyng the relatons n Ug.

18 A graded categorcal g representaton nvolves the same setup except that each category D µ s graded, some shfts appear n the badontness of E, F, and the natural transformatons X, T have degrees as ndcated n Remark Modular representaton theory of symmetrc groups Gong back to the work of Lascoux-Leclerc-Thbon [LLT] and Gronowsk [Gr], the prme motvatng example of a categorcal g-representaton concerns modular representatons of symmetrc groups. In fact, ths categorcal acton has proved to be very mportant n understandng modular representaton theory. Fx a prme p and an algebracally closed feld k of characterstc p. We wll be nterested n the category Rep(S n ) of fnte-dmensonal representatons of S n over k. These categores wll provde an acton of the affne Le algebra ŝl p. The basc functors we consder between these categores are the nducton and restrcton functors correspondng to the natural embeddng S n 1 S n. Recall the Young-Jucys-Murphy elements Y m := (1 m) + + (m 1 m) ks n. A fundamental result s that the egenvalues of Y m actng on a representaton M le n the prme subfeld Z/p k. We wll dentfy of Z/p as the set I of smple roots of our Kac-Moody algebra ŝl p. Let Z/p. Usng the Young-Jucys-Murphy elements, we defne functors E and F of -restrcton and -nducton as follows. If M Rep(S n ), we let E (M) denote the generalzed -egenspace of Y n. Snce Y n commutes wth the acton S n 1, we see that E (M) s an S n 1 representaton. Smlarly, we defne F (M) to be the generalzed -egenspace of Y n+1 actng on Ind S n+1 S n M. Symmetrc polynomals n Y 1,..., Y n span the centre of ks n. Thus we may regard a central character γ : ks n k as an element of (Z/p) n /S n, whch we thnk of as the set of n-element multsubsets of Z/p. Thus for each µ = (µ 0,..., µ p 1 ) N p such that µ = n, we can consder the category Rep(S n ) µ of representatons M of S n whose generalzed central character s gven by the multset γ(µ) := {0 µ 0,..., (p 1) µ p 1 }. Theorem 4.1. The category n Rep(S n ) carres a categorcal ŝl p-acton. More precsely, we get the categorcal ŝl p-acton as follows We defne D ω0 µ α = Rep(S n ) µ for each µ N p (where n = µ ). For each Z/p, we defne E, F as above. The dot X and crossng T are defned wth the help of Y n and the transposton (n 1 n). The fact that these categores carry a nave categoral ŝl p -acton was proven by Lascoux-Leclerc-Thbon [LLT] and by Gronowsk [Gr]. The above statement of an actual categorcal ŝl p-acton was proven by Chuang-Rouquer [R3, Theorem 4.23]. In ths theorem, we work wth a verson of Uŝl p defned over the feld k (rather than C).

19 Let us be more precse about the defntons of X and T. Consder the functor Res Sn S n p. Ths functor wll have endomorphsms Y n p+1,..., Y n and (n p +1 n p+2),..., (n 1 n). It s easly seen that they defne an acton of a degenerate affne Hecke algebra H p on Res Sn S n p. For any µ wth µ = n, the functor E p : Rep(S n ) µ Rep(S n p ) µ pα s a drect summand of Res Sn S n p and thus E p carres an acton of H p. Theorem 3.16 from [R2] explans how we can convert ths to an acton of the nl affne Hecke algebra H p (a smlar result was obtaned by Brundan-Kleshchev [BK]). Usng ths result, we can construct the categorcal ŝl p-acton. For more detals, see Secton of [R2] Cyclotomc quotents There s a natural way to construct categorfcatons of rreducble representatons of g usng cyclotomc quotents of KLR algebras. For each domnant weght λ = n ω X + and each ν NI, let R ν (λ) be the quotent of R ν by the deal generated by all dagrams of the form n m Let V(λ) µ = R λ µ (λ)-pmod be the category of proectve modules over the cyclotomc quotents. Note that there s an acton of U g on V(λ) comng from maps R λ µ (λ) R ν R λ (µ ν) (λ) whch are gven by horzontal concatenaton (here U g s defned n the same fashon as U + g). In partcular, we have functors F : V(λ) V(λ). The followng result was conectured by Khovanov-Lauda [KL1] and was proved by Kang-Kashwara [KK, Ka] and Webster [W1]. Theorem 4.2. The functors F admt badonts E and ths defnes a categorcal g-acton on V(λ). Moreover, V(λ) categorfes the rreducble representaton V (λ) of hghest weght λ. Rouquer has proved [R3] that a slght generalzaton of V(λ) s the unversal categorcal g-representaton wth hghest weght λ. Also, Lauda-Vazran [LV] that one can construct the crystal of V (λ) usng the smple modules over the algebras R ν (λ). Remark 4.3. Webster [W1] has generalzed ths constructon. For any sequence λ 1,..., λ n, he has constructed certan dagrammatc algebras R ν (λ 1,..., λ n ) whose categores of proectve modules admt a categorcal g-acton as above. Ths constructon categorfes the tensor product representaton V (λ 1 ) V (λ n ).

20 Geometrc examples It s natural to generalze the constructon of the categorcal sl 2 -acton on sheaves on Grassmannans (Theorem 1.10). The generalzaton uses Nakama quver varetes. For each domnant weght λ, there exsts a (dsconnected) Nakama quver varety Y (λ) = µ X Y (λ, µ). It s a modul space of framed representatons of a doubled quver wth preproectve relaton. Nakama [N] has constructed an acton of g on H (Y (λ)). Ths motvated the queston of constructng categorcal actons of g on categores defned usng Y (λ). The varety Y (λ, µ) s almost a cotangent bundle n fact, t can be vewed as an open subset of a cotangent bundle to a certan stack M(λ, µ). Ths motvated Zheng [Z] to defne a certan category of constructble sheaves D(λ, µ) on M(λ, µ) whch should carry a categorcal g acton. Example 4.4. In the case g = sl 2 and λ = n, then Y (n, n 2k) = T G(k, C n ) and D(n, n 2k) = D b c(g(k, C n )). Theorem 4.5. The categores D(λ, µ) carry a categorcal g-acton. Zheng [Z] proved a weaker verson of ths theorem (he only establshed a nave categorcal acton). The above statement was proven by Rouquer [R3] usng Theorem 3.4. Webster [W2] also explaned that the category D(λ, µ) can be vewed (under Remann- Hlbert correspondence and Hamltonan reducton) as a category of modules over a deformaton quantzaton of Y (λ, µ). [A] REFERENCES A. ARABIA Cycles de Schubert et cohomologe équvarante de K/T, Invent. Math. 85 (1986), [BFK] J. BERNSTEIN, I. FRENKEL, M. KHOVANOV A categorfcaton of the Temperley-Leb algebra and Schur quotents of U(sl 2) va proectve and Zuckerman functors, Selecta Math. (N.S.) 5 (1999), [B] [BK] J. BRUNDAN Quver Hecke algebras and categorfcaton; arxv: J. BRUNDAN, A. KLESHCHEV Blocks of cyclotomc Hecke algebras and Khovanov-Lauda algebras, Invent. Math. 178 (2009), [C] S. CAUTIS Equvalences and stratfed flops, Composto Math. 148 (2012), [CKL] S. CAUTIS, J. KAMNITZER, A. LICATA Derved equvalences for cotangent bundles of Grassmannans va categorcal sl(2) actons, to appear n J. rene angew. Math.; arxv: [CL] S. CAUTIS, A. LAUDA Implct structure n 2-representatons of quantum groups; arxv:

21 [CR] [G] [Gr] [KK] [Ka] J. CHUANG, R. ROUQUIER Derved equvalences for symmetrc groups and sl 2-categorfcaton, Ann. of Math. (2) 167 (2008), V. GINZBURG Geometrc methods n the representaton theory of Hecke algebras and quantum groups, NATO Adv. Sc. Inst. Ser. C Math. Phys. Sc. 514, I. GROJNOWSKI Affne sl p controls the modular representaton theory of the symmetrc groups and related Hecke algebras; math/ S. KANG, M. KASHIWARA Categorfcaton of hghest weght modules va Khovanov-Lauda-Rouquer algebra, Invent. Math. 190 (2012), M. KASHIWARA Badontness n cyclotomc Khovanov-Lauda-Rouquer algebras, Publ. Res. Inst. Math. Sc. 48 (2012), [KL1] M. KHOVANOV, A. LAUDA A dagrammatc approach to categorfcaton of quantum groups I, Represent. Theory 13 (2009), [KL2] M. KHOVANOV, A. LAUDA A dagrammatc approach to categorfcaton of quantum groups II, Trans. Amer. Math. Soc. 363 (2011), [KL3] M. KHOVANOV, A. LAUDA A categorfcaton of quantum sl(n), Quantum Topol. 1 (2010), [LLT] A. LASCOUX, B. LECLERC, and J.-Y. THIBON Hecke algebras at roots of unty and crystal bases of quantum affne algebras, Comm. Math. Phys. 181 (1996), [La] A. LAUDA A categorfcaton of quantum sl(2), Adv. Math. 225 (2010), [LV] [Lu1] [Lu2] [N] [R1] [R2] A. LAUDA, M. VAZIRANI Crystals from categorfed quantum groups, Adv. Math. 228 (2011), G. LUSZTIG Canoncal bases arsng from quantzed envelopng algebras, J. Amer. Math. Soc. 3 (1990), G. LUSZTIG Quvers, perverse sheaves, and quantzed envelopng algebras, J. Amer. Math. Soc 4 (1991), H. NAKAJIMA Instantons on ALE spaces, quver varetes, and Kac-Moody algebras, Duke Math. J. 76 (1994), R. ROUQUIER Catégores dérvées et géométre bratonnelle (d après Bondal, Orlov, Brdgeland, Kawamata et al.), Sémnare Bourbak (2004/05), Exp. n 946, Astérsque 307 (2006), R. ROUQUIER 2-Kac-Moody algebras; arxv: [R3] R. ROUQUIER Quver Hecke algebras and 2-Le algebras, Algebra Colloq. 19 (2012), [ST] P. SEIDEL, R. THOMAS Brad group actons on derved categores of coherent sheaves, Duke Math J. 108 (2001),

22 [VV] [W1] M. VARAGNOLO, É. VASSEROT Canoncal bases and KLR-algebras, J. rene angew. Math. 659 (2011), B. WEBSTER Knot nvarants and hgher representaton theory I: dagrammatc and geometrc categorfcaton of tensor products; arxv: [W2] B. WEBSTER A categorcal acton on quantzed quver varetes; arxv: [Z] H. ZHENG Categorfcaton of ntegrable representatons of quantum groups; arxv: Joel KAMNITZER Unversty of Toronto Department of Mathematcs 40 St. George Street Toronto, Ontaro Canada M5S 2E4 E-mal : kamntz@math.toronto.edu