CATEGORICAL ACTIONS AND CRYSTALS

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1 CATEGORICAL ACTIONS AND CRYSTALS JONATHAN BRUNDAN AND NICHOLAS DAVIDSON Abstract. Ths s an expostory artcle developng some aspects of the theory of categorcal actons of Kac-Moody algebras n the sprt of works of Chuang Rouquer, Khovanov Lauda, Webster, and many others. 1. Introducton Ths work s a contrbuton to the study of categorfcatons of Kac-Moody algebras and ther ntegrable modules. The subect has ts roots n Lusztg s constructon of canoncal bases of quantum groups usng geometry of quver varetes [Lu1] (whch happened around 1990). It s ntmately connected to the rch combnatoral theory of crystal bases ntated at the same tme by Kashwara [K1]. In the decade after that, several other examples were studed related to the representaton theory of the symmetrc group and assocated Hecke algebras [LLT, A, G] (buldng n partcular on deas of Bernsten and Zelevnsky [BZ]), ratonal representatons of the general lnear group [BK], and the Bernsten-Gelfand-Gelfand category O assocated to the general lnear Le (super)algebra [BFK, B1]. The frst serous attempt to put these examples nto a unfed axomatc framework was undertaken by Chuang and Rouquer [CR]. They bult a powerful structure theory for studyng categorcal actons of sl 2, whch they appled notably to prove Broué s Abelan Defect Conecture for the symmetrc groups. Another maor breakthrough came n 2008, when Khovanov and Lauda [KL1, KL2, KL3] and Rouquer [R1] ndependently ntroduced some new algebras called quver Hecke algebras, and used them to construct Kac-Moody 2-categores assocated to arbtrary Kac-Moody algebras. The defntons of Kac-Moody 2-categores gven by Khovanov and Lauda and by Rouquer look qute dfferent, so that for a whle subsequent works splt nto two dfferent schools accordng to whch defnton they were followng. In fact, Rouquer s and Khovanov and Lauda s defntons are equvalent, as was establshed by the frst author [B3]. In ths (mostly expostory) artcle, we wll revst some of Rouquer s foundatonal defntons n the lght of [B3]. We do ths usng the dagrammatc formalsm of Khovanov and Lauda wherever possble. From the outset, we have systematcally ncorporated the better choce of normalzaton for the second aduncton of the Kac-Moody 2-category suggested by [BHLW]. For a survey wth greater emphass on the connectons to geometry, we refer the reader to Kamntzer s text [Kam]. Another of our goals s to extend several of the exstng results so that they may be appled n some more general stuatons. To explan the novelty, we need some defntons. Let k be an algebracally closed feld and Vec be the category of (small) vector spaces. A fnte-dmensonal category s a small k-lnear category A all of whose morphsm spaces are fnte-dmensonal. Let Mod-A denote the functor category Hom(A op, Vec) of rght modules over A. We say that A s Artnan f all of the fntely generated obects and the fntely cogenerated obects n Mod-A have fnte length (see also Remark 2.1). A locally Schuran category s an Abelan category that s equvalent to Mod-A for some 2010 Mathematcs Subect Classfcaton: 17B10, 18D10. Research supported n part by NSF grant DMS

2 2 J. BRUNDAN AND N. DAVIDSON fnte-dmensonal category A. If n addton A s Artnan, then the full subcategory of Mod-A consstng of all obects of fnte length s a Schuran category n the sense of [BLW, 2.1]. In the Abelan settng, the general structural results about 2-representatons of Kac- Moody 2-categores obtaned n [CR, R1, R2] typcally only apply to categores n whch all obects have fnte length and whose rreducble obects satsfy Schur s Lemma. If one wants there to be enough proectves and nectves too, ths means that one s workng n a Schuran category n the sense ust defned. The man new contrbuton of ths paper s to extend some of these structural results to locally Schuran categores. The motvaton for dong ths from a Le theoretc perspectve s as follows. Let g be a symmetrzable Kac-Moody algebra wth Chevalley generators {e, f I}, weght lattce P, etc... Recall that a g-module V s ntegrable f t decomposes nto weght spaces as V = P V, and each e and f acts locally nlpotently. In order to categorfy an ntegrable module wth fnte-dmensonal weght spaces, t s reasonable to hope that one can use a fnte-dmensonal category whose blocks are fnte-dmensonal algebras, n whch case all subsequent constructons can be performed n the Schuran category consstng of fnte-dmensonal modules over these algebras. Examples nclude the mnmal categorfcaton L mn (κ) of the ntegrable hghest weght module L(κ) of (domnant) hghest weght κ defned already by Khovanov, Lauda and Rouquer va cyclotomc quver Hecke algebras, and the mnmal categorfcatons L mn (κ 1,..., κ n ) of tensor products L(κ 1 ) L(κ n ) of ntegrable hghest weght modules ntroduced by Webser n [W2]. In [W1], Webster also nvestgated categorfcatons of more general tensor products nvolvng both ntegrable lowest weght and hghest weght modules; see also [BW] for the constructon of canoncal bases n such mxed tensor products. Away from fnte type, these modules have nfnte-dmensonal weght spaces. The canddates for ther mnmal categorfcatons suggested by Webster are fnte-dmensonal categores whch are not Artnan n general, so that the locally Schuran settng becomes essental. In type A, there are some closely related examples arsng from the cyclotomc orented Brauer categores of [BCNR], whch n level one are Delgne s categores Rep(GL t ) (e.g. see [EHS]). These also ft nto the framework of ths artcle. Here s a gude to the organzaton of the remander of the artcle. In Secton 2, we set up the basc algebrac foundatons of locally Schuran categores. Everythng here s ether well known (e.g. see [M]), or t s an obvous extenson of classcal results. However our language s new. Secton 3 s an exposton of the defnton of Kac-Moody 2-category, based manly on [B3]. We also dscuss brefly the graded verson of the Kac-Moody 2-category. Ths s mportant as t makes the connecton to quantum groups, although we wll not emphasze t elsewhere n the artcle. Secton 4 begns wth a revew of Rouquer s theory of 2-representatons of Kac-Moody 2-categores. We recall hs defnton of the unversal categorfcaton L(κ) of L(κ) from [R2, 4.3.3]. The mnmal categorfcaton L mn (κ) s a certan fnte-dmensonal specalzaton of L(κ); t can be realzed equvalently n terms of cyclotomc quver Hecke algebras. We also ntroduce a 2-representaton L(κ κ), whch s expected to play the role of unversal categorfcaton for the tensor product L(κ κ) := L (κ ) L(κ) of the ntegrable lowest weght module L (κ ) of (ant-domnant) lowest weght κ wth the ntegrable hghest weght module L(κ) (see Constructon 4.13). The mnmal categorfcaton L mn (κ κ) from [W1, Proposton 5.6] s a certan fnte-dmensonal specalzaton of L(κ κ). After that, we focus on nlpotent categorcal actons on locally Schuran categores. Any such structure has an assocated crystal n the sense of Kashwara; for example, the crystal assocated to L mn (κ) s the hghest weght crystal B(κ). Ths has

3 CATEGORICAL ACTIONS AND CRYSTALS 3 already found many strkng applcatons n classcal representaton theory; e.g. see [FK] (the oldest) and [DVV] (the most recent at the tme of wrtng). Acknowledgements. We thank Ben Webster for sharng hs deas n [W3], and for suggestng the reducton argument used n the proof of Theorem Also we thank Aaron Lauda for gvng us the opportunty to wrte ths survey, and the referee for many helpful suggestons. Notaton. Throughout, we work over an algebracally closed feld k. Ths means that all (2-)categores and (2-)functors wll be assumed to be k-lnear by default. 2. Locally Schuran categores In ths secton, we ntroduce our language of locally Schuran categores Locally untal algebras. A locally untal algebra s an assocatve (not necessarly untal) algebra A equpped wth a small famly (1 x ) x X of mutually orthogonal dempotents such that A = 1 y A1 x. x,y X A locally untal homomorphsm (resp. somorphsm) between two locally untal algebras s an algebra homomorphsm (resp. somorphsm) whch takes dstngushed dempotents to dstngushed dempotents. Also, we say that A s a contracton of B f there s an algebra somorphsm A B sendng each dstngushed dempotent n A to a sum of dstngushed dempotents n B. We say that A s locally Noetheran (resp. locally Artnan) f all of the left deals A1 x and all of the rght deals 1 y A satsfy the Ascendng Chan Condton (resp. the Descendng Chan Condton). One can also defne analogs of left (resp. rght) Noetheran or Artnan for locally untal algebras, requrng ust that all the left deals A1 x (resp. the rght deals 1 y A) satsfy the approprate chan condton. Unlke n the untal settng, locally left/rght Artnan does not mply locally left/rght Noetheran (but see Lemma 2.8 below). The followng example of a locally untal algebra that s locally left Artnan but not locally left Noetheran s taken from the end of [M, 3]: consder the locally untal algebra of upper trangular matrces over k wth rows and columns ndexed by the totally ordered set N { }, all but fntely many of whose entres are zero. All modules over a locally untal algebra wll be assumed to be locally untal wthout further menton; for a rght module V ths means that V = x X V 1 x as a drect sum of subspaces. If V s any A-module satsfyng ACC, t s clearly fntely generated. Conversely, assumng that A s locally Noetheran (resp. locally Artnan), fntely generated modules satsfy ACC (resp. DCC). We deduce n the locally Noetheran case that submodules of fntely generated modules are fntely generated. Let Mod-A be the category of all rght A-modules. We ll also need the followng full subcategores of Mod-A: lfdmod-a consstng of all locally fnte-dmensonal modules,.e. rght modules V wth dm V 1 x < for all x X; fgmod-a consstng of all fntely generated modules; pmod-a consstng of all fntely generated proectve modules. Replacng rght wth left everywhere here, we obtan analogous categores A-Mod, A-lfdMod, A-fgMod and A-pMod of left modules. There are contravarant equvalences : lfdmod-a A-lfdMod, # : pmod-a A-pMod defned as follows: the dual V of a locally fnte-dmensonal rght module V s the left module x X Hom k(v 1 x, k); the dual P # of a fntely generated proectve rght

4 4 J. BRUNDAN AND N. DAVIDSON module P s the left module Hom A (P, A). If V and P are left modules nstead, ther duals V and # P are the rght modules defned analogously. Remark 2.1. The data of a locally untal algebra A s the same as the data of a small category A wth obect set X and morphsms Hom A (x, y) := 1 y A1 x. In ths ncarnaton, locally untal algebra homomorphsms correspond to functors. A rght A- module becomes a functor A op Vec, and then a module homomorphsm s a natural transformaton of functors. We say A s Noetheran (resp. Artnan) f A s locally Noetheran (resp. locally Artnan) n the sense already defned. All of the other notons ntroduced n ths subsecton can be recast n ths more categorcal language too, as was done n [M]. For example, the proectve module 1 x A corresponds to the functor Hom A (, x) : A op Vec. Then the Yoneda Lemma asserts that there s a fully fathful functor A pmod-a sendng x ob A to 1 x A, and a Hom A (x, y) to the homomorphsm 1 x A 1 y A defned by left multplcaton by a 1 y A1 x. Ths extends canoncally to an equvalence of categores A pmod-a, (2.1) where A denotes the addtve Karoub envelope of A, that s, the dempotent completon of the addtve envelope of A. For locally untal algebras A and B wth dstngushed dempotents (1 x ) x X and (1 y ) y Y, respectvely, an (A, B)-bmodule M = x X,y Y 1 xm1 y determnes an adont par (T M, H M ) of functors T M := A M : Mod-A Mod-B, H M := x X Hom B (1 x M, ) : Mod-B Mod-A. Here are a couple of useful facts about tensorng wth bmodules. Frst, there s a natural somorphsm V A Hom B (Q, M) Hom B (Q, V A M), v f (q v f(q)) (2.2) for all rght A-modules V and fntely generated proectve rght B-modules Q; cf. [AF, 20.10]. Also, gven another locally untal algebra C and a rght exact functor E : Mod-B Mod-C commutng wth drect sums, EM s an (A, C)-bmodule, and there s a natural somorphsm of rght C-modules V A EM E(V A M), v n E(f v )(n) (2.3) for all rght A-modules V, where f v : M V A M s the rght C-module homomorphsm m v m. The proof of ths nvolves a reducton to the case V = A usng the Fve Lemma. Defnton 2.2. We say that an (A, B)-bmodule M s left rgd f t has a left dual n the 2-category of bmodules; see [EGNO, 2.10] for our conventons here. It means that there exsts a (B, A)-bmodule M #, an (A, A)-bmodule homomorphsm coev : A M B M #, and a (B, B)-bmodule homomorphsm ev : M # A M B, such that the compostons M # M can coev 1 A A M M B M # A M 1 ev M B B can M, can M # A A 1 coev M # A M B M # ev 1 B B M # can M # are the denttes. In other words, T M # s rght adont to T M. We say that M s rght rgd f t has a rght dual,.e. there exsts a (B, A)-bmodule # M such that T# M s left

5 CATEGORICAL ACTIONS AND CRYSTALS 5 adont to T M. Fnally, we say M s rgd f t s both left and rght rgd, and sweet 1 f n addton M # = # M as (B, A)-bmodules. The followng s essentally [EGNO, Ex ]. Lemma 2.3. Let A and B be locally untal algebras wth dstngushed dempotents (1 x ) x X and (1 y ) y Y, respectvely. Let M be an (A, B)-bmodule. (1) The bmodule M s left rgd f and only f 1 x M s fntely generated and proectve as a rght B-module for each x X. In that case, M # = x X (1 xm) #. (2) It s rght rgd f and only f M1 y s fntely generated and proectve as a left A-module for each y Y. In that case, # M = y Y # (M1 y ). Proof. (1) Suppose that M possesses a left dual M #. Then T M has a rght exact rght adont T M #, so T M sends proectves to proectves. Hence, 1 x M = T M (1 x A) s proectve for each x X. Let coev(1 x ) = n x =1 v x, f x, for v x, 1 x M and f x, (M # )1 x. Then any v 1 x M s equal to (1 ev)(v x, f x, v) n =1 v x,a. Ths shows that 1 x M s fntely generated. Conversely, suppose that each 1 x M s fntely generated and proectve as a rght module. Then (2.2) mples that H M = TM # where M # := x X (1 xm) #. Hence, we have constructed a bmodule M # such that T M # s rght adont to T M, provng that M s left rgd. (2) Smlar (workng wth left modules nstead of rght ones). By a proectve generatng famly for an Abelan category C, we mean a small famly (P (x)) x X of compact 2 proectve obects such that for each V ob C there s some x X wth Hom C (P (x), V ) 0. Just lke n [F, Exercse 5.F], one can show that an Abelan category C s equvalent to Mod-A for some locally untal algebra A f and only f C possesses arbtrary drect sums and has a proectve generatng famly; see also [M, Theorem 3.1]. We ust need ths n the followng specal case, whch s the locally untal analog of the classcal Morta Theorem: Theorem 2.4. Let B be a locally untal algebra. Suppose that (P (x)) x X s a proectve generatng famly for Mod-B. Let A := Hom B (P (x), P (y)), x,y X vewed as a locally untal algebra wth dstngushed dempotents (1 x := 1 P (x) ) x X. Let P := x X P (x), whch s an (A, B)-bmodule. (1) The functors T P = A P and H P = x X Hom B(1 x P, ) are quas-nverse equvalences between the categores Mod-A and Mod-B. (2) We have that H P = TQ where Q := P #. Thus, we have constructed a sweet (A, B)-bmodule P and a sweet (B, A)-bmodule Q such that P B Q = A and Q A P = B as bmodules. Proof. The fact that H P T P = 1Mod-A follows from (2.2). Then one deduces that T P H P = 1Mod-B too by a standard argument; cf. [AF, 22.2]. Fnally Lemma 2.3(1) mples that H P = TQ. Corollary 2.5. For locally untal algebras A and B, the followng are equvalent: 1 The language sweet bmodule appears n [K, 2.6], but (n vew of Lemma 2.3) the ones defned there are ust what we call rgd bmodules, snce t s not assumed that # M = M #. Note though that the mportant examples constructed n [K] do satsfy ths extra hypothess, so they are sweet n our sense too. 2 For categores of the form Mod-A for some locally untal algebra A, a proectve s compact f and only f t s fntely generated.

6 6 J. BRUNDAN AND N. DAVIDSON (1) the categores Mod-A and Mod-B are equvalent; (2) the categores pmod-a and pmod-b are equvalent; (3) the categores A-pMod and B-pMod are equvalent; (4) the categores A-Mod and B-Mod are equvalent. Proof. (1) (2). The restrcton of an equvalence Mod-A Mod-B gves an equvalence pmod-a pmod-b. (2) (1). Let F : pmod-a pmod-b be an equvalence of categores. Let (1 x ) x X be the dstngushed dempotents n A. Then (P (x) := F (1 x A)) x X s a proectve generatng famly for Mod-B such that A = x,y X Hom B(P (x), P (y)). Now apply Theorem 2.4. (3) (4). Ths s the same as (1) (2) wth A and B replaced by the opposte algebras. (2) (3). Ths follows as pmod-a (resp. pmod-b) s contravarantly equvalent to A-pMod (resp. B-pMod). Two locally untal algebras A and B are sad to be Morta equvalent f the condtons of Corollary 2.5 are satsfed. For example, f A s a contracton of B, then the categores Mod-A and Mod-B are obvously somorphc. Hence, A and B are Morta equvalent. For another smple example, let N be any (possbly nfnte but small) set and M N (k) be the algebra of N N matrces wth entres n k, all but fntely many of whch are zero. Ths s a locally untal algebra wth dstngushed dempotents gven by the dagonal matrx unts {e, N}. Applyng Theorem 2.4 wth B := k, X := N and takng each P (x) to be a copy of k, we see that M N (k) s Morta equvalent to the ground feld k. Remark 2.6. Suppose that A and B are the categores assocated to locally untal algebras A and B as n Remark 2.1. We say that A and B are Morta equvalent f ther addtve Karoub envelopes A and B are equvalent. In vew of Corollary 2.5 and (2.1), ths s equvalent to the algebras A and B beng Morta equvalent as above. The fnal theorem n ths subsecton s concerned wth adont functors. Agan ths s classcal n the untal settng. Theorem 2.7. Let B, P = x X P (x) and A be as n Theorem 2.4, so that H P : Mod-B Mod-A s an equvalence of categores. Suppose we are gven a functor E : Mod-B Mod-B. Then E possesses a rght adont f and only f t s rght exact and commutes wth drect sums. In that case, let M := Hom B (P (x), EP (y)) x,y X vewed as an (A, A)-bmodule n the natural way. Then the dagram H P Mod-B Mod-A E T M Mod-B H P commutes up to a canoncal somorphsm. Mod-A Proof. It s standard that functors possessng a rght adont are rght exact and commute wth drect sums. Conversely, suppose that E s rght exact and commutes wth drect sums. Usng (2.3) then (2.2), we get that H P E T P = HP T EP = THP (EP ), whch s somorphc to T M as H P (EP ) = M. Thus H P E T P = TM. Composng on the rght wth the quas-nverse H P of T P, we deduce that H P E = T M H P. Ths

7 CATEGORICAL ACTIONS AND CRYSTALS 7 proves the fnal part of the theorem. Hence E has a rght adont as T M has the rght adont H M Fnte-dmensonal categores. Let A be a locally untal algebra wth dstngushed dempotents (1 x ) x X. We assume n ths subsecton that A s locally fntedmensonal,.e. each subspace 1 y A1 x s fnte-dmensonal. Equvalently, the assocated category A from Remark 2.1 s a fnte-dmensonal category,.e. t s a small k-lnear category all of whose morphsm spaces are fnte-dmensonal. All of the rght deals 1 x A and the left deals A1 x are locally fnte-dmensonal. Hence, so are ther duals (A1 x ) and (1 x A). Consequently, all fntely generated modules are locally fnte-dmensonal, as are all fntely cogenerated modules. Lemma 2.8. For a locally fnte-dmensonal locally untal algebra A, locally Artnan mples locally Noetheran. Proof. As A1 x satsfes DCC, ts dual (A1 x ) satsfes ACC. Hence, (A1 x ) s fntely generated, so satsfes DCC. Hence, A1 x satsfes ACC. Smlarly each 1 x A has ACC. Here are varous other basc facts about modules over a locally fnte-dmensonal locally untal algebra A. For the most part, these are proved by mmckng the usual proofs n the settng of fnte-dmensonal algebras, so we wll be qute bref. Fx representatves {L(b) b B} for the somorphsm classes of rreducble rght A-modules 3. (L1) If V s fntely generated (resp. locally fnte-dmensonal) and W s locally fntedmensonal (resp. fntely cogenerated) then Hom A (V, W ) s fnte-dmensonal. (L2) Schur s Lemma holds: End A (L(b)) = k for each b B. (L3) Any fntely generated (resp. fntely cogenerated) module satsfes the Krull- Schmdt Theorem. (L4) The category Mod-A s a Grothendeck category,.e. t s Abelan, t possesses arbtrary drect sums, drect lmts of short exact sequences are exact, and there s a generator (namely, the regular module A tself). Hence, by the general theory of Grothendeck categores, every A-module has an nectve hull; moreover, a module V s fntely cogenerated f and only f ts socle soc(v ),.e. the sum of the rreducble submodules of V, s an essental submodule of V of fnte length. (L5) For b B, let A b := A/ Ann A (L(b)), whch s a locally untal algebra wth dstngushed dempotents (1 x ) x X that are the mages of the ones n A. Also let M b := x,y X Hom k(l(b)1 x, L(b)1 y ), vewed as a locally untal algebra wth multplcaton that s the opposte of composton. Note that M b s a contracton of M N (k) where N := {(x, ) x X, 1 dm L(b)1 x }. Then the natural rght acton of A on L(b) nduces a locally untal somorphsm A b Mb. Hence, A b s a contracton of a locally untal matrx algebra. Next, let J := b B Ann A(L(b)) be the Jacobson radcal of A. The map A/J b B A b, a + J (a + Ann A (L(b))) b B s a well-defned algebra somorphsm. Hence, A/J s a contracton of a (possbly nfnte) drect sum of locally untal matrx algebras. It follows that A/J s semsmple,.e. every A/J-module s completely reducble. Moreover, J s the smallest two-sded deal of A wth ths property. (L6) For a rght A-module V, ts radcal rad(v ) := V J s the ntersecton of all of ts proper maxmal submodules; ts head hd(v ) := V/ rad(v ) s ts largest completely reducble quotent. Applyng to the statements made n (L4), we deduce that every fntely generated A-module has a proectve cover; moreover, 3 We use the notaton B here as ultmately ths set wll carry a crystal structure; cf. Defnton 4.30.

8 8 J. BRUNDAN AND N. DAVIDSON V s fntely generated f and only f rad(v ) s a superfluous submodule and hd(v ) s of fnte length 4. (L7) Let P (b) be a proectve cover of L(b). For any rght A-module V, the composton multplcty [V : L(b)] s defned as usual to be the supremum of the multplctes #{ = 1,..., n V /V 1 = L(b)} taken over all fltratons 0 = V 0 < < V n = V and all n N. By Schur s Lemma, we have that [V : L(b)] = dm Hom A (P (b), V ) N { }. Notng that Hom A (1 x A, L(b)) = L(b)1 x, the proectve module 1 x A decomposes as 1 x A = P (b) dm L(b)1 x. b B All but fntely many summands on the rght hand sde are zero, so there are only fntely many b B such that L(b)1 x 0. Hence, for any V, we have that dm V 1 x = b B[V : L(b)] dm L(b)1 x. In partcular, we get from ths that V s locally fnte-dmensonal f and only f [V : L(b)] < for all b B. (L8) Gven b B, we choose x X so that L(b)1 x 0. The decomposton of 1 x A derved n (L7) mples that there exsts a prmtve dempotent 1 b 1 x A1 x such that P (b) = 1 b A. Then I(b) := (A1 b ) s an nectve hull of L(b). For any V, we have that [V : L(b)] = dm Hom A (V, I(b)). (L9) Suppose we are gven a famly (A ) I of locally fnte-dmensonal locally untal algebras, wth the dstngushed dempotents n A ndexed by X. Then A := I A s a locally fnte-dmensonal locally untal algebra wth dstngushed dempotents ndexed by X := I X. Moreover there s an equvalence of categores I Mod-A Mod-A whch sends an obect (V ) I of I Mod-A to the A-module I V. (L10) Suppose B = I B s a partton such that that Hom A (P (b), P (c)) = 0 for all b B, c B and. For x X, we can wrte 1 x unquely as a sum of mutually orthogonal dempotents 1 x = I 1 (x,) so that 1 (x,) A = b B P (b) dm L(b)1x. Let A := x,y X 1 (y,)a1 (x,), whch s tself a locally untal algebra wth dempotents (1 (x,) ) x X and rreducbles represented by {L(b) b B }. Then we have that A = I A. Hence, A s a contracton of I A. If none of the B can be parttoned any further n ths way, we call ths the block decomposton of A, and refer to ndecomposable subalgebras A as blocks Locally Schuran categores. Later n the artcle, we wll be nterested n categorcal actons on categores of the followng form: Defnton 2.9. We say that a category C s locally Schuran f t s equvalent to Mod-A for some locally fnte-dmensonal locally untal algebra A. Gven a locally Schuran category C, Theorem 2.4 gves a recpe for constructng a locally fnte-dmensonal locally untal algebra A such that C s equvalent to Mod-A: choose a proectve generatng famly (P (x)) x X for C; set A := Hom C (P (x), P (y)) (2.4) x,y X 4 One can gve a drect proof of ths usng the fact that J s locally nlpotent n the sense that eje s a nlpotent deal of the fnte-dmensonal algebra eae for any dempotent e A.

9 CATEGORICAL ACTIONS AND CRYSTALS 9 vewed as a locally untal algebra wth dstngushed dempotents (1 x := 1 P (x) ) x X ; then the functor H := x X Hom C (P (x), ) : C Mod-A (2.5) s an equvalence of categores. Often t s convenent to proceed by choosng representatves {L(b) b B} for the somorphsm classes of rreducble obect n C and lettng P (b) (resp. I(b)) be a proectve cover (resp. an nectve hull) of L(b). Then we call (P (b)) b B a mnmal proectve generatng famly for C, and C s equvalent to Mod-B where B := Hom C (P (b), P (c)). (2.6) b,c B Ths a basc locally untal algebra: the rreducble B-modules are all one dmensonal. An obect V n C s locally fnte-dmensonal f and only f all ts composton multplctes are fnte. We let pc fgc lfdc be the full subcategores of C consstng of fntely generated proectve, fntely generated, and locally fnte-dmensonal obects; for A as n (2.4), these are equvalent to the subcategores pmod-a, fgmod-a and lfdmod-a of Mod-A. We say that C s Noetheran f all fntely generated (resp. cogenerated) obects satsfy ACC (resp. DCC); equvalently, the algebra A from (2.4) s locally Noetheran. We say that C s Artnan f all fntely generated (resp. cogenerated) obects satsfy DCC (resp. ACC); equvalently, the algebra A s locally Artnan. We say that C s fnte f there are only fntely many somorphsm classes of rreducble obect; equvalently, the basc algebra B from (2.6) s fnte-dmensonal. By Lemma 2.8, Artnan mples Noetheran. If C s Artnan then fgc s a Schuran category n the sense of [BLW, 2.1]: t s Abelan, all obects are of fnte length, there are enough proectves and nectves, and the endomorphsm algebras of the rreducble obects are one dmensonal. Moreover, for V ob C, the followng are equvalent: V s fntely generated; V s fntely cogenerated; V has fnte length Sweet endofunctors. We wll consder categorcal actons on locally Schuran categores nvolvng functors of the followng form: Defnton Let C be a locally Schuran category. We say that an endofunctor E of C s sweet f there s an endofunctor F whch s badont to E. Recallng the defnton of sweet bmodule from Defnton 2.2, the followng theorem gves an algebrac characterzaton of sweet endofunctors. Theorem Let E be an endofunctor of a locally Schuran category C. Fx an equvalence H : C Mod-A as n (2.5). Then E s sweet f and only f there s a sweet bmodule M such that H E = T M H. In that case, we have that M = Hom C (P (x), EP (y)). (2.7) x,y X Moreover, E s exact, contnuous and cocontnuous, and t preserves the sets of locally fnte-dmensonal, fntely generated, fntely cogenerated, proectve and nectve obects. Proof. If M s a sweet bmodule such that H E = T M H, then E s sweet snce T M s a sweet endofunctor of Mod-A and H s an equvalence. Conversely, suppose that E possesses a badont F. Theorem 2.7 shows that H E = T M H for M as n (2.7); smlarly H F = T N H for some bmodule N. Then T M and T N are badont. Hence,

10 10 J. BRUNDAN AND N. DAVIDSON N s both rght and left dual to M,.e. M s a sweet bmodule. It follows at once that E and F both send fntely generated obects to fntely generated obects, as T M and T N clearly do. Snce E has a badont, t s exact, contnuous and cocontnous. Also F s exact, so E preserves proectves and nectves. To see that E preserves locally fnte-dmensonal obects, we observe for locally fnte-dmensonal V that dm Hom C (P (b), EV ) = dm Hom C (F P (b), V ) < for all b B. Smlarly F preserves locally fnte-dmensonal obects. Fnally, snce E preserves fntely generated obects, we have that dm Hom C (EP (b), L(c)) = dm Hom C (P (b), F L(c)) = [F L(c) : L(b)] s zero for all but fntely many c. Hence, dm Hom C (L(c), EI(b)) = dm Hom C (F L(c), I(b)) = [F L(c) : L(b)] s zero for all but fntely many c. Ths mples that EI(b) s a fnte drect sum of I(c) s. Hence, E preserves fntely cogenerated obects, and smlarly for F. Lemma Suppose that F and G are sweet endofunctors of a locally Schuran category C. Let η : F G be a natural transformaton. If η L : F L GL s an somorphsm for each rreducble obect L ob C, then η s an somorphsm. Proof. We may assume that C = Mod-B for a basc locally untal algebra B. Ths means that the rreducble B-modules are parametrzed by the same set B as ndexes ts dstngushed dempotents, and [V : L(b)] = dm Hom B (P (b), V ) = dm V 1 b for each b B. The man step s to prove that η V : F V GV s an somorphsm for each locally fnte-dmensonal B-module V. Assumng ths, the lemma may be deduced as follows: gven any B-module V, consder a two-step proectve resoluton Q P V 0; snce P and Q are drect sums of fntely generated proectves, and F and G commute wth arbtrary drect sums, the locally fnte-dmensonal result shows that η P and η Q are somorphsms. Hence, η V s an somorphsm too by the Fve Lemma. So now suppose that V s locally fnte-dmensonal. It suffces to show for each fxed a B that the restrcton of η V defnes a lnear somorphsm between (F V )1 a and (GV )1 a. Let X := {b B (F L(b))1 a 0} = {b B Hom B (P (a), F L(b)) 0}. Fxng a left adont E to F, we have that X = {b B Hom B (EP (a), L(b)) 0}. Snce EP (a) s a fntely generated proectve, we deduce from ths that X s fnte; moreover, EP (a) a drect sum of ndecomposable proectves of the form P (b) for b X. Now we proceed by nducton on n := b X dm V 1 b N. In case n = 0, we have that Hom B (P (a), F V ) = Hom B (EP (a), V ) = 0. Hence, (F V )1 a = 0; smlarly, (GV )1 a = 0. So the desred concluson that (F V )1 a = (GV )1a s trval. For the nducton step, we take a vector 0 v V 1 b for some b X. Let W := vb and W := rad(w ), so that we have a fltraton 0 W < W V wth W/W = L(b). By nducton, η W and η V/W restrct to somorphsms (F W )1 a (GW )1 a and (F V/F W )1 a (GV/GW )1a. Also η W/W s an somorphsm as W/W s rreducble. Hence, η V defnes an somorphsm (F V )1 a (GV )1a as requred Serre quotents. Fnally n ths secton, we revew brefly the standard notons of Serre subcategory and Serre quotent category n the settng of locally Schuran categores. Let C be a locally Schuran category wth rreducble obects represented by {L(b) b B} as above. Let B be any subset of B and C be the full subcategory of C consstng of all the obects whose rreducble subquotents are somorphc to L(b) for b B. It s a Serre subcategory of C,.e. t s closed under takng subobects, quotents and

11 CATEGORICAL ACTIONS AND CRYSTALS 11 extensons. Moreover t s tself a locally Schuran category wth rreducble obects represented by {L(b) b B }. To see ths, defne B accordng to (2.6) so that C s equvalent to Mod-B. Then C s equvalent to Mod-B where B s the quotent of B by the two-sded deal generated by the dempotents {1 b b B \ B }. The exact ncluson functor ι : C C corresponds to the natural nflaton functor from Mod-B to Mod-B, and t has a left adont ι! (resp. a rght adont ι ) whch sends an obect to ts largest quotent (resp. subobect) belongng to C. We have that ι! ι = 1 C = ι ι. The Serre quotent C/C s an Abelan category equpped wth an exact quotent functor π : C C/C satsfyng the followng unversal property: f F : C D s any exact functor to an Abelan category D then there s a unque functor F : C/C D such that F = F π. In fact C/C s another locally Schuran category wth rreducbles represented by {πl(b) b B\B }. Agan ths s easy to see n terms of the algebra B: the category C/C s equvalent to modules over the algebra ebe := b,c B\B 1 bb1 c. The quotent functor π corresponds to the obvous truncaton functor e : Mod-B Mod-eBe sendng a B-module V to V e := b B\B V 1 b. Consequently, π has a left adont π! : C/C C and a rght adont π : C/C correspondng to the functors ebe eb and b B\B Hom ebe(b1 b, ), respectvely. We have that π π! = 1 C/C = π π. Lemma In the above setup, assume that we are gven V, W ob C such that V s fntely generated, W s fntely cogenerated, and all consttuents of hd(v ) and soc(w ) are of the form L(b) for b B \ B. Then the functor π nduces an somorphsm Hom C (V, W ) Hom C/C (πv, πw ). Proof. The count of aduncton defnes a morphsm f : π! πv V. By the assumptons on V, f s an epmorphsm. Moreover f becomes an somorphsm on applyng π, hence ker f belongs to C. Usng also the assumptons on W, we deduce that Hom C/C (πv, πw ) = Hom C (π! πv, W ) = Hom C (V, W ). 3. Kac-Moody 2-categores In ths secton, we revew Rouquer s defnton of Kac-Moody 2-category from [R1]. Then, followng [B3], we explan the relatonshp between ths and the 2-category ntroduced by Khovanov and Lauda n [KL3], and dscuss the graded verson. Note our exposton uses a slghtly dfferent normalzaton for the second aduncton compared to [B3] based on the dea of [BHLW] Kac-Moody data. Let I be a fnte ndex set 5 and A = ( d ), I be a generalzed Cartan matrx, so d = 2, d 0 for, and d = 0 d = 0. We assume that A s symmetrzable, so that there exst postve ntegers (d ) I such that d d = d d for all, I. Pck a fnte-dmensonal complex vector space h and lnearly ndependent subsets {α I} and {h I} of h and h, respectvely, such that h, α = d for all, I. Let P := { h h, Z for all I}, Q := I Zα, P + := { h h, N for all I}, Q + := I Nα. We refer to P and Q as the weght lattce and the root lattce, respectvely. We vew P as an nterval-fnte poset va the usual domnance orderng: µ µ Q +. 5 Wth mnor adustments to the basc defntons, everythng here can be extended to nfnte I too; type A s partcularly mportant n applcatons.

12 12 J. BRUNDAN AND N. DAVIDSON Let g be the assocated Kac-Moody algebra wth Cartan subalgebra h. Thus, g s the Le algebra generated by h and elements e, f ( I) subect to the usual Serre relatons: for h, h h and, I we have that [h, h ] = 0, [e, f ] = δ, h, (3.1) [h, e ] = h, α e, [h, f ] = h, α f, (3.2) (ad e ) d+1 (e ) = 0, (ad f ) d+1 (f ) = 0. (3.3) Let U(g) be ts unversal envelopng algebra. Actually t s often more convenent to work wth the dempotented verson U(g) of U(g), whch s a certan locally untal algebra wth dstngushed dempotents (1 ) P. It s defned by analogy wth [Lu2, 23.1] (whch treats the quantum case). As well as beng an algebra, U(g) s a (U(g), U(g))- bmodule wth h1 = h, 1 = 1 h, e 1 = 1 +α e and f 1 = 1 α f. The elements {1, e 1, f 1 I, P } generate U(g) subect to the relatons derved from (3.1) (3.3). Weght modules for g,.e. g-modules V such that V = P V, are ust the same as (locally untal) U(g)-modules. By an upper (resp. lower) ntegrable module, we mean a weght module wth fntedmensonal weght spaces all of whose weghts le n a fnte unon of sets of the form Q + (resp. + Q + ). By the classcal theory from [Kac, Chapters 9 10], a g-module s upper ntegrable f and only f t s somorphc to a fnte drect sum of the rreducble modules 6 L(κ) := U(g)1 κ / e 1 κ, f 1+ h,κ 1 κ I (3.4) for κ P + ; these are the ntegrable hghest weght modules. Smlarly, a g-module s lower ntegrable f and only f t s a fnte drect sum of the (rreducble) ntegrable lowest weght modules L (κ ) := U(g)1 κ / e 1 h,κ 1 κ, f 1 κ I. (3.5) for κ P +. Generalzng (3.4) (3.5), we let L(κ κ) := U(g)1 κ+κ / e 1 h,κ 1 κ+κ, f 1+ h,κ 1 κ+κ I (3.6) for κ P + and κ P +. These modules are not so well studed, but they play an mportant role n Lusztg s constructon of canoncal bases for U(g) from [Lu2, Part IV]. They are ntegrable modules but they may have nfnte-dmensonal weght spaces outsde of fnte type, so that they are nether upper nor lower ntegrable. The next lemma s the classcal counterpart of [Lu2, Proposton ]. Lemma 3.1. There s a g-module somorphsm L(κ κ) L(κ ) L(κ) such that u 1 κ+κ u( 1 κ 1 κ ) for u U(g). The followng lemma about annhlators wll be useful later on. Lemma 3.2. We have that κ P + κ P + Moreover f g s of fnte type then Ann U(g) (L(κ κ)) = {0}. P + Ann U(g) (L()) = {0}. 6 All of these assertons depend on the assumpton that A s symmetrzable.

13 CATEGORICAL ACTIONS AND CRYSTALS 13 Proof. The proof of the frst statement reduces usng Lemma 3.1 to checkng that the maps U (g) L(κ), u (u 1 κ ) κ P +, κ P + U + (g) κ P + L (κ ), u (u 1 κ ) κ P + are nectve, where U ± (g) are the postve and negatve parts of U(g) generated by the e and f, respectvely. These are well-known facts; e.g. they can be deduced n a non-classcal way from [Lu2, Proposton ]. The second statement (whch s even better known) follows from the frst snce each L(κ κ) s a fnte drect sum of L() s n vew of Lemma 3.1 and complete reducblty Strct 2-categores. Let Cat be the category of (small) k-lnear categores and k-lnear functors. It s a monodal category wth tensor functor : Cat Cat Cat defned on obects (= categores) by lettng C C be the category wth obects that are pars (, ) ob C ob C, morphsms Hom C C ((, ), (µ, µ )) := Hom C (, µ) k Hom C (, µ ), and composton law defned by (g g ) (f f ) := (g f) (g f ). The defnton of on morphsms (= functors) s obvous: F F s the functor that sends (, ) (F, F ) and g g F g F g. Defnton 3.3. A strct 2-category s a category enrched n Cat. Thus, for obects, µ n a strct 2-category C, there s gven a category Hom C (, µ) of morphsms from to µ, whose obects F, G are the 1-morphsms of C, and whose morphsms x : F G are the 2-morphsms of C. For example, Cat can be vewed as a strct 2-category Cat wth 2-morphsms beng natural transformatons. Gven a strct 2-category C, we use the shorthand Hom C (F, G) for the vector space Hom HomC (,µ)(f, G) of all 2-morphsms x : F G. Let us also brefly recall the strng calculus for 2-morphsms n C; e.g. see [L2, 2]. We represent a 2-morphsm x Hom C (F, G) by the pcture µ G x F The vertcal composton y x of x wth another 2-morphsm y Hom C (G, H) s obtaned by vertcally stackng pctures: H y. µ G x. F Gven 2-morphsms x : F H, y : G K between 1-morphsms F, H : µ, G, K : µ ν, we denote ther horzontal composton by yx : GF KH, and represent t by horzontally stackng pctures: ν K y µ H x. G F

14 14 J. BRUNDAN AND N. DAVIDSON When confuson seems unlkely, we wll use the same notaton for a 1-morphsm F as for ts dentty 2-morphsm. Wth ths conventon, we have that yh Gx = yx = Kx yf, or n pctures: ν K H y µ x. = G F ν K H y µ x. = G F ν K H y µ x.. Ths s the nterchange law; t means that dagrams for 2-morphsms are nvarant under rectlnear sotopy. We note that any strct 2-category C has an addtve envelope constructed by takng the addtve envelope of each of the morphsm categores n C. The addtve Karoub envelope Ċ s the strct 2-category obtaned by takng dempotent completons after that. Fnally, we defne the Grothendeck rng K 0 (Ċ) :=,µ ob C G K 0 (HomĊ(, µ)), (3.7) where the latter K 0 denotes the usual splt Grothendeck group of an addtve category. Horzontal composton nduces a multplcaton makng K 0 (Ċ) nto a locally untal rng wth dstngushed dempotents (1 ) ob C Quver Hecke categores and the nl Hecke algebra. The data of a strct monodal category C s equvalent to that of a strct 2-category C wth one obect; the obects and morphsms n C correspond to the 1-morphsms and 2-morphsms n C. For strct monodal categores, we wll use the same dagrammatc formalsm as explaned n the prevous subsecton; the only dfference s that there no need to label the regons of the dagrams by obects snce there s only one. In the next defnton, we ntroduce the quver Hecke category, whch s half of the Kac-Moody 2-category U(g) to be defned n the next subsecton. Everythng from ths pont on depends on the Kac-Moody data from 3.1 plus some addtonal parameters: we fx unts t k such that t = 1 and d = 0 t = t ; scalars 7 s pq k for 0 < p < d, 0 < q < d such that s pq = sqp. Defnton 3.4. The (postve) quver Hecke category H s the strct monodal category generated by obects I and morphsms : and : subect to the followng relatons: 0 f =, = t t d + t d + s pq 0<p<d 0<q<d p q F f d = 0, otherwse, = δ,, = δ,, 7 In [B3] and elsewhere, scalars s pq are ncorporated nto the relatons also for p = 0 or q = 0; we don t allow so much freedom here because t makes t mpossble to prove that dots are nlpotent n the cyclotomc quotents dscussed below (see Lemma 4.16).

15 k k = CATEGORICAL ACTIONS AND CRYSTALS 15 r,s 0 r+s=d 1 r t s + k 0<p<d 0<q<d r,s 0 r+s=p 1 s pq r q f = k, k 0 otherwse. (We depct the nth power of under vertcal composton by labellng the dot wth n.) There s also the (negatve) quver Hecke category H generated by obects I and morphsms : and : subect to the relatons obtaned by reversng the drectons of all the arrows n the above, then swtchng the order of the terms on the left hand sdes of the second, thrd and fourth relatons. In fact, H s somorphc to H, but the dfferent normalzaton of generators s sometmes more convenent. For obects = n 1 I n and = m 1 I m, there are no morphsms n H unless m = n. The endomorphsm algebra H n := Hom H (, ) (3.8), I n s the (postve) quver Hecke algebra whch was ntroduced ndependently n [R1] and [KL1]. There s also the negatve verson H n :=, I Hom n H (, ), whch s somorphc to H n wth a dfferent normalzaton of generators. In the specal case that I s a sngleton, the quver Hecke algebra H n s the nl Hecke algebra NH n, whch plays a crucal role n the general theory. Numberng strands by 1,..., n from rght to left, let us wrte X for the element of NH n correspondng to a dot on the th strand, and T for the element correspondng to the crossng of the th and ( + 1)th strands. Let S n be the symmetrc group wth ts usual smple reflectons s 1,..., s n 1, length functon l and longest element w n. It acts naturally on the polynomal algebra Pol n := k[x 1,..., X n ]; we wrte Sym n for the subalgebra of nvarants. The followng are well known; e.g. see [KL1, 2], [R2, 2] or [B2, 2]. (N1) There s a fathful representaton of NH n on Pol n defned by X f := X f and T f := s(f) f X X +1 (the th Demazure operator). (N2) For any w S n, let T w be the correspondng element of NH n defned va a reduced expresson for w. Then NH n s a free left Pol n -module on bass {T w w S n }. In partcular, Pol n NH n. (N3) The algebra Pol n s a free Sym n -module on bass {b w w S n } where b w := ( 1) l(w) T w X n 1 1 X n 2 2 X n 1. Moreover b wn = 1. (N4) There s an algebra somorphsm NH n EndSymn (Pol n ) nduced by the acton of NH n on Pol n. Hence, NH n = Mn! (Sym n ) and Z(NH n ) = Sym n. (N5) The element π n := ( 1) l(wn) X n 1 1 X n 2 2 X n 1 T wn acts on the bass for Pol n from (N3) by π n b w = δ w,1 b w. Hence, t s a prmtve dempotent n NH n, and NH n = (NHn π n ) n! as a left NH n -module. In the remander of the subsecton, we wsh to gve a frst ndcaton of the power of the quver Hecke relatons. Let C be some category whch s addtve and dempotentcomplete. Suppose that we are gven a categorcal acton of the quver Hecke category H on C,.e. there s a strct monodal functor Φ : H End(C) (3.9) where End(C) denotes the strct monodal category of all endofunctors of C. Let E := Φ() and x := Φ ( ) End(E ). Snce E s k-lnear (as always), t s addtve, hence s

16 16 J. BRUNDAN AND N. DAVIDSON t nduces an endomorphsm e := [E ] of the splt Grothendeck group K 0 (C). More generally, for I and n 1, we can obvously dentfy the nl Hecke algebra NH n wth End H ( n ); then the mage under Φ of the dempotent π n from (N5) gves us an dempotent π,n End(E n ). Let E(n) := π,n E n,.e. t s the endofunctor of C that sends an obect V to the mage of (π,n ) V End C (E n V ), and a morphsm f : V W to the restrcton of E n f. The followng lemma shows that ths categorfes the dvded power e (n) := e n /n!. ( ) Lemma 3.5. We have that E n n! =. E (n) Proof. By (N5), the dentty element of NH n s a sum of n! prmtve dempotents, each of whch s conugate to π n. Lemma 3.6. Suppose for some V ob C that there s a monc polynomal f(t) of degree n such that f((x ) V ) = 0. Then E (n+1) V = 0. In partcular, f C s fnte-dmensonal, then all e act locally nlpotently on K 0 (C). Proof. The second statement follows from the frst: f C s fnte-dmensonal then, for any V ob C, we have that (x ) V s an element of the fnte-dmensonal algebra End C (E V ). Hence, t certanly satsfes some polynomal relaton. To prove the frst statement, we frst note the followng dentty n NH n+1 : ( 1) l(wn+1) π n+1 f(x 1 )X n 1 2 X n T wn+1 = π n+1. Ths holds because πn+1 2 = π n+1 and moreover T wn+1 X1 m X2 n 1 X n T wn+1 = 0 for m < n by degree consderatons. Now as above we dentfy NH n+1 wth End H ( (n+1) ), apply Φ to our dentty, then evaluate the resultng natural transformatons at V. By assumpton, Φ(f(X 1 )) V = E nf((x ) V ) = 0. Hence, the left hand sde vanshes, and we deduce that Φ(π n+1 ) V = 0. Ths s the dentty endomorphsm of E (n+1) V, so the latter obect s somorphc to zero. Perhaps most strkng of all, we have the followng, whch s an mmedate consequence of the even stronger categorcal Serre relatons from [R1, Proposton 4.2]: Lemma 3.7. The endomorphsms e of K 0 (C) satsfy the Serre relatons from (3.3) Kac-Moody 2-categores. We are ready to formulate Rouquer s defnton of the Kac-Moody 2-category U(g); cf. [R1, 4.1.3]. Defnton 3.8. The Kac-Moody 2-category U(g) s the strct 2-category 8 wth obects P ; generatng 1-morphsms E 1 : + α and F 1 : α for each I and P, whose dentty 2-morphsms wll be represented dagrammatcally by +α and α E E 1 E E 1,, respectvely; and generatng 2-morphsms : E 1 E 1, : : 1 F E 1 and : E F 1 1. The generatng 2-morphsms are subect to the followng relatons. Frst, we have the postve quver 8 Some authors requre t s addtve from the outset but we don t assume ths.

17 CATEGORICAL ACTIONS AND CRYSTALS 17 Hecke relatons (cf. Defnton 3.4): 0 f =, = = = k k t d t f d = 0, + t d = δ,, r s + t r,s 0 k r+s=d 1 Next we have the rght aduncton relatons + s pq 0<p<d 0<q<d 0<p<d 0<q<d r,s 0 r+s=p 1 s pq p q r s k otherwse, q f = k, 0 otherwse. =, whch mply that F 1 +α s a rght dual of E 1. Fnally there are some nverson = relatons. To formulate these, defne a new 2-morphsm settng := Then the nverson relatons assert that the followng are somorphsms: h, 1 n=0 h, 1 m=0 n., : E F 1 F E 1 by : E F 1 F E 1 f, : E F 1 F E 1 1 h, f h, 0, m : E F 1 1 h, F E 1 f h, 0, the last two beng 2-morphsms n the addtve envelope of U(g). Remark 3.9. More formally, the nverson relatons mean that there are some addtonal generatng 2-morphsms : F E 1 E F 1, : 1 E F 1 and n F E 1 1 for 0 n < h, and 0 m < h, such that the followng hold: ( ) 1 = f, m :

18 18 J. BRUNDAN AND N. DAVIDSON h, 1 n=0 h, 1 m=0 = n ( m = ( h, 1 n=0 h, 1 m=0 n ) 1 f h, 0, ) 1 m f h, 0. As usual wth obects defned by generators and relatons, one then needs to play the game of dervng consequences from the defnng relatons. Here we record some whch were establshed n [B3]; we also cte below the more recent exposton n [BE2] snce that uses exactly the same normalzaton as here. (K1) Negatve quver Hecke relatons. Defne 2-morphsms : F 1 F 1 and : F F 1 F F 1 by settng :=, :=. k On rotatng the postve quver Hecke relatons clockwse through 180, one sees that these satsfy the negatve quver Hecke relatons: 0 f =, = = k = f d = 0, d t + t r,s 0 r+s=d 1 = δ, t k d, + r s + s pq 0<p<d 0<q<d 0<p<d 0<q<d r,s 0 r+s=p 1 s pq p q k r s otherwse, q f = k, 0 otherwse. (K2) Second aduncton. We next ntroduce 2-morphsms : 1 E F 1 and : F E 1 1. The defnton of these was suggested already by Rouquer n [R1, 4.1.4]. Followng the dea of [BHLW] we wll normalze them n a dfferent way whch depends on an addtonal choce of unts c ; k for each I and P such that c +α; = t c ;. Fxng such a choce from now on, we set := c ; f h, > 0, h, 1 c ; h, f h, 0,

19 CATEGORICAL ACTIONS AND CRYSTALS 19 := c 1 ; c 1 ; h, 1 f h, < 0, h, f h, 0. Then by [B3, Theorem 4.3] (or [BE2, Proposton 6.2] wth the present normalzaton) we have the left aduncton relatons: =, Ths means that F 1 +α s also a left dual of E 1. (K3) Ptchfork relatons and cyclcty. The followng relatons follow from the defntons and [B3, Theorem 5.3] (or [BE2, Proposton 4.1] and [BE2, Proposton 7.2]): =. =, =, =, =, =, =. It follows that =. Moreover, the 2-morphsms and equal to ther left mates: are cyclc.e. ther rght mates are =, =. Ths s the man advantage of the normalzaton of the second aduncton from [BHLW] as chosen n (K2). (K4) Infnte Grassmannan relatons. Let Sym be the algebra of symmetrc functons over k. Recall Sym s generated both by the elementary symmetrc functons e n (n 1) and by the complete symmetrc functons h n (n 1). Adoptng the conventon that e 0 = h 0 = 1 and e n = h n = 0 for n < 0, these two famles of generators are related by the equaton ( 1) r e r h s = 0 for all n > 0. (3.10) r+s=n

20 20 J. BRUNDAN AND N. DAVIDSON Take I, P and set h := h,. Then the nfnte Grassmannan relatons assert that there s a well-defned homomorphsm such that β ; (h n ) = c 1 ; β ; (e n ) = ( 1) n c ; β ; : Sym End U(g) (1 ) n+h 1 f n > h, (3.11) n h 1 f n > h. (3.12) Ths was proved orgnally by Lauda 9 n [L1, Proposton 8.2]; see [BE2, Proposton 5.1] where t s establshed usng our normalzaton. It motvated Lauda s ntroducton also of certan negatvely dotted bubbles, whch are 2-morphsms n End U(g) (1 ) defned so that (3.11) (3.12) hold for all n Z. (K5) Dual nverson relatons. The followng 2-morphsms are nvertble: h, 1 n=0 h, 1 m=0 m n : F E 1 E F 1 f, : F E 1 1 h, E F 1 f h, 0, : F E 1 E F 1 1 h, f h, 0. Ths may be deduced from the defntons above usng also the followng relatons proved n [B3, Corollary 3.3] (or [BE2, Corollary 5.2]): n = r 0 r, n r 2 m = r 0 m r 2. r Note here we are usng the negatvely dotted bubbles. Another consequence of the last relatons dsplayed, plus the descrpton of the leftward crossng gven n (K3), s that all 2-morphsms n U(g) are generated (under both vertcal and horzontal composton) by upward dots, upward crossngs, and leftward and rghtward cups and caps. (K6) Curl relatons. For n 0 we have: n = r 0 n r 1 r, n = r 0 r n r 1. These are proved e.g. n [BHLW, Lemma 3.2] or [BE2, Corollary 5.4]. (K7) Bubble sldes. For the next relatons, we adopt the followng convenent shorthand: n+ := n+ h, 1, n+ := n h, 1. (3.13) 9 In fact, Lauda showed for g = sl2 that ths homomorphsm s an somorphsm. In general, the product of the homomorphsms β ; over all I should gven an somorphsm I Sym End U(g) (1 ), but the proof of ths asserton depends on the Nondegeneracy Condton dscussed later n the subsecton.