HEISENBERG CATEGORIFICATION AND HILBERT SCHEMES

Transcription

1 HEISENBERG CATEGORIFICATION AND HILBERT SCHEMES SABIN CAUTIS AND ANTHONY LICATA Astract. Gven a fnte sugroup Γ SL 2 (C) we defne an addtve 2-category H Γ whose Grothendeck group s somorphc to an ntegral form h Γ of the Hesenerg algera. We construct an acton of H Γ on derved categores of coherent sheaves on Hlert schemes of ponts on the mnmal resolutons Ĉ 2 /Γ. Contents 1. Introducton Analogy wth Kac-Moody Le algera actons The 2-category H Γ Actons of H Γ Khovanov s Hesenerg categorfcaton Organzaton Acknowledgements 4 2. Quantum Hesenerg algeras The McKay correspondence The quantum Hesenerg algera assocated to Γ The Fock space 6 3. The category H Γ The algera B Γ The category H Γ The category H Γ Man Theorem # Remarks on relatons n H Γ Acton of H Γ Categores Functors P and Q Natural transformatons Man Theorem # Proof of Theorem Composton of X s Adont relatons Dots and adunctons Ptchfork relatons Dots and crossngs Composton of crossngs relatons Counter-clockwse crcles and curls Graphcal calculus and H Γ The 2-category H Γ The functor η : H Γ H Γ 32 1

2 2 SABIN CAUTIS AND ANTHONY LICATA 6.3. Proposton 2 for H Γ Relaton to Hlert schemes Proof of Man Theorem An aelan 2-representaton of H Γ Superalgeras and supermodules The algeras Bn Γ Functors P and Q Natural Transformatons The man Theorem Relaton to wreath products k[γ n S n ] Open Prolems Relaton to Kac-Moody 2-categores U(g) Degenerate affne Hecke algeras The structure of End H Γ(d) Hochschld cohomology of Hlert schemes 46 Appendx A. The case Γ Z 2 46 References Introducton The well-documented relatonshp etween the Hesenerg algera and Hlert schemes of ponts on a surface S nvolves an acton of the Hesenerg algera on the cohomology n H (Hl n (S)), [N1, G]. Inspred y other constructons from geometrc representaton theory, one expects ths Hesenerg acton on cohomology to lft to algerac K 0 -theory, though n general lftng ths acton s not straghtforward. In the specal case when S C 2 one can use the natural C acton together wth localzaton technques to defne Hesenerg algera actons on localzed equvarant K-theory (see [FT, SV]). After K-theory there s yet another level where one consders the derved categores of coherent sheaves n DCoh(Hl n (S)). Then one should defne a categorcal oect related to the Hesenerg algera whch acts on these trangulated categores. More precsely, the Hlert schemes {Hl n (S)} n N naturally form a 2-category Hl(S): (1) The oects of Hl(S) are the natural numers N; (2) The 1-morphsms from m to n are compactly supported oects n DCoh(Hl m (S) Hl n (S)). Composton of 1-morphsms s gven y convoluton. (3) For P 1, P 2 DCoh(Hl m (S) Hl n (S)), the space of 2-morphsms from P 1 to P 2 s the space Ext (P 1, P 2 ). The composton of 2-morphsms s the natural composton of Exts. So to gve a categorcal Hesenerg acton one should defne a 2-category H S and a 2-functor H S Hl(S). The Grothendeck group of H S should e somorphc to the Hesenerg algera so that passng to K-theory one gets a Hesenerg acton on n K 0 (Hl n (S)). In ths paper we construct such a 2- category H S H Γ and a 2-functor H Γ Hl(S) when S Ĉ2 /Γ s the mnmal resoluton of the quotent of C 2 y a non-trval fnte sugroup Γ SL 2 (C) (Theorems 1 and 2). In partcular, we otan a (quantum) Hesenerg algera acton on n K 0 (Hl n (S)) Analogy wth Kac-Moody Le algera actons. There are parallel stores for Hesenerg algeras h Γ and for Kac-Moody Le algeras U(g). In [N2] and [N3] Nakama constructs actons of the envelopng algera U(g) on the cohomology and K-theory of quver varetes, generalzng earler work of Gnzurg for U(sl n ) [CG]. On the other hand, Khovanov-Lauda [KL] defne a 2-category U(g)

3 HEISENBERG CATEGORIFICATION AND HILBERT SCHEMES 3 whose Grothendeck group s somorphc to (an ntegral verson of) the quantzed envelopng algera. There s also ndependent and very smlar work of Rouquer [R] n the same drecton. In [CKL] the authors, together wth Joel Kamntzer, construct a geometrc categorcal g acton on the derved category of coherent sheaves on quver varetes. Ths acton (at least conecturally) nduces an acton of the 2-categores U(g) on the derved categores of quver varetes. The 2-categores U(g) and these actons on the derved categores of quver varetes are analogous to our Hesenerg 2-categores H Γ and ther acton on the derved categores of Hlert schemes The 2-category H Γ. By the McKay correspondence, somorphsm classes of fnte sugroups Γ SL 2 (C) are parametrzed y smply-laced affne Dynkn dagrams (.e. dagrams of type Ân, D n, Ê 6, Ê 7 and Ê8). These dagrams also parametrze smple smply-laced affne Le algeras g, each of whch has a dstngushed homogeneous Hesenerg sualgera h Γ g. The somorphsm class of the Hesenerg algera h Γ does not depend on Γ. However, the choce of Γ determnes a presentaton of h Γ whch we descre n secton 2.2, and a 2-category H Γ, whch we defne n secton 3. The 2-morphsms n H Γ are defned usng a planar graphcal calculus remnscent of the graphcal calculus used y Khovanov-Lauda [KL] n ther categorfcaton of quantum groups. Our frst theorem (1) says that the Grothendeck group K 0 (H Γ ) s somorphc to the Hesenerg algera h Γ n ts Γ-presentaton. Note that the Hesenerg algeras h Γ are somorphc to each other as astract algeras, ut the 2-categores H Γ for dfferent Γ are not equvalent to one another. In ths sense, dfferent actons of the Hesenerg algera are often shadows of essentally dfferent categorcal actons. Ths s n contrast to the case of Kac-Moody Le algeras, where the categorfcatons n [KL] and [R] seem to e more rgd Actons of H Γ. A drect relatonshp etween the fnte group Γ SL 2 (C) and the Hesenerg algera h Γ can e formulated n two closely related ways (one algerac and the other geometrc). In the algerac settng (followng [FJW]) one consders the wreath products Γ n S n of Γ wth the symmetrc group S n. Then one constructs the asc representaton of h Γ on the Grothendeck groups of the module categores C[Γ n S n ] mod. Geometrcally one consders the Hlert schemes Hl n (Ĉ2 /Γ) and constructs the asc representaton of h Γ on ther cohomology. We wll work n a settng that shares features wth oth the algerac and geometrc constructons aove y consderng drectly the ounded derved categores D(A Γ n gmod) of fntely generated graded A Γ n modules. Here A Γ n s the algera A Γ n : [(Sym (V ) Γ) (Sym (V ) Γ) (Sym (V ) Γ)] S n whch nherts the natural gradng from Sym (V ) (here V C 2 ). We recall why D(A Γ n gmod) s equvalent to derved categores of Hlert schemes n secton 6.4 and explan the relatonshp to C[Γ n S n ] mod n secton 8.6. Our second man theorem (Theorem 2) constructs a natural acton of the Hesenerg category H Γ on n D(A Γ n gmod). The Grothendeck group K 0 (A Γ n mod) s somorphc to K 0 (C[Γ] n S n mod). On the other hand, K 0 (A Γ n mod) K 0 (Hl n (Ĉ2 /Γ)) s somorphc to the cohomology of Hl n (Ĉ2 /Γ). Thus the category n D(A Γ n mod) gves a common categorfcaton of the spaces used y Nakama, Gronowsk and Frenkel-Jng-Wang Khovanov s Hesenerg categorfcaton. Our 2-category H Γ takes as ts nput data the emedded fnte sugroup Γ SL 2 (C), and the defnton of H Γ can e generalzed to an artrary emedded fnte group Γ SL n (C). The assocated Hesenerg 2-category wll then act naturally on n D(A Γ n mod) where A Γ n s defned as aove except that now V s the standard representaton of SL n (C). In fact, such a constructon already makes sense when Γ s trval and dm(v ) 0. In ths case the defnton of the assocated Hesenerg category s due to Khovanov [K].

4 4 SABIN CAUTIS AND ANTHONY LICATA Unsurprsngly, some of our constructons share many features wth (and take a good deal of nspraton from) hs constructon. In Khovanov s constructon, lke ours, the 1-morphsms are generated y two elements P and Q whch are adont up to shft. The dfference appears n the structure of 2-morphsms. Here the rôles of Γ and V ecome apparent. In partcular, our categores nhert a non-trval Z gradng from the natural gradng on the exteror algera Λ (V ). Ths gradng makes the Grothendeck group of H Γ nto a Z[t, t 1 ] module whch categorfes a quantum deformaton of the Hesenerg algera Organzaton. Secton 2 defnes the (quantum) Hesenerg algera h Γ. Our choce of generators for h Γ dffers somewhat from the choces made elsewhere n the lterature. Secton 3 defnes the 2-category H Γ whch categorfes h Γ. We also defne a map from h Γ to the Grothendeck group of H Γ. Ths map s shown to e an somorphsm n secton 7 (Theorem 1). Sectons 4 and 5 are concerned wth defnng an acton of H Γ on n D(A Γ n mod). In Secton 6 we study a slghtly dfferent 2-category H Γ usng an alternatve smplfed graphcal calculus. H Γ s Morta equvalent to H Γ n the sense that the spaces of 2-morphsms n H Γ and H Γ are Morta equvalent (susequently the 2-representaton theores of H Γ and H Γ are equvalent). We construct a natural functor η : H Γ H Γ whch nduces an somorphsm at the level of Grothendeck groups (so oth H Γ and H Γ categorfy h Γ ). The 2-category H Γ s of ndependent nterest ut another motvaton for ntroducng t s to facltate the graphcal computatons needed to prove Theorem 1. In secton 8 we sketch a second, smpler (non-derved) acton of H Γ, related to the acton of secton 4 y Koszul dualty. Secton 9 contans a dscusson of currently unanswered questons and futher drectons. The appendx: most of the paper assumes the Dynkn dagram assocated to Γ SL 2 (C) s smply-laced, whch smplfes notaton ut fals to cover the case Γ Z 2. In secton A we collect the requred defntons for Γ Z 2 (wth these modfcatons the man theorems of the paper hold for any nontrval Γ). The defnton of the Hesenerg category for the trval sugroup of SL 2 (C) s of nterest n ts own rght, ut s not dscussed drectly n ths paper Acknowledgements. The authors would lke to thank Mkhal Khovanov for several useful conversatons and for gvng us an early verson of hs paper [K]. We would also lke to thank Igor Frenkel for sharng wth us hs unpulshed notes wth Khovanov and Malkn on categorfcaton and vertex operator algeras [FKM]. Ths proect egan when oth authors were at the program Homology Theory of Knots and Lnks at MSRI n We thank MSRI for ts hosptalty and great workng envronment. S.C. was supported y Natonal Scence Foundaton Grant / Quantum Hesenerg algeras 2.1. The McKay correspondence. Fx an algeracally closed feld k of characterstc zero. Let Γ SL 2 (k) denote a fnte sugroup (n our dscusson ths can nclude Γ k the Cartan sugroup of dagonal matrces). For notatonal convenence, the ody of the paper assumes that Γ Z 2. We deal wth ths extra case n appendx A. Denote y V the standard two dmensonal representaton of Γ. Under the McKay correspondence the fnte sugroup Γ corresponds to an affne Dynkn dagram wth vertex set I Γ and edge set E Γ. By defnton each vertex I Γ s ndexed y an rreducle representaton V of Γ and two vertces, I Γ are oned y k edges where k s the numer of tmes V appears as a drect summand of V V. Notce that snce V s self dual ths relaton s symmetrc n and.

5 HEISENBERG CATEGORIFICATION AND HILBERT SCHEMES 5 For nstance, when G Z/nZ then we get the affne Dynkn dagram Ãn 1, whch s an n cycle. When Γ k the assocated dagram s the Dynkn dagram à of the Le algera sl. Defne the quantum Cartan matrx C Γ to e the matrx wth entres, ndexed y, I Γ gven y t + t 1 f, 1 f are connected y an edge 0 f are not connected y an edge Note that at t 1 the matrx C Γ ecomes the extended Cartan matrx of type ADE (ths s the McKay correspondence etween nontrval fnte sugroups Γ SL 2 (C) and smply-laced affne Dynkn dagrams) The quantum Hesenerg algera assocated to Γ. Fx an algeracally closed feld k of characterstc zero. We defne the Hesenerg algera h Γ assocated to Γ to e the untal k[t, t 1 ] algera wth generators p (n), q (n) for I Γ and ntegers n 0 and relatons (1) p (n) p (m) (2) q (n) q (m) (3) q (n) p (m) k 0 p (m) p (n) for all, I Γ, q (m) q (n) for all, I Γ, [k + 1]p (m k) q (n k) for all I Γ, (4) q (n) p (m) p (m k) q (n k) for all I Γ wth, 1 k 0 (5) q (n) p (m) p (m) q (n) for all I Γ wth, 0. Here [k + 1] t k + t k t k 2 + t k denotes the quantum nteger, and n the aove relatons we have set p (0) q (0) 1 and p (k) q (k) 0 when k < 0 (thus the summatons n the aove relatons are all fnte.) Most mathematcal lterature aout the Hesenerg algera uses a slghtly dfferent presentaton than the one aove. Typcally, the (undeformed) Hesenerg algera assocated to Γ s presented as k algera generated y elements a (m), m Z and I Γ, together wth and a central element c; the defnng relatons are [a (m), a (n)] mδ m, n, c, where, s the (, ) entry of the Cartan matrx. A dfferent set of generators of the Hesenerg algera s then otaned as the homogeneous components n z of the halves of vertex operators exp( 1 m a (m)z m ) and exp( 1 m a ( m)z m ). m 1 m 1 Our defnng relatons aove are a t-deformaton of the relatons etween these homogeneous components. Moreover, snce the representatons of h Γ we consder n ths paper are all of level 1, so that c acts y 1, we have also found t convenent to replace c y 1 n the defnton of h Γ. The verson of the Hesenerg algera we use s actually an dempotent modfcaton of h Γ (ths s smlar to the appearance of dempotent versons of U q (g) n the categorfcaton of quantum groups [KL], [R]). In the dempotent modfcaton, the unt 1 s replaced y a collecton of orthogonal dempotents {1 m } m Z, wth 1 k+m p (m) p (m) 1 k δ k+m,n 1 n p (m) 1 k

6 6 SABIN CAUTIS AND ANTHONY LICATA and 1 k m q (m) q (m) 1 k δ k m,n 1 n q (m) 1 k. The remanng defnng relatons n the untal algera h Γ gve, for each k Z, a defnng relaton of the dempotent modfcaton. Namely, take the orgnal relaton and add the dempotent 1 k at the end of the left and rght hand sde, e.g. q (n) p (m) 1 k p (m) q (n) 1 k for all I Γ wth, 0 and all k Z. Note that these relatons do not depend essentally on the partcular dempotent 1 k. As a result, we ause notaton slghtly and also denote oth the untal algera and ts dempotent modfcaton y h Γ. One feature of the dempotent-modfed h Γ s that t can e consdered as a k[t, t 1 ]-lnear category (ust as any k-algera wth a collecton of dempotents can e consdered as a k-lnear category.) The oects of ths category are then the ntegers, whle the space of morphsms from n to m s the k[t, t 1 ] module 1 n h Γ 1 m. Snce h Γ s already a category, ts categorfcaton H Γ defned n secton 3 wll e a 2-category The Fock space. Let h Γ h Γ denote the sualgera generated y the q (n) 1 k for all I Γ, k 0 and n 0. Let trv 0 denote the trval representaton of h Γ, where 1 0 acts e the dentty and 1 k acts y 0 for k < 0. The h Γ module F Γ Ind hγ (trv h 0 ) gven y nducng the trval representaton Γ from h Γ to h Γ s called the Fock space representaton of h Γ. The Fock space F Γ s naturally somorphc to the space of polynomals n the commutng varales } IΓ,m 0. If we grade the Fock space y declarng deg(p (m) ) m then the dempotent 1 l h Γ acts y proectng onto the degree l suspace nsde F Γ (m) k[{p },m ] (n partcular, when l < 0 the dempotent 1 l acts y 0.) We wll construct categorfcatons of ths representaton n sectons 4 and 8. {p (m) 3. The category H Γ 3.1. The algera B Γ. To defne the 2-category H Γ whch categorfes h Γ we frst need to defne the algera B Γ and fx notaton nvolvng dempotents n ths algera. Snce Γ acts on V t also acts on the exteror algera Λ (V ). Let B Γ : Λ (V ) Γ e ther semdrect product, whch contans oth Λ (V ) and k[γ] as sualgeras. Ths sem-drect product s also called the smash product of k[γ] and Λ (V ), and s sometmes denoted Λ (V )#k[γ] n Hopf algera lterature, though we prefer the term sem-drect product and avod the # notaton. Explctly, an element n B Γ s a lnear comnaton of terms (v, γ) where v Λ (V ) and γ Γ. The multplcaton n B Γ s gven y (v, γ) (v, γ ) (v (γ v ), γγ ). The natural Z gradng on Λ (V ) extends to a gradng of B Γ y puttng k[γ] B Γ n degree zero. Ths makes B Γ nto a superalgera. We denote the degree of an element B Γ y. If we fx a ass {v 1, v 2 } of V and let ω : v 1 v 2 2 (V ) then B Γ has a homogeneous ass over k gven y {(1, γ), (v 1, γ), (v 2, γ), (ω, γ)} γ Γ. Defne a k-lnear trace tr : B Γ k y settng tr((ω, γ)) δ γ,1 1 and tr((1, γ)) tr((v 1, γ)) tr((v 2, γ)) 0. The trace tr s supersymmetrc (for any a, B Γ we have tr(a) ( 1) a tr(a)) and non-degenerate. Ths also nduces a trace on k[γ] va tr(γ) : tr((ω, γ)). Ths corresponds to the usual trace on k[γ] dvded y Γ (n ths way tr(1) 1). For a fxed k-ass B of B Γ, let B denote the ass of B Γ dual to B wth respect to the assocated non-degenerate lnear form a, : tr(a). We denote the dual vector of B y B.

7 HEISENBERG CATEGORIFICATION AND HILBERT SCHEMES 7 Remark 3.1. We may thnk of elements of B Γ as k[γ]-module homomorphsms, B Γ Homk[Γ] (k[γ], Λ (V ) k[γ]). The aove somorphsm s gven y the followng composton B Γ HomB Γ(B Γ, B Γ ) Hom B Γ(Ind BΓ k[γ]k[γ], B Γ ) Hom k[γ] (k[γ], Res k[γ] B Γ ) B Γ Hom k[γ] (k[γ], Λ (V ) k[γ]) Idempotents. Let V 1,..., V m denote the dstnct rreducle representatons of Γ. By Maschke s theorem, the group algera k[γ] decomposes as a drect product of matrx algeras k[γ] M n1 (k)... M nm (k) where the dstnct rreducle representatons V 1,..., V m are each realzed as an rreducle representaton of one of the matrx algeras M n (k). Let f 1,..., f m denote the dstnct parwse orthogonal central dempotents of k[γ]; each f s the dentty matrx n the matrx algera M n (k), and 1 f k[γ]. The aove s not, n general, a mnmal decomposton of 1 as a sum of orthogonal dempotents, snce the dempotents f themselves are not mnmal f dm(v ) > 1. For each and s 1,..., n, let e,s denote the matrx unt of M n (k) whose (s, s) entry s equal to 1 and whose other entres are 0. Then the e,s are mnmal orthogonal dempotents n k[γ], wth n f e,s. s1 An mportant role wll e played y the (super)algera B Γ n : (B Γ... B Γ ) S n. The aove tensor product s understood n the super sense, and the acton of S n s y superpermutatons: f s k S n s the smple transposton (k, k + 1), then s k ( 1... k k+1... n ) ( 1) k k k+1 k... n. The degree zero sualgera k[γ n S n ] B Γ n contans all the dempotents of B Γ n. Ths sualgera s somorphc to a drect product of matrx algeras k[γ n S n ] M l1 (k)... M ls (k) wth one matrx algera for each somorphsm class of rreducle k[γ n S n ]-module. Just as the somorphsm classes of rreducle k[s n ]-modules are parametrzed y parttons of n the somorphsm classes of rreducle k[γ n S n ]-modules are parametrzed y partton-valued functons of I Γ (.e. I Γ -tuples of parttons of n) The category H Γ. We defne an addtve, k-lnear, Z-graded 2-category H Γ as follows. The oects of H Γ are ndexed y the ntegers Z. The 1-morphsms are generated y P and Q, where for each n, P denotes a 1-morphsm from n to n + 1 and Q s a 1-morphsm from n + 1 to n. Thus a 1-morphsm of H Γ s a fnte composton (sequence) of P s and Q s. The dentty 1-morphsm of n s denoted 1 (the empty sequence). Remark 3.2. Techncally we should wrte P (n) and (n)q to dentfy the doman and range of P and Q. However, the propertes of P and Q do not depend on n so we usually omt ths extra parameter n order to smplfy notaton. We could have chosen to defne H Γ as a monodal 1-category nstead of as a 2-category, ut we prefer to consder H Γ as a 2-category ecause ths etter parallels the story for Kac-Moody Le algeras n [KL, R].

8 8 SABIN CAUTIS AND ANTHONY LICATA The space of 2-morphsms etween two 1-morphsms s a Z-graded k-algera generated y sutale planar dagrams modulo local relatons. The dagrams consst of orented compact one-manfolds mmersed nto the plane strp R [0, 1] modulo sotopes fxng the oundary. A sngle upward orented strand denotes the dentty 2-morphsm d : P P whle a downward orented strand denotes the dentty 2-morphsm d : Q Q. An upward-orented lne s allowed to carry dots laeled y elements B Γ. For example, s an element of Hom H Γ (P, P ), whle s an element of Hom H Γ (P Q, QP ). Note that the doman of a 2-morphsm s specfed at the ottom of the dagram and the codoman s specfed at the top, and compostons of 2-morphsms are read from ottom to top. The local relatons we mpose are the followng. Frst we have relatons nvolvng the movement of dots along the carrer strand. We allow dots to move freely along strands and through ntersectons:. The U-turn 2-morphsms are adunctons makng P and Q adont up to a gradng shft that wll e defned later n ths secton. Collson of dots s controlled y multplcaton n the algera B Γ : Note that dots compose, so to speak, n the drecton of the arrow. Dots on dstnct strands supercommute when they move past one another: ( 1) In addton to specfyng how dots collde and slde we mpose the followng local relatons n the Γ graphcal calculus:

9 HEISENBERG CATEGORIFICATION AND HILBERT SCHEMES 9 (6) (7) B (8) tr(). 0. In the frst equaton on lne 7 aove, the summaton s taken over a ass B of B Γ ths morphsm s easly seen to e ndependent of the choce of ass. We assgn a Z-gradng on the space of planar dagrams y defnng deg 0 deg deg 1 deg deg 1 and y defnng the degree of a dot laeled y to e the degree of n the graded algera B Γ. When equpped wth these assgnments all of the graphcal relatons are graded. Thus Hom H Γ (, ) s a Z-graded vector space and composton of morphsms s compatle wth the gradng. Because of the relaton dctatng how dots slde through crossngs, there s a natural map from the semdrect product B Γ n : (B Γ B Γ ) S n to the endomorphsm algera Hom H Γ (P n, P n ) whose mage s the sualgera spanned y rad-lke dagrams (.e. dagrams wth no local maxma or local mnma). Notce that, f we denote y T k upward crossngs of the k and (k + 1)st adacent strands and y X k () dots laeled B Γ on the kth strand then ths sualgera s generated y T 1,..., T n 1 and X 1 (),..., X n () for B Γ. The T s and Xs are suect to the followng relatons: T 2 k 1 and T k T l T l T k for k l > 1, T k T k+1 T k T k+1 T k T k+1 for all k 1,..., n 2 T k X k () X k+1 ()T k for all k 1,..., n 1. X k ()X k ( ) X k ( ) and X k ()X l ( ) ( 1) X l ( )X k () f k l.

10 10 SABIN CAUTIS AND ANTHONY LICATA We wll wrte T and X() nstead of T k and X k () when the suscrpt s understood. In fact, t wll follow from the constructon of secton 8 that the map Bn Γ End H Γ (P n ) s nectve, so that the suspace of endomorphsms spanned y rad-lke dagrams s somorphc Bn Γ (see Remark 9.1) The category H Γ. We defne H Γ to the e Karou envelope (also known as the dempotent completon) of H Γ. By defnton, the oects of H Γ are also ndexed y Z whle a 1-morphsm conssts of a par (R, e) where R s a 1-morphsm of H Γ and e : R R s an dempotent endomorphsm e2 e. Morphsms from (R, e) to (R, e ) are morphsms g : R R such that ge g and e g g. The dempotent e defnes the dentty morphsm of (R, e). Snce H Γ s a graded 2-category, the (splt) Grothendeck group K 0 (H Γ ) of H Γ s a k[q, q 1 ]-lnear category where multplcaton y q corresponds to the shft 1 (we wll assume Grothendeck groups are tensored wth the ase feld k). The oects of K 0 (H Γ ) are the same as the oects of H Γ, namely the ntegers. The space of morphsms Hom K0(H Γ)(n, m) s the splt Grothendeck group of the addtve category Hom HΓ (n, m). Composton of 1-morphsms n H Γ gves the Grothendeck group K 0 (H Γ ) the structure of a k-algera Man Theorem #1. The frst man theorem of ths paper s that H Γ categorfes h Γ : Theorem 1. There s a canoncal somorphsm of algeras π : h Γ K0 (H Γ ). Ths theorem wll e proven n Secton 7. In the rest of ths secton we want to sketch how the morphsm π comes aout. To do ths consder varous dempotent 2-morphsms n the category H Γ. Snce M n (k) k[γ] B Γ End H Γ (P ), any dempotent e M n (k) gves rse (for any n Z) to a 1-morphsm (P, e) n Hom HΓ (n, n + 1). Let P (P, f ) H Γ, where the f are the parwse orthogonal central dempotents of C[Γ] descred n secton Lemma 1. There s an somorphsm of 1-morphsms n the category H Γ P m P 1 Proof. The dempotents f defne a 2-morphsm from m P 1 to P n H Γ. Snce 1 m 1 f and f f δ, f, ths 2-morphsm s an somorphsm. The 1-morphsms P can e further decomposed n H Γ whenever the dempotents f are not mnmal. Recall the mnmal dempotents e,1,... e,n from secton Note that for fxed the e,, whch are dstnct dempotents n M n (k), defne somorphc representatons C[Γ]e,1 C[Γ]e,l of C[Γ]. In partcular, the oects (P, e, ) for dstnct are all somorphc n H Γ. Let P denote the 1-morphsm (P, e,1 ) H Γ. Then we have the followng Lemma 2. There s an smorphsm n H Γ n P 1 P n where n s the dmenson (as a k vector space) of the k[γ]-module V. Proof. The proof s ust as n Lemma 1 usng the fact that f n 1 e,.

11 HEISENBERG CATEGORIFICATION AND HILBERT SCHEMES 11 We wll prove later n secton 7 that the oects P H Γ are ndecomposale. The canoncal adunctons n H Γ nvolvng P and Q (the cups and caps) also allow us to decompose Q, snce any dempotent e End H Γ (P ) gves rse to an dempotent e End H Γ (Q). We defne 1- morphsms Q H Γ y Q (Q, e,1 ). It follows mmedately y aduncton that Q s ndecomposale too. A asc consequence of the relatons etween 2-morphsms n H Γ s the followng result proved n secton 6. Proposton 1. For, I Γ, we have Q P { P Q d 1 d 1 f, P Q d k f are connected y k 0 edges where denotes the gradng shft n H Γ. More generally, we defne 1-morphsms P (n) P (n) and Q (n) y (P n, e trv,,1 ) and Q (n) (Q n, e trv,,1 ) where the dempotent e trv,,1 s a mnmal dempotent n M n (k) n S n k[γ n S n ] correspondng to the trval representaton of S n. In secton 6 we prove the followng proposton. Proposton 2. For, I Γ we have somorphsms n H Γ : Q (n) P (m) Q (n) P (m) P (n) P (m) Q (n) Q (m) P (m) P (n) for all, I Γ, Q (m) Q (n) for all, I Γ, k 0 P (m k) Q (n k) H (P k ) for all I Γ, k 0 P (m k) Q (n k) for all I Γ wth, 1, Q (n) P (m) P (m) Q (n) for all I Γ wth, 0 where H (P k ) denotes the graded cohomology of proectve spaces P k shfted so that t les n n degrees k, k + 2,..., k 2, k. Notce that n the thrd equaton aove the graded dmenson of H (P k ) s the quantum nteger [k + 1]. By conventon H (P k ) 0 f k < 0. Also, P (l) Q (l) 0 when l < 0 so the summand correspondng to a k > mn(n, m) s zero (so the drect sum on the rght s fnte). It follows mmedately from Proposton 2 that there s a well-defned morphsm gven y sendng p (n) to the class [P (n) π : h Γ K 0 (H Γ ) ] and sendng Q (m) to the class [Q (m) ]. That the map π s an somorphsm s a rgorous formulaton of the nformal statement H Γ categorfes the Hesenerg algera Remarks on relatons n H Γ. We end ths secton wth some remarks on the relatons from lnes (6) - (8). The second relaton n (8) s natural for degree reasons; n fact mposng ths relaton s equvalent to declarng that the oect P does not have negatve degree endomorphsms. Moreover, f we eleve that the dentty should not have negatve degree endomorphsms then the left relaton n (8) s mmedate unless a has degree 2, that s, unless a s a lnear comnaton of elements of the form (v 1 v 2, γ). Snce (v 1 v 2, γ) (v 1, γ) (γ 1 v 2, 1),

12 12 SABIN CAUTIS AND ANTHONY LICATA from movng γ 1 v 2 around the crcle we get (v 1 v 2, γ) ( γ 1 v 2 v 1, γ) where the mnus sgn s a consequence of passng the dot laeled (v 1, γ) y the dot laeled (γ 1 v 2, 1). Snce γ acts nontrvally on V, t follows that the couterclockwse crcle wth a sold dot laeled y (v 1 v 2, γ) s zero unless γ 1. To understand the relaton mposed when γ 1, we add a cup at the ottom of the frst relaton n (7). Smplfyng, we otan v 1 v 2 It follows that f the closed dagram whch appears aove s a scalar then t must e equal to one. Alternatvely, f we multply the upward strand y v 1 v 2 and close off we get that the counterclockwse crcle wth a sold dot laeled y v 1 v 2 s an dempotent. So, f ths dempotent were not equal to one, then the dentty map would reak up nto a drect sum of multple 1-morphsms n the dempotent completon. Ths would result n extra unwanted 1-morphsms n our Hesenerg category H Γ Further relatons. Although left-twst curls on an upward pontng strand are zero rght-twst curls (whch have degree 2) and are not necessarly zero. As shorthand, we wll draw rght twst curls as hollow dots: : To pass a sold dot past a hollow dot nvolves sldng the sold dot along the rght-twst curl. Thus sold dots transform commute wth hollow dots. Hollow dots have an nterestng affne Hecke type relaton wth crossngs whch nvolves the creaton of laeled sold dots,. + B + B where agan the summatons are over the ass B. These relatons are remnscent of relatons n the degenerate affne Hecke algera assocated to the symmetrc group and t s wreath products, see for nstance [RS, W, WW]. The proof of these relatons s an easy calculaton usng the defnng relatons for 2-morphsms n H Γ.

13 HEISENBERG CATEGORIFICATION AND HILBERT SCHEMES 13 Fnally, t also follows drectly from the graphcal relatons that the trple pont move, whch was defned to hold when all three strands are orented up, n fact holds n all possle orentatons. 4. Acton of H Γ In ths secton we construct an acton of H Γ on the derved categores of fnte dmensonal graded modules over certan algeras A Γ n. Ths nduces an acton on derved categores of coherent sheaves on Hlert schemes and allows us to show n the next secton that our categores H Γ are non-degenerate (susequently provng Theorem 1) Categores. Snce Γ SL 2 (k) acts on V k 2 t acts naturally on Sym (V ) and we denote y Sym (V ) Γ ther sem-drect product (also known as the smash product Sym (V )#k[γ]). For nstance, f Γ Z/nZ then we can dentfy Sym (V ) wth k[x, y] and the acton s ζ (x, y) (ζx, ζ 1 y). Sym (V ) Γ s a k-algera, and an element of Sym (V ) Γ s a lnear comnaton of elements of the form (f, γ) where f Sym (V ) and γ Γ wth multplcaton gven y For each non-negatve nteger n defne (f, γ) (f, γ ) (fγ f, γγ ). A Γ n : [(Sym (V ) Γ) (Sym (V ) Γ) (Sym (V ) Γ)] S n where S n s the symmetrc group actng y permutng the n terms n the product. Notce that here we use the ordnary tensor product and not the super tensor product used to defne Bn. Γ These algeras nhert the natural gradng from Sym (V ). We denote y D(A Γ n gmod) the ounded derved category of fnte dmensonal, graded (left) A Γ n-modules. We have maps k[γ] A Γ 1 p k[γ] where the frst denotes the natural ncluson of the group algera k[γ] and the second s the proecton whch takes Sym >0 (V ) to zero. Thus any k[γ]-module s also an A Γ 1 -module and vce versa Functors P and Q. Consder the natural ncluson A Γ n A Γ 1 A Γ n+1 makng use of the emeddng S n S n S 1 S n+1. Ths gves A Γ n+1 and A Γ n k[γ] the structure of a module over A Γ n A Γ 1. Defne the (A Γ n+1, A Γ n) module and the (A Γ n, A Γ n+1) module P Γ (n) : A Γ n+1 A Γ n A Γ (AΓ 1 n k[γ]) (n)q Γ : (A Γ n k[γ]) A Γ n A Γ 1 Defne the functor P(n) : D(A Γ n gmod) D(A Γ n+1 gmod) y P(n)( ) : P Γ (n) A Γ n ( ). Smlarly, we defne (n)q : D(A Γ n+1 gmod) D(A Γ n gmod) y AΓ n+1. (n)q( ) : (n)q Γ A Γ n+1 ( )[ 1]{1}

14 14 SABIN CAUTIS AND ANTHONY LICATA where [ ] denotes the cohomologcal shft whle { } the gradng shft. Note that the relaton etween these shfts and the shft n H Γ s that 1 [1]{ 1}. We wll usually omt the (n) and ust wrte P or Q Natural transformatons. In order to defne an H Γ acton on n D(A Γ n gmod) we also need to defne the followng natural transformatons: (1) X(v) : P P[1]{ 1} and X(v) : Q Q[1]{ 1} for any v V (2) X(γ) : P P and X(γ) : Q Q for any γ Γ (3) T : PP PP, T : QQ QQ, T : QP PQ and T : PQ QP (4) ad : QP d[ 1]{1} and ad : PQ d[1]{ 1} (5) ad : d QP[ 1]{1} and ad : d PQ[1]{ 1}. These natural transformatons defne the acton of the followng 2-morphsms n H Γ (as usual we read the dagrams from the ottom up): (1) v and v (2) γ and γ (3),, and (4) and (5) and Prelmnares. Snce we wll deal wth tensor products of complexes we refly revew some standard conventons. Gven two complexes A A d A+1... and B B d B+1... the tensor product A B s the complex wth terms A B and dfferental A B d 1+( 1) 1 d A +1 B A B +1. Gven a map f : A A [k] of complexes nduced y f : A A +k we get a map f 1 : A B A B [k] nduced y (f 1) f 1 : A B A +k B. Smlarly, gven a map g : B B [k] nduced y g : B B +k we get a map 1 g nduced y (1 g) ( 1) k g 1 : A B A B +k. Notce that f k 1 then (f 1) and (1 g) ant-commute Defnton of X End(P ). We defne a map X(γ) : k[γ] k[γ] of left A Γ 1 -modules va multplcaton on the rght γ γ γ. Ths nduces a map X(γ) : P Γ (n) P Γ (n) of (A Γ n+1, A Γ n)-modules. To defne X(v) we have to work harder. Consder A Γ 1 V where the tensor product s over k (so A Γ 1 only acts on the left factor y multplcaton on the left). Then the multplcaton map Sym (V ) V Sym (V ) nduces a map d : A Γ 1 V A Γ 1 of left A Γ 1 -modules va (f, γ) w (fγ w, γ) where f Sym (V ), γ Γ and w V. Smlarly, we have a map d : A Γ 1 2 V A Γ 1 V gven y (f, γ) (w 1 w 2 ) (f(γw 1 ), γ) w 2 (f(γw 2 ), γ) w 1.

15 HEISENBERG CATEGORIFICATION AND HILBERT SCHEMES 15 Usng these maps we can defne the followng free resoluton of the left A Γ 1 -module k[γ]: (9) 0 A Γ 1 2 V d A Γ 1 V d A Γ 1 k[γ]. Suppose v V. We need to defne a map X(v) : k[γ] k[γ][1]{ 1} nsde D(A Γ 1 gmod). To do ths defne φ v : A Γ 1 V A Γ 1 { 1} to e the map nduced y (f, γ) w w, v (f, γ), where, : V V k s the natural parng. It s easy to check ths map s a map of graded left A Γ 1 -modules. Smlarly, defne to e the map nduced y φ v : A Γ 1 2 V A Γ 1 V { 1} (f, γ) (w 1 w 2 ) w 1, v (f, γ) w 2 w 2, v (f, γ) w 1. Usng these maps we can wrte down the followng commutatve dagram (10) 0 A Γ 1 2 V d A Γ 1 V d A Γ 1 A Γ 1 2 V { 1} d A Γ 1 V { 1} d A Γ 1 { 1} 0 φ v The commutatvty of the mddle square s an easy exercse (note that the dfferental s d n the second row snce shftng the complex y one negates the dfferental). The map from (10) nduces a map X(v) : k[γ] k[γ][1]{ 1} and susequently a map X(v) : P Γ (n) P Γ (n)[1]{ 1} (n the derved category) of graded (A Γ n+1, A Γ n)-modules Defnton of X End(Q). If γ Γ then multplcaton on the left nduces a map X(γ) : k[γ] k[γ] of rght A Γ 1 -modules va γ γ γ and susequently a map X(γ) : (n)q Γ (n)q Γ of (A Γ n, A Γ n+1)-modules. The map X(v) : (n)q Γ (n)q Γ [1]{ 1} s defned y usng a resoluton of Q Γ as aove. We consder V A Γ 1 where A Γ 1 only acts on the second factor from the rght. We have the maps of rght A Γ 1 -modules va d : V A Γ 1 A Γ 1 and d : 2 V A Γ 1 V A Γ 1 w (f, γ) (wf, γ) and (w 1 w 2 ) (f, γ) w 1 (w 2 f, γ) w 2 (w 1 f, γ). Usng these maps we have a free resoluton of the rght A Γ 1 -module k[γ] as n (9) (11) 0 2 V A Γ 1 d V A Γ 1 φ v d A Γ 1 k[γ]. Then we can wrte a map X(v) : k[γ] k[γ][1]{ 1} of graded rght A Γ 1 -modules as n equaton (10) except that we use the maps gven y ψ v : V A Γ 1 A Γ 1 { 1} and ψ v : 2 V A Γ 1 V A Γ 1 { 1} w (f, γ) v, w (f, γ) and (w 1 w 2 ) (f, γ) v, w 1 w 2 (f, γ) v, w 2 w 1 (f, γ).

16 16 SABIN CAUTIS AND ANTHONY LICATA Usng these maps we can agan wrte down the followng commutatve dagram (12) 0 2 V A Γ 1 d V A Γ 1 d A Γ 1 ψ v 2 V A Γ 1 { 1} d V A Γ 1 { 1} d A Γ 1 { 1} 0. Ths nduces a map X(v) : k[γ] k[γ][1]{ 1} and susequently a map X(v) : (n)q Γ (n)q Γ [1]{ 1} (n the derved category) of graded (A Γ n, A Γ n+1)-modules Defnton of T End(P 2 ). To defne T we need a map Now P Γ (n + 1) A Γ n+1 P Γ (n) P Γ (n + 1) A Γ n+1 P Γ (n). P Γ (n + 1) A Γ n+1 P Γ (n) A Γ n+2 A Γ n+1 A Γ 1 (AΓ n+1 k[γ]) A Γ n A Γ 1 (AΓ n k[γ]) A Γ n+2 A Γ n A Γ 1 AΓ (AΓ 1 n k[γ] k[γ]) ) A Γ n+2 A Γ n A Γ AΓ 2 n (A Γ 2 A Γ1 A Γ1 k[γ] k[γ]. Now defne a map T : A Γ 2 A Γ 1 A Γ (k[γ] k[γ]) AΓ 1 2 A Γ 1 A Γ (k[γ] k[γ]) y 1 (13) a (γ γ ) as 1 (γ γ) where a A Γ 2 and s 1 S 2 s the transposton (1, 2). Let us check that ths nduces a well defned map. Suppose a (a, a ) where a, a A Γ 1. Then (14) T (a (γ γ )) T (1 (a γ a γ )) s 1 (a γ a γ). On the other hand (a, a )s 1 (γ γ) s 1 (a, a ) (γ γ) s 1 (a γ, a γ) whch agrees wth (14). Ths shows that T s well defned. Notce that the nduced map s gven y T : A Γ n+2 A Γ n A Γ 1 AΓ 1 (AΓ n k[γ] k[γ]) A Γ n+2 A Γ n A Γ 1 AΓ 1 (AΓ n k[γ] k[γ]) 1 (1 γ γ ) s n+1 (1 γ γ) where s n+1 (n + 1, n + 2). The map T : QQ QQ s defned n exactly the same way. To defne T : QP PQ we need a map (A Γ n k[γ]) A Γ n A Γ 1 AΓ n+1 A Γ n A Γ 1 (AΓ n k[γ]) A Γ n A Γ n 1 A Γ 1 (AΓ n 1 k[γ] k[γ]) A Γ n 1 A Γ 1 AΓ n of (A Γ n, A Γ n)-modules. To defne ths map t suffces to say where to take (1 γ) a (1 γ ) where a A Γ n+1 equals 1 or s n (n, n + 1). If a 1 then we map t to zero whle Fnally, to defne T : PQ QP we need a map (1 γ) s n (1 γ ) 1 (1 γ γ) 1. A Γ n A Γ n 1 A Γ 1 (AΓ n 1 k[γ] k[γ]) A Γ n 1 A Γ 1 AΓ n (A Γ n k[γ]) A Γ n A Γ 1 AΓ n+1 A Γ n A Γ 1 (AΓ n k[γ]) of (A Γ n, A Γ n)-modules. Ths map s unquely defned y 1 (1 1 1) 1 (1 1) s n (1 1). ψ v

17 HEISENBERG CATEGORIFICATION AND HILBERT SCHEMES Defnton of ad : QP d[ 1]{1}. We need a map of (A Γ n, A Γ n)-modules Now ad : (n)q Γ A Γ n+1 P Γ (n) A Γ n. (n)q Γ A Γ n+1 P Γ (n) (A Γ n k[γ]) A Γ n A Γ 1 AΓ n+1 A Γ n A Γ 1 (AΓ n k[γ]). Notce that k[s n+1 ] as a k[s n ]-module s free and generated y 1 and s n (n, n + 1). So the map k[s n+1 ] k[s n ] of k[s n ]-modules gven y 1 1 and s n 0 nduces a map A Γ n+1 A Γ n A Γ 1 of A Γ n A Γ 1 -modules. Susequently we otan a map (15) (16) (n)q Γ A Γ n+1 P Γ (n) (A Γ n k[γ]) A Γ n A Γ 1 (AΓ n A Γ 1 ) A Γ n A Γ 1 (AΓ n k[γ]) (A Γ n k[γ]) A Γ n A Γ 1 (AΓ n k[γ]). Usng the composton k[γ] A Γ 1 k[γ] k[γ] tr k where the frst map s multplcaton and the second s the trace map (normalzed so that tr(1) 1) we get a morphsm (A Γ n k[γ]) A Γ n A Γ 1 (AΓ n k[γ]) A Γ n A Γ n A Γ n A Γ n. Composng wth (15) defnes ad : (n)q Γ A Γ n+1 P Γ (n) A Γ n Defnton of ad : PQ d[1]{ 1}. We need a map of (A Γ n+1, A Γ n+1)-modules Now P Γ (n) A Γ n (n)q Γ A Γ n+1[2]{ 2}. P Γ (n) A Γ n (n)q Γ A Γ n+1 A Γ n A Γ 1 (AΓ n k[γ] k[γ]) A Γ n A Γ 1 AΓ n+1 so we ascally need a map h : k[γ] k[γ] A Γ 1 [2]{ 2} of graded (A Γ 1, A Γ 1 )-modules. Then we defne ad as the composton ad : A Γ n+1 A Γ n A Γ 1 (AΓ n k[γ] k[γ]) A Γ n A Γ 1 AΓ n+1 h A Γ n+1 A Γ n A Γ AΓ 1 n+1[2]{ 2} A Γ n+1[2]{ 2} where the second map s multplcaton. To defne h we use the resolutons (9) and (11) of k[γ]. Tensorng the two resolutons we see that h s defned y a map (A Γ 1 2 V ) A Γ 1 (A Γ 1 V ) (V A Γ 1 ) A Γ 1 ( 2 V A Γ 1 ) A Γ 1 { 2}. We defne the map from each summand as (1) ((f, γ) (w w )) (f, γ ) (f, γ)(f, γ )ω(w w ) (2) ((f, γ) w) (w (f, γ )) (f, γ)(f, γ )ω(w w ) (3) (f, γ) ((w w ) (f, γ )) (f, γ)(f, γ )ω(w w ) where ω : 2 V k s our fxed somorphsm. In order for ths map to e well defned (.e. a map of complexes) we need the composton (A Γ 1 2 V ) A Γ 1 (A Γ 1 V ) (V A Γ 1 ) A Γ 1 ( 2 V A Γ 1 ) A Γ 1 { 2} to e zero. (A Γ 1 2 V ) (V A Γ 1 ) (A Γ 1 V ) ( 2 V A Γ 1 )

18 18 SABIN CAUTIS AND ANTHONY LICATA Let us check that the composton (A Γ 1 2 V ) (V A Γ 1 ) A Γ 1 { 2} s zero. The frst map s (f, γ) (w w ) (w (f, γ )) ((f, γ) (w w )) (f w, γ ) + Composng wth the second map we get ((fγ w, γ) w ) (w (f, γ )) ((fγ w, γ) w) (w (f, γ )). (f(γf )(γw ), γγ )ω(w w ) + (f(γf )(γw), γγ )ω(w w ) (f(γf )(γw ), γγ )ω(w w ). Snce the maps are lnear n the w s t suffces to check ths s zero when w w, when w w and when w w. In all cases one of the three terms s mmedately zero and the other two cancel out. Smlarly, the other composton maps ((f, γ) w) ((w w ) (f, γ )) to (f(γw)(γf ), γγ )ω(w w ) (f(γw )(γf ), γγ )ω(w w ) + (f(γw )(γf ), γγ )ω(w w ) whch also equals zero Defnton of ad : d QP[ 1]{1}. We need a map of (A Γ n, A Γ n)-modules whch translates to a map A Γ n (n)q Γ A Γ n+1 P Γ (n)[ 2]{2} (17) A Γ n[2]{ 2} (A Γ n k[γ]) A Γ n A Γ 1 AΓ n+1 A Γ 1 A Γ n (k[γ] AΓ n). We frst replace each k[γ] wth ts resoluton usng (9) and (11). Then to defne the map n (17) we ust need to defne a map A Γ n{ 2} (A Γ n ( k V A Γ 1 )) A Γ n A Γ AΓ 1 n+1 A Γ 1 A Γ ((AΓ n 1 l V ) A Γ n) k+l2 of graded (A Γ n, A Γ n)-modules. We send 1 (1 1) 1 ((1 w 1 w 2 ) 1) (1 (w 1 1)) 1 ((1 w 2 ) 1) + (1 (w 2 1)) 1 ((1 w 1 ) 1) (1 (w 1 w 2 1) 1 (1 1) where w 1, w 2 s our chosen ass of V. Ths unquely determnes the map. To see ths s well defned (.e. a map of complexes) we need to check that the composton k+l1 (AΓ n ( k V A Γ 1 )) A Γ n A Γ AΓ 1 n+1 A Γ 1 A Γ ((AΓ n 1 l V ) A Γ n) A Γ n{ 2} k+l2 (AΓ n ( k V A Γ 1 )) A Γ n A Γ 1 AΓ n+1 A Γ 1 A Γ n ((AΓ 1 l V ) A Γ n) s zero. The check of ths fact s a straght-forward exercse Defnton of ad : d PQ[1]{ 1}. We need a map of graded (A Γ n+1, A Γ n+1)-modules A Γ n+1 P Γ (n) A Γ n (n)q Γ or equvalently a map We send (18) 1 A Γ n+1 A Γ n+1 A Γ n A Γ 1 (AΓ n k[γ] k[γ]) A Γ n A Γ 1 AΓ n+1. n s... s n (1 γ γ 1 ) s n... s 0 γ Γ where s (, + 1) S n+1. Here s... s n 1 f 0 y conventon.

19 HEISENBERG CATEGORIFICATION AND HILBERT SCHEMES 19 To check ths module map s well defned we need to show that n n as... s n (1 γ γ 1 ) s n... s s... s n (1 γ γ 1 ) s n... s a 0 γ Γ 0 γ Γ for any a A Γ n+1. If a (A Γ 1 ) n+1 A Γ n+1 then n as... s n (1 γ γ 1 ) s n... s 0 γ Γ n s... s n (1 γ γ 1 ) s n... s 0 γ Γ n s... s n (1 γ γ 1 ) s n... s 0 γ Γ n s... s n (1 γ γ 1 ) s n... s a 0 γ Γ where (s n... s ) a. The second equalty follows snce γ Γ γ γ 1 les n the centre of k[γ] k k[γ]. Snce A Γ n+1 s generated y k[s n+1 ] as an ((A Γ 1 ) n+1, (A Γ 1 ) n+1 )-module t remans to show that n s k s... s n (1 γ γ 1 ) s n... s 0 γ Γ n s... s n (1 γ γ 1 ) s n... s s k 0 γ Γ for k 1,..., n. The left hand sde s the sum over γ Γ of k 1 s... s n s k 1 (1 γ γ 1 ) s n... s + s k s k+1... s n (1 γ γ 1 ) s n... s k s k+1... s n (1 γ γ 1 ) s n... s k + n k+2 s... s n s k (1 γ γ 1 ) s n... s ecause s k s... s n s... s n s k 1 f k 1 and s k s... s n s... s n s k f k + 2. A smlar calculaton of the rght sde yelds the same expresson so ad s well defned Man Theorem #2. The second man theorem of ths paper s the followng: Theorem 2. The natural transformatons X, T and ad satsfy the Hesenerg 2-relatons and gve a categorcal Hesenerg acton of H Γ on n 0 D(A Γ n gmod). We shall check all the Hesenerg relatons requred for the proof of Theorem 2 n the next secton. Remark 4.1. In type A (when Γ s a cyclc group) one can defne everythng k -equvarantly where the k SL(V ) s the torus whch commutes wth Γ. More precsely, the acton of k on V nduces an acton on Sym (V ) and nstead of A Γ n one uses Â Γ n : [(Sym (V ) (Γ k )) (Sym (V ) (Γ k )) (Sym (V ) (Γ k ))] S n. It s easy to check that all the 1-morphsms and 2-morphsms defned aove are compatle wth ths extra k -structure. Ths means Theorem 2 stll holds so we get an acton of H Γ on n 0 D(ÂΓ n gmod) (see also [FJW2]).

20 20 SABIN CAUTIS AND ANTHONY LICATA 5. Proof of Theorem Composton of X s. We can defne X() : P P[ ]{ } for any homogeneous B Γ y composng a sequence of X(γ) and X(v) where γ Γ and v V. In order for ths to e well defned we need to check that (1) X(γ)X(γ ) X(γγ ) for any γ, γ Γ (2) X(γ)X(v) X(γ v)x(γ) for any γ Γ and v V (3) X(v )X(v) X(v)X(v ). (1) The frst asserton s clear snce Γ acts on P Γ (n) A n+1 An A 1 (A n k[γ]) y rght multplcaton on k[γ]. (2) To see the second asserton we replace k[γ] y ts free resoluton (9). In the free resoluton Γ acts on the rght only on the frst factor of A Γ 1 V. Wrtng out the composton we see that two pars of compostons should agree: A Γ 1 V γ A Γ 1 V φv A Γ 1 { 1} and A Γ 1 V φγv A Γ 1 { 1} γ A Γ 1 { 1} A Γ 1 2 V γ A Γ 1 2 V φ v A Γ 1 V { 1} and A Γ 1 2 V φ γv A Γ 1 V { 1} γ A Γ 1 V { 1}. In the frst par the frst composton s equal to whle the second composton s equal to (f, γ ) w (f, γ γ) (γ 1 w) γ 1 w, v (f, γ γ) (f, γ ) w w, γv (f, γ ) w, γv (f, γ γ) and these are the same snce, s nvarant under the acton of Γ. Smlarly, n the second par the frst composton s equal to (f, γ ) (w 1 w 2 ) (f, γ γ) (γ 1 w 1 γ 1 w 2 ) whle the second composton s γ 1 w 2, v (f, γ γ) (γ 1 w 1 ) + γ 1 w 1, v (f, γ γ) (γ 1 w 2 ) (f, γ ) (w 1 w 2 ) w 2, γv (f, γ ) w 1 + w 1, γv (f, γ ) w 2 w 2, γv (f, γ γ) (γ 1 w 1 ) + w 1, γv (f, γ γ) (γ 1 w 2 ) whch s clearly the same as the frst composton. (3) To see the thrd asserton we agan replace k[γ] y ts free resoluton (9) and then we need to show that the compostons A Γ 1 2 V φ v A Γ 1 V { 1} φv A Γ 1 { 2} and A Γ 1 2 V φ v A Γ 1 V { 1} φ v A Γ 1 { 2} dffer precsely y a mnus sgn. The frst composton s (f, γ) (w 1 w 2 ) w 2, v (f, γ) w 1 + w 1, v (f, γ) w 2 w 1, v w 2, v (f, γ) + w 2, v w 1, v (f, γ) and smlarly the second composton s (f, γ) (w 1 w 2 ) w 1, v w 2, v (f, γ) + w 2, v w 1, v (f, γ). These two compostons clearly dffer y multplcaton y 1. Ths shows that B Γ Λ (V ) Γ acts on P and a smlar proof shows that t also acts on Q.

21 HEISENBERG CATEGORIFICATION AND HILBERT SCHEMES X s on dfferent strands. We also need to check that elements of B Γ on dfferent strands supercommute, namely (IX())(X( )I) ( 1) (X( )I)(IX()) for any homogeneous, B Γ. Ths follows from the dscusson at the egnnng of ths secton. X() s a map of degree of complexes and that such maps supercommute. In other words, we are usng that whenever f : A A + f and g : A A + g are maps of complexes then (f 1)(1 g) ( 1) f g (1 g)(f 1) as maps of complexes A B A B [ f + g ] Adont relatons. We now check that the followng compostons nvolvng adunctons are all equal to the dentty: (1) P Iad PQP[ 1]{1} adi P and Q adi QPQ[ 1]{1} Iad Q (2) P adi PQP[1]{ 1} Iad P and Q Iad QPQ[1]{ 1} adi Q. Ths wll prove the graphcal relatons n oth possle orentatons The composton P Iad PQP[ 1]{1} adi P. To show ths s the dentty t suffces to check that the composton C D {2} C s the dentty for 0, 1, 2 where C : A Γ n+1 A Γ n A Γ (AΓ 1 n (A Γ 1 V )) D : A Γ n+1 A Γ n A Γ (AΓ 1 n (A Γ 1 V )) A Γ n (A Γ n ( k V A Γ 1 )) A Γ n A Γ AΓ 1 n+1 A Γ 1 A Γ ((AΓ n 1 l V ) A Γ n). k+l2 We wll prove the case n 0 whch s the same as the general case ut smplfes the notaton. In ths case the composton s ( ) (A Γ 1 V ) f (A Γ 1 V ) A Γ 0 ( k V A Γ 1 ) A Γ 1 (A Γ 1 l V ) {2} g (A Γ 1 V ). k+l2 The mage of any element under f conssts of three terms. The only term that wll e non-zero after applyng g wll e the term If 0 and a A Γ 1 then (A Γ 1 V ) A Γ 0 ( 2 V A Γ 1 ) A Γ 1 (A Γ 1 V ). a a ((w 1 w 2 1) 1) (a 1) ω(w 1 w 2 ) a whch proves that the composton s the dentty when 0. If 1 then (f, γ) w ((f, γ) w) ( (w 1 1) (1 w 2 ) + (w 2 1) (1 w 1 )) ((f, γ) w 2 )ω(w w 1 ) + ((f, γ) w 1 )ω(w w 2 )

22 22 SABIN CAUTIS AND ANTHONY LICATA where a (f, γ). Snce all maps are lnear n the w s t suffces to check the two cases w w 1 and w w 2. In the frst case the frst term s zero and we get whle n the second case If 2 and a A Γ 1 then (f, γ) w 1 ((f, γ) w 1 )ω(w 1 w 2 ) (f, γ) w 1 (f, γ) w 2 ((f, γ) w 2 ) ω(w 2 w 1 ) (f, γ) w 2. a (w w ) (a (w w )) 1 (1 (w 1 w 2 )) a (w 1 w 2 )ω(w w ) and t s easy to check ths equals a (w w ) so that the composton s the dentty. Ths concludes the proof of the frst relaton n (1). The fact that the composton Q adi QPQ[ 1]{1} Iad Q s also the dentty follows smlarly The composton P adi PQP[1]{ 1} Iad P. Ths s the same as the composton A Γ n+1 A Γ n A Γ 1 (AΓ n k[γ]) A Γ n+1 A Γ n A Γ 1 (AΓ n k[γ] k[γ]) A Γ n A Γ 1 AΓ n+1 A Γ n A Γ 1 (AΓ n k[γ]) A Γ n+1 A Γ n A Γ (AΓ 1 n k[γ]). Usng (18) the frst map s gven y n 1 (1 1) s... s n (1 γ γ 1 ) s n... s (1 1). 0 γ Γ Composng wth the second map all terms are zero unless 0 (recall that s... s n 1 f 0) n whch case we get the composton 1 (1 1) γ Γ 1 (1 γ γ 1 ) 1 (1 1) γ Γ 1 A Γ n A Γ 1 (1 γ) A Γ n (tr(γ 1 )) 1 A Γ n A Γ 1 (1 1) A Γ n tr(1) 1 (1 1) where two get the last two equaltes we use that tr(γ 1 ) 0 unless γ 1 n whch case tr(1) 1. Ths proves that the composton P adi PQP[1] Iad P s the dentty. The fact that Q Iad QPQ[1] adi Q s the dentty follows smlarly Dots and adunctons. Next we study how natural transformatons of P and Q nteract wth the adunctons. We wll show that the followng pars of maps are equal for any B Γ (for smplcty we omt shfts n ths secton) (1) QP IX() QP ad d and QP X()I QP ad d (2) PQ X()I PQ ad d and PQ IX() PQ ad d (3) d ad PQ X()I PQ and d ad PQ IX() PQ (4) d ad QP IX() QP and d ad QP X()I QP.

23 HEISENBERG CATEGORIFICATION AND HILBERT SCHEMES 23 Equaltes (1) and (2) are equvalent to the followng defnng relatons for 2-morphsms n H Γ : Notce that relatons (3) and (4) aove follow formally from (1) and (2) va the adontness propertes. Susequently we only check (1) and (2). Also, snce any s the composton of v s and γ s t suffces to consder the cases when γ and v Relaton (1) when γ. Recall that (19) (n)q Γ A Γ n+1 P Γ (n) (A Γ n k[γ]) A Γ n A Γ 1 AΓ n+1 A Γ n A Γ 1 (AΓ n k[γ]). Now the map QP IX(γ) QP n (1) s (1 γ 1 ) a (1 γ 2 ) (1 γ 1 ) a (1 γ 2 γ) where we can assume a 1 or a (n, n + 1). If a 1 then QP ad d takes ths to tr(γ 1 γ 2 γ)1 A Γ n. If a (n, n + 1) then ths s mapped to zero. On the other hand, QP X(γ)I QP n (1) s (1 γ 1 ) a (1 γ 2 ) (1 γγ 1 ) a (1 γ 2 ) whch s then mapped to tr(γγ 1 γ 2 )1 A Γ n f a 1 and to zero f a (n, n+1). So the two compostons agree. One can prove (2) when γ smlarly Relaton (1) when v. To wrte down the composton QP IX(v) QP ad d we resolve oth copes of k[γ] n (19). We get a map of complexes whch s determned y the composton C 1 Iφ v 0 ad C 2 C 3 A Γ n where C 1 : k+l1 ((A Γ n ( k V A Γ 1 )) A Γ n A Γ 1 AΓ n+1 A Γ n A Γ 1 ((AΓ n (A Γ 1 l V )) C 2 : (A Γ n A Γ 1 ) A Γ n A Γ 1 AΓ n+1 A Γ n A Γ 1 (AΓ n A Γ 1 ) C 3 : (A Γ n k[γ]) A Γ n A Γ 1 AΓ n+1 A Γ n A Γ 1 (AΓ n k[γ]). The second map s nduced y the natural proecton A Γ 1 k[γ]. If k 1 the frst map s zero and f k 0 t s gven y (1 a) a (1 (a w)) (1 a) a (1 a w, v ) C 2 where a A Γ n+1 s ether 1 or the transposton (n, n+1). The only case when ths term s not mapped to zero s when a 1 and oth a, a le n degree zero. So let a γ and a γ, n whch case (20) ad (Iφ v )((1 γ) 1 (1 (γ w)) 1 w, v tr(γγ ) A Γ n. Smlarly, QP X(v)I QP ad 0 ψ vi d s determned y the composton C 1 C 2 C 3 Ths s zero f k 0. If k 1, the composton s also zero except on terms of the form (21) (1 (w γ)) 1 (1 γ ) (1 v, w γ) 1 (1 γ ) 1 v, w tr(γγ ) A Γ n. ad A Γ n.