A categorification of quantum sl n

Transcription

1 A categorfcaton of quantum sl n Aaron Lauda Jont wth Mkhal Khovanov Columba Unversty January 20th, 2009 Avalable at lauda/talks/kyoto Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

2 Rngs R(ν) Gven a graph Γ = consder brad-lke dagrams wth dots whose strands are labelled by the vertces I of the graph Γ. Let ν = I ν, for ν = 0, 1, 2,... ν keeps track of how many strands of each color occur n a dagram = = 0 The rng R(ν) as the set of planar dagrams colored by ν, modulo planar brad-lke sotopes and the followng local relatons: Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

3 Local relatons I = 0 = = = Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

4 Local relatons II = f k k k = + f = = f k k k k k Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

5 Local relatons III = f = otherwse, k k some of,, k may be equal Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

6 Gradng q gradng shft deg = 2 deg = 2 f = 0 f 1 f The R(ν) relatons are homogeneous wth respect to ths gradng. Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

7 Grothendeck groups R = ν Æ[I] R(ν) K 0 (R) := ν Æ[I] K 0 (R(ν)) where K 0 (R(ν)) s the Grothendeck group of the category R(ν) pmod of graded proectve fntely-generated R(ν)-modules. K 0 (R(ν)) has generators [M] over all obects of R(ν) pmod and defnng relatons [M] = [M 1 ] + [M 2 ] f M = M 1 M 2 [M{s}] = q s [M] s K 0 (R(ν)) s a [q, q 1 ]-module. Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

8 There are nducton and restrcton functors correspondng to nclusons R(ν) R(ν ) R(ν + ν ) Ind ν+ν ν,ν : R(ν) R(ν ) pmod R(ν + ν ) pmod Res ν+ν ν,ν : R(ν + ν ) pmod R(ν) R(ν ) pmod Summng over all ν,ν gves functors Ind: (R R) pmod R pmod Res: R pmod (R R) pmod These map proectves to proectves [Ind]: K 0 (R) K 0 (R) K 0 (R) [Res]: K 0 (R) K 0 (R) K 0 (R) Wrte [Ind](x 1, x 2 ) for x 1, x 2 K 0 (R) as x 1 x 2 Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

9 Proposton [Ind] turns K 0 (R) nto an assocatve untal [q, q 1 ]-algebra. [Res] turns K 0 (R) nto a coassocatve countal [q, q 1 ]-coalgebra. Proof follows from transtvty of nducton and restrcton. Example R(ν) R(ν ) R(ν ) R(ν + ν ) R(ν ) R(ν + ν + ν ) Res ν+ν +ν ν+ν,ν Res ν+ν,ν ν,ν,ν (M) = Res ν+ν +ν ν,ν,ν (M) R(ν) R(ν ) R(ν ) R(ν) R(ν + ν ) R(ν + ν + ν ) Res ν+ν +ν ν,ν +ν Res ν,ν +ν ν,ν,ν (M) = Res ν+ν +ν ν,ν,ν (M) Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

10 Defne a symmetrc blnear form on [I] by = 2 for all I = 1 f = 0 f For ν Æ[I] and x K 0 (R(ν)) wrte x = ν Æ[I] Twsted balgebra structure Equp K 0 (R) K 0 (R) wth the algebra structure (x 1 x 2 )(x 1 x 2 ) = q x 2 x 1 x 1 x 1 x 2x 2 for homogeneous x 1, x 2, x 1, x 2. Then [Res] s an algebra homomorphsm from K 0 (R) to K 0 (R) K 0 (R). We say that K 0 (R) s a twsted balgebra. Proof uses a verson of Mackey s nducton-restrcton theorem. Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

11 Blnear form on K 0 (R) Let k be a feld and work wth k-lnear combnatons of dagrams. For any two R(ν)-modules M and N defne Hom(M, N) := the k-vector space of gradng preservng morphsms HOM(M, N) := a Hom(M, N{a}) the -graded k-vector space of all morphsms Graded HOM gves a semlnear form (,): K 0 (R) K 0 (R) É(q) ([P],[Q]) := gdmhom(p, Q) Example ([E ],[E ]) = 1 1 q 2 ([E ],[E ]) = 0 f Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

12 The balgebra U + Recall the É(q)-algebra U + wth generators E for I and relatons E E = E E f E E E = E (2) E + E (2) E f The tensor square U + U + s an assocatve algebra wth twsted multplcaton (x 1 x 2 )(x 1 x 2 ) = q x 2 x 1 x 1 x 1 x 2x 2 for homogeneous x 1, x 2, x 1, x 2 n U+. U + s a coalgebra wth comultplcaton r : U + U + U + determned by r : E E E and the requrement that r s an algebra homomorphsm. These structures descend to the ntegral form U + balgebra over [q, q 1 ] makng t a twsted Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

13 There s a unque É(q)-semlnear form on U + such that (1, 1) = 1 and (E, E ) = δ, (1 q 2 ) 1 for all, I, (x, yy ) = (r(x), y y ) for x, y, y U +, (xx, y) = (x x, r(y)) for x, x, y U +. We restrct ths semlnear form to U + Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

14 Work over a feld k. Theorem (A. Lauda, M.K arxv: ) There s an somorphsm of twsted balgebras: γ : U + E (a 1) 1 E (a 2) 2... E (a k) k K [ 0(R) ] E (a 1) 1 E (a 2) 2... E (a k) k multplcaton multplcaton gven by [Ind] comultplcaton comultplcaton gven by [Res] The semlnear form on U + maps to the HOM form on K 0(R) (x, y) = (γ(x), γ(y)) Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

15 Inectvty of γ Inectvty of the map γ: U + K 0(R) uses that U + s the quotent of a free assocatve algebra by the radcal of the semlnear form. Ths follows from the quantum verson of the Gabber-Kac theorem (proof, due to Lusztg for an arbtrary graph, uses perverse sheaves). Surectvty of γ Surectvty follows by mrrorng the work of Kleshchev, Gronowsk and Vazran. Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

16 Conecture When the graph Γ s a tree and the feld k =, then under the map U + K 0(R) Lusztg-Kashwara canoncal bass elements go to symbols of ndecomposable proectves n the Grothendeck group. If Γ contans an odd cycle then one should consder the sgn modfed verson of the rngs R(ν). Proof due to Brundan and Kleshchev when Γ s a chan or a cycle. Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

17 Sgned verson of R(ν) Choose an orentaton on the graph Γ for each orented edge, nvertble elements τ and τ of k For such a choce of data defne the k-algebra R τ (ν) by dagrams modulo sotopes and local relatons, analogous to the defnton of R(ν). All local relatons of R(ν) hold n R τ (ν) except for the followng modfed relatons: Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

18 = τ τ τ τ f f = τ f = τ f Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

19 R τ (ν) depends only on the products τ τ 1 over all edges of Γ. When Γ s a tree, there s an somorphsm of algebras R(ν) = R τ (ν) gven by rescalng the generators. When Γ contans a sngle cycle, the algebras R τ (ν) depend on a sngle parameter. R τ pmod also categorfes U + Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

20 Cartan datum A Cartan datum (I, ) conssts of a fnte set I and a symmetrc blnear form on [I] takng values n subect to the condtons {2, 4, 6,... } d := 2 {0, 1, 2,...} for any n I We can buld a graph Γ from a Cartan datum: ff 0 Example For the smply-laced case = 2 = 1 = 0 for all I f f The numbers d are mportant n what follows. Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

21 R(ν)-relatons, non-smply laced case deg = deg = = = for = = Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

22 0 f = = f = 0 d + d f 0 = a+b=d 1 a b f = otherwse, k k some of,, k may be equal Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

23 Example For the Cartan datum B 2 = 2 = 4 = = 2 d = 2 d = 1 = + Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

24 Example (contnued) = + = + = Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

25 We want to categorfy the quantum Serre relatons defnng U + a+b=d 1 ( 1) a E (a) E E (b) = 0 We demonstrate the proof usng the B 2 example Example ( = 2 = 4 = = 2 ) d = 2 d = 1 The new Serre relaton s ( 1) a E (a) E E (b) = E E (3) E E E (2) + E (2) E E E (3) E = 0 a+b=d 1 We construct an somorphsm E E (3) E (2) E E = E E E (2) E (3) E Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

26 We smplfy notaton and wrte dempotents e m defnng E m as follows e 2 := e 3 := e 4 := We can prove relatons lke e n 1 = e n e n Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

27 Example Consder the seres of maps E E (3) E (2) E E α + (0,3) α + (2,1) α (2,1) E (3) E E E E (2) α (3,0) α (1,2) α + (1,2) E E (3) E (2) E E α + (0,3) := e 2 α + (1,2) := e 2 α + (2,1) := e 3 α (1,2) := e 3 α (2,1) := e 2 α (3,0) := e 2 Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

28 To show that we have an somorphsm E E (3) E (2) E E = E E E (2) E (3) E t s enough to show = Id = 0 = 0 n the dagram α + (0,3) E E (3) E E E (2) α (1,0) α + (1,2) E (2) α (2,0) (Ths s how the proof goes n general) α + (2,0) E E E (3) α (3,0) E Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

29 Here s an example showng that gong rght twce s zero snce α + (0,3) E E (3) E E E (2) α + (1,2) E (2) E E = 0 e 2 = = = 0 e 2 Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

30 Here s an example showng that gong rght twce s zero snce α + (0,3) E E (3) E E E (2) α + (1,2) E (2) E E = 0 e 2 = = = 0 e 2 Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

31 Here s an example showng that gong rght twce s zero snce α + (0,3) E E (3) E E E (2) α + (1,2) E (2) E E = 0 e 2 = = = 0 e 2 Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

32 Here s an example showng that gong rght twce s zero snce α + (0,3) E E (3) E E E (2) α + (1,2) E (2) E E = 0 e 2 = = = 0 e 2 Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

33 α + (0,3) For example, to show E E (3) E E E (2) α (1,2) α+ (0,3) = Id E E (3) α (1,2) e 2 = = + e 3 e 3 But e 3 = = 0 Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

34 α + (0,3) For example, to show E E (3) E E E (2) α (1,2) α+ (0,3) = Id E E (3) α (1,2) e 3 e 2 = e 3 = e 3 + e 3 But e 3 = = 0 Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

35 α + (0,3) For example, to show E E (3) E E E (2) α (1,2) α+ (0,3) = Id E E (3) α (1,2) e 3 e 2 = e 3 = e 3 + e 3 But e 3 = = 0 Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

36 α + (0,3) For example, to show E E (3) E E E (2) α (1,2) α+ (0,3) = Id E E (3) α (1,2) e 3 e 2 = e 3 = e 3 + e 3 But e 3 = = 0 Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

37 e 3 = = + = + = + 0 = = e 3 = Id (3) E E Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

38 e 3 = = + = + = + 0 = = e 3 = Id (3) E E Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

39 e 3 = = + = + = + 0 = = e 3 = Id (3) E E Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

40 e 3 = = + = + = + 0 = = e 3 = Id (3) E E Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

41 e 3 = = + = + = + 0 = = e 3 = Id (3) E E Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

42 e 3 = = + = + = + 0 = = e 3 = Id (3) E E Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

43 For the general case, we use a α + a,b (, ) {}}{{}}{ b a α a,b (, ) {}}{{}}{ b e,a+1 e,b 1 e,a 1 e,b+1 }{{} }{{} a+1 b 1 }{{} }{{} a 1 b+1 We have a dagram of proectve modules and gradng-preservng homomorphsms E (a 1) E E (b+1) α + (a 1,b+1) α (a,b) E (a) E E (b) α + (a,b) α (a+1,b 1) E (a+1) E E (b 1) Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

44 The relatons n R(ν) categorfy the quantum Serre relatons defnng U + a+b=d 1 ( 1) a E (a) E E (b) Theorem (A. Lauda, M.K., arxv: ) For arbtrary Cartan datum (I, ) and assocated Kac-Moody algebra g there s an somorphsm of twsted balgebras. Corollary U + (g) K 0(R) The bass of ndecomposables n R gves rse to a bass n U + (g) where the structure constants are n Æ[q, q 1 ]. Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

45 Conectured categorfcaton of U q (g) Combne defnton of rngs R(ν) categorfyng U + (g) wth the defnton of the 2-category U(sl 2 ) categorfyng U q (sl 2 ) leads to a defnton of a 2-category U q (g) assocated to an arbtrary Kac-Moody algebra. Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

46 A root datum of type (I, ) conssts of free fntely generated abelan groups: the weght lattce X, and the root lattce Y (Lusztg s notaton) together wth a perfect parng, : Y X ; nclusons I X ( X ) and I Y ( ) such that, X = 2 := d for all, I. Ths mples, X = 2 for all. We wrte X to denote the mage of n X. Example For root datum assocated to the Dynkn graph of sl n. Smple roots Y correspond to vertces of the Dynkn graph. A weght X s an n 1-tuple = ( 1,..., n 1 ) wth :=, Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

47 The quantum group U assocated to a root datum as above s the untal assocatve É(q)-algebra gven by generators E, F, K µ for I and µ Y, subect to the relatons: ) K 0 = 1, K µ K µ = K µ+µ for all µ,µ Y, ) K µ E = q µ, X E K µ for all I, µ Y, ) K µ F = q µ, X F K µ for all I, µ Y, K K v) E F F E = δ, where K ± = K ±( /2), v) For all q q 1 a+b=, X +1 a+b=, X +1 ( 1) a E (a) ( 1) a F (a) E E (b) = 0 F F (b) = 0 Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

48 The É(q)-algebra U s obtaned from U by adonng a collecton of orthogonal dempotents 1 for each X, such that 1 1 = δ, 1 K µ 1 = 1 K µ = q µ, 1 E 1 = 1 +X E F 1 = 1 X F The algebra U decomposes as drect sum of weght spaces U = 1 U1., X The algebra U s the [q, q 1 ]-subalgebra of U generated by products of dvded powers E (a) 1 and F (a) 1, and has a smlar weght decomposton U = 1 ( U )1, X Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

49 We keep track of products of E s and F s usng fnte sequence of elements of I wth sgns. That s, = (ǫ 1 1,ǫ 2 2,...,ǫ m m ) where ǫ 1,...,ǫ m {+, } and 1,..., m I Wrte E + for E and E for F so that E := E ǫ1 1 E ǫ2 2...E ǫm m U, E 1 := E ǫ1 1 E ǫ2 2...E ǫm m 1 U. Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

50 Defnton Let (Y, X,,,... ) be a root datum of type (I, ). U s an addtve k-lnear 2-category. The 2-category U conssts of obects: for X. The homs U(, ) between two obects, are addtve k-lnear categores consstng of: obects of U(, ): a 1-morphsm n U from to s a formal fnte drect sum of 1-morphsms E 1 {t} = 1 E 1 {t} for any t and sgned sequence SSeq such that = + X. morphsms of U(, ): for 1-morphsms E 1 {t}, E 1 {t } U, hom sets U(E 1 {t}, E 1 {t }) of U(, ) are graded k-vector spaces gven by lnear combnatons of degree t t dagrams, modulo certan relatons, bult from compostes of: Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

51 Generatng 2-morphsms I. Degree zero dentty 2-morphsms 1 x for each 1-morphsm x n U; the dentty 2-morphsms 1 E+ 1 {t} and 1 E 1 {t}, for I, are represented graphcally by 1 E+ 1 {t} 1 E 1 {t} + X X deg 0 deg 0 Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

52 Generatng 2-morphsms II. More generally, for a sgned sequence = ǫ 1 1 ǫ ǫ m m, the dentty 1 E 1 {t} 2-morphsm s represented as 1 2 m + X 1 2 m where the strand labelled α s orented up f ǫ α = + and orented down f ǫ α =. Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

53 Generatng 2-morphsms III. For each X the 2-morphsms 2-morphsm: + X + X Degree: 2-morphsm: Degree: c +, c, c +, c, wth c ±, := 2 (1 ±, ) Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

54 The 2-morphsms are subect to the followng local relatons: Relatons I. 1 +X E + 1 and 1 E 1 +X are badont, up to gradng shfts: + X + X = + X = + X + X = + X + X = + X Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

55 Relatons + X = + X = + X = = Aaron Lauda Jont wth Mkhal Khovanov (Columba Unversty) A categorfcaton of quantum sl n January 20th, / 60

56 All dotted bubbles of negatve degree are zero. That s, α = 0 f α <, 1 α = 0 f α <, 1 for all α + Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

57 Fake bubbles Just lke the sl 2 case, fake bubbles are formal symbols nductvely defned by the equaton ( + t + + ) t α +, 1, 1+1, 1+α ( + + ) t α + = 1 and the addtonal condton 1, 1 = 1, 1+α = 1 f, = 0. Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

58 More sl2, = f=0, f =, 1+f, g=0, 1+g, g = = + + f 1 +f 2 +f 3 =, 1 g 1 +g 2 +g 3 =, 1, 1+f 2 f 3 f1 g1, 1+g 2 g 3 Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

59 NlHecke relatons: = 0, = = = For = = Aaron Lauda Jont wth Mkhal Khovanov (Columba Unversty) A categorfcaton of quantum sl n January 20th, / 60

60 The R(ν)-relatons For = d + d f = 0, f 0. = = Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

61 More R(ν)-relatons Unless = k and 0 = k k For 0 = d 1 a=0 a d 1 a Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

62 Composton The addtve k-lnear composton functor U(, ) U(, ) U(, ) s gven on 1-morphsms of U by E 1 {t } E 1 {t} E 1 {t + t } for X =, and on 2-morphsms of U by uxtaposton of dagrams Aaron Lauda Jont wth Mkhal Khovanov (Columba Unversty) A categorfcaton of quantum sl n January 20th, / 60

63 For any two sgned sequences and and weght there s a homogenous spannng set B,, for HOM U (E 1, E 1 ) If B,, s a bass then ( ) gdm HOM U (E 1, E 1 ) = q deg(s) = E 1, E 1 π s B,, where π s a proportonalty constant that only depends on the root datum. If the B,, form a bass we say that the graphcal calculus s nondegenerate. Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

64 Example The spannng set B (,+,+, ),(+, ), whose elements are a 2 + a 1 a 3 + +, 1+α 1, 1+α 2, 1+α 3, 1+α 4 k k, 1+α 5 } {{ } bubble monomal a 1 a 3 + a 3 + +, 1+α 1, 1+α 2, 1+α 3, 1+α 4 k k, 1+α 5 } {{ } bubble monomal Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

65 The 2-category U s an addtve 2-category. The Karouban envelope U s the smallest 2-category whch contans U and has splttng of dempotent 2-morphsms. U(n, m) := Kar ( U(n, m) ) where Kar ( U(n, m) ) s the usual Karouban envelope of the addtve category U(n, m). An obects of Kar ( U(n, m) ) s a 1-morphsm x : n m together wth an dempotent 2-morphsm e: x x. Thnk of ths as the mage of the dempotent e n x. Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

66 Theorem (M. Khovanov, A.L., arxv: ) For any feld k and any root datum there s a surectve homomorphsm γ: U (g) K 0 ( U q (g)) 1 [1 ] E 1 [E 1 ] for any dvded power sgned sequence. The map γ s nectve f the graphcal calculus for the root datum and feld k s nondegenerate, that s, f certan spannng sets B,, for k-vector space of 2-morphsms HOM U (E 1,E 1 ) are a bass Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

67 Theorem (M. Khovanov, A.L., arxv: ) For any feld k and any root datum there s a surectve homomorphsm γ: U (g) K 0 ( U q (g)) 1 [1 ] E 1 [E 1 ] for any dvded power sgned sequence. The map γ s nectve f the graphcal calculus for the root datum and feld k s nondegenerate, that s, f certan spannng sets B,, for k-vector space of 2-morphsms HOM U (E 1,E 1 ) are a bass Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

68 Theorem (M. Khovanov, A.L., arxv: ) The graphcal calculus for root datum of sl n and any feld k s nondegenerate. Hence, K 0 ( U q (sl n )) = U (sl n ) and U q (sl n ) can be vewed as a categorfcaton of U q (sl n ). (For the proof we construct a famly of 2-representatons where elements of the spannng sets map to lnearly ndependent operators on vector spaces n ths 2-representaton) Indecomposable 1-morphsms up to somorphsm and gradng shft consttute a bass of K 0 ( U q (sl n )) = U (sl n ) whch mght depend on the ground feld k. Multplcaton n ths bass s n Æ[q, q 1 ] Graded 2Hom HOM U (x, y) categorfes the semlnear form x, y Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

69 Theorem (M. Khovanov, A.L., arxv: ) The graphcal calculus for root datum of sl n and any feld k s nondegenerate. Hence, K 0 ( U q (sl n )) = U (sl n ) and U q (sl n ) can be vewed as a categorfcaton of U q (sl n ). (For the proof we construct a famly of 2-representatons where elements of the spannng sets map to lnearly ndependent operators on vector spaces n ths 2-representaton) Indecomposable 1-morphsms up to somorphsm and gradng shft consttute a bass of K 0 ( U q (sl n )) = U (sl n ) whch mght depend on the ground feld k. Multplcaton n ths bass s n Æ[q, q 1 ] Graded 2Hom HOM U (x, y) categorfes the semlnear form x, y Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

70 Theorem (M. Khovanov, A.L., arxv: ) The graphcal calculus for root datum of sl n and any feld k s nondegenerate. Hence, K 0 ( U q (sl n )) = U (sl n ) and U q (sl n ) can be vewed as a categorfcaton of U q (sl n ). (For the proof we construct a famly of 2-representatons where elements of the spannng sets map to lnearly ndependent operators on vector spaces n ths 2-representaton) Indecomposable 1-morphsms up to somorphsm and gradng shft consttute a bass of K 0 ( U q (sl n )) = U (sl n ) whch mght depend on the ground feld k. Multplcaton n ths bass s n Æ[q, q 1 ] Graded 2Hom HOM U (x, y) categorfes the semlnear form x, y Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

71 Rescale, nvert the orentaton, and send : Consder the operaton on the dagrammatc calculus that rescales the -crossng,, for all I and X, nverts the,, orentaton of each strand and sends. Relatons are nvarant. + +k+k k k k + +k + ω = µ µ + +k + k somorphsm ω: U U 1 µ E 1 {t} 1 µ E 1 {t} É(q)-lnear algebra homomorphsm ω: U U 1 1 E 1 F 1 F 1 E 1 Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

72 Rescale, reflect across the y-axs, : Consder the operaton that rescales the -crossng,, for all I and X, reflects a dagram across,, the y-axs, and sends to. Relatons are nvarant. σ + +k+k k µ + +k + = k +k +k + µ + +k + 2-somorphsm σ: U U op 1 µ E s1 E s2 E sm 1 E sm 1 {t} 1 E sm E sm 1 E s2 E s1 1 µ {t} contravarant on 1-morphsms É(q)-lnear algebra ant-automorphsm σ: U Uop X E 1 1 E 1 X 1 F 1 +X 1 X F 1 Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

73 Reflect across the x-axs and nvert orentaton: ψ + +k+k k {t } µ + +k + {t} = + +k + { t} µ + +k +k k { t } The degree shfts on the rght hand sde are requred for ths transformaton to preserve the degree of a dagram. 2-somorphsm ψ: U U co 1 µ E 1 {t} 1 µ E 1 { t} contravarant on 2-morphsms É(q)-antlnear algebra homomorphsm ψ: U U 1 1 q a E 1 q a E 1 q a F 1 q a F 1 Aaron Lauda Jont wth Mkhal Khovanov (Columba A categorfcaton Unversty) of quantum sl n January 20th, / 60

74 Rotaton by 180 (takng rght adonts): For each 1 µ x1 U denote ts rght adont by 1 y1 µ. The symmetry of rotaton by 180 can also be realzed by the 2-functor that sends a 1-morphsm 1 µ x1 to ts rght adont 1 y1 µ and each 2-morphsm ζ : 1 µ x1 1 µ x 1 to ts mate under the adunctons 1 µ x1 1 y1 µ and 1 µ x 1 1 y 1 µ x ζ y ζ y = ζ x y y 2-somorphsm τ : U U coop 1 µ x1 1 y1 µ contravarant on 1-morphsm and 2-morphsms É(q)-antlnear algebra anthomomorphsm τ : U U op 1 1 E 1 q 1, F 1 +X F 1 +X q 1+, E 1 Aaron where Lauda Jont thewth degree Mkhal Khovanov shft t (Columba for A categorfcaton the Unversty) rghtofadont quantum sl n January 20th, / 60