Categorification of quantum groups

Transcription

1 Categorfcaton of quantum groups Aaron Lauda Jont wth Mkhal Khovanov Columba Unversty June 29th, 2009 Avalable at lauda/talks/ Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

2 The goal: categorfy U + q (g) The quantum envelopng algebra U q (g) of a symmetrzable Kac-Moody Le algebra g has a decomposton U q (g) = U q U q (h) U + q U + q has the structure of a balgebra: try to categorfy the balgebra U + q The plan: defne a new algebra R Decategorfcaton (Grothendeck group) R mod K 0 (R mod) ( category of fntely generated graded proectve modules = U + q (g) ) Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

3 Why categorfy quantum groups? Categorfed representaton theory should provde new nsghts for ordnary representaton theory, especally relatng to postvty and ntegralty propertes. Algebrac/combnatoral analog of perverse sheaves. Conectured applcatons to low-dmensonal topology Representaton theoretc explanaton of Khovanov homology Categorfcaton of the Reshetkhn-Turaev quantum knot nvarants. Crane-Frenkel conectured categorfed quantum groups would gve 4-dmensonal TQFTs Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

4 U + q U q(g) g = sl n E = e,+1 = Le algebra relatons: [E, E ] = 0 > 1 [E,[E, E ]] = 0 = 1 Envelopng algebra relatons for U + (sl n ) E E = E E > 1 2E E E = E 2 E + E E 2 = ± 1 Quantum envelopng algebra U + q (sl n ) E E = E E > 1 quantum 2 (q + q 1 )E E E = E 2 E + E E 2 = ± 1 Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

5 U + q (sl n ) has a generator E for each vertex of the Dynkn graph E 1 E 2 E n 1 U + q for any Γ Let Γ be an unorented graph wth set of vertces I. Γ = U + q s the É(q)-algebra wth: generators: E I relatons: E E = E E (q + q 1 )E E E = E 2 E + E E 2 f f U + q s Æ[I] graded wth deg(e ) =. Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

6 Integral form of U + q Defne quantum ntegers and quantum factorals: [a] := qa q a q q 1 [a]! := [a][a 1]...[1] Example [1] = 1 [2] = q2 q 2 q q 1 [3] = q3 q 3 q q 1 = q + q 1 = q q 2 The algebra U + s the [q, q 1 ]-subalgebra of U q + products of quantum dvded powers: E (a) := E a [a]! generated by all Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

7 Snce we can wrte the U + q relaton = E 2 q + q 1 E (2) (q + q 1 )E E E = E 2 E + E E 2 f as E E E = E (2) E + E E (2) f Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

8 Categorfcaton of U + q Assocated to 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 Form an abelan group by takng -lnear (or k-lnear) combnatons of dagrams: Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

9 Multplcaton s gven by stackng dagrams on top of each other when the colors match: = = 0 Defnton Gven ν Æ[I] defne 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 Categorfcaton Unversty) of quantum groups June 29th, / 33

10 Local relatons I = 0 = = = Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

11 Local relatons II = f k k k = + f = = f k k k k k Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

12 Local relatons III = f = otherwse, k k some of,, k may be equal Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

13 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 Categorfcaton Unversty) of quantum groups June 29th, / 33

14 Example If ν = 0 then R(0) = wth unt element gven by the empty dagram. If ν = for some vertex, then a dagram s a lne wth some number a 0 of dots on t. a := Hence, R() = [x] where the somorphsm maps a a x a Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

15 R ν s the assocatve, F-algebra on generators 1, x a,, ; ψ b, for 1 a m, 1 b m 1 and Seq(ν) subect to the followng relatons for, Seq(ν): 1 1 = δ, 1, x a, = 1 x a, 1, ψ a, = 1 sa()ψ a, 1, x a, x b, = x b, x a,, 0 f r = r+1 ψ a,sa()ψ a, = ( 1 ) f (α a, α a+1 ) = 0, x a, a+1 a, + x a+1, a a+1, 1 f (α a, α a+1 ) 0 and a a+1 ψ b,sa()ψ a, = ψ a,sb ()ψ b, f a b > 1, ψ a,sa+1 s a()ψ a+1,sa()ψ a, ψ a+1,sas a+1 ()ψ a,sa+1 ()ψ a+1, = a, a+1 1 x = a, r x a, a+1 1 r a+2, f a = a+2 and (α a, α a+1 ) 0 r=0 0 otherwse, ψ a, x b, x sa(b),s a()ψ a, = 1 f a = b and a = a+1 1 f a = b + 1 and a = a+1 0 otherwse. Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

16 Let R = ν R(ν). For each product of E s n U q + dempotent n R: we have an E E E k E E E l 1 kl := k l Ths gves rse to a proectve module E E E k E E E l := R1 kl = R( k + l)1 kl correspondng to the dempotent 1 kl above. Example For a gven I we wrte E m correspondng to the dempotent 1 m = for the proectve module R(m) = NH m, where m :=... Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

17 Example Consder R1 k = R( + + k)1 k The proectve module E E E k := R( + + k)1 k conssts of lnear combnatons of dagrams that have the sequence k at the bottom Dagram.e. and R( + + k)1 k k But / R( + + k)1 k Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

18 We can construct maps between proectve modules by addng dagrams at the bottom Example We get a module map from E E E k := R( + + k)1 k to E k E E := R( + + k)1 kl as follows: R1 k R1 k : Dagram k k k Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

19 Gven a graded module M and a Laurent polynomal f = f a q a [q, q 1 ] wrte M f or M to denote the drect sum over a of f a copes of M{a} Example Snce [3] = q q 2 [q, q 1 ], for a graded module M M = M{2} M{0} M{ 2} [3] f Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

20 Example (n = 2) E (2) = E2 or E 2 q+q 1 = (q + q 1 )E (2) Recall that = so that e 2 = s an dempotent. E (2) s the proectve module for ths dempotent E (2) := R(2)e 2 {1} E 2 = E (2) {1} E (2) { 1} Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

21 Categorfcaton of E E = E E E E = E E f E E = E E f E E E E E E These maps are somorphsms snce = = f Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

22 Categorfcaton of E E E = E (2) E + E E (2) = f Let e = (e ) 2 = = + = = = e e = 1 e = s dempotent too (e ) 2 = e Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

23 Orthogonalty e e = e e = 0 and 1 = e + e mply E E E = E E E e E E E e But = e = = dempotent e 2 of proecton onto E (2) e = = dempotent e 2 of proecton onto E (2) Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

24 Therefore, so that the relaton E E E e = E E (2) E E E e (2) = E E = f together wth the other relatons mply E E E = E E (2) E (2) E Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

25 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 Categorfcaton Unversty) of quantum groups June 29th, / 33

26 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 Categorfcaton Unversty) of quantum groups June 29th, / 33

27 Work over a feld k. Theorem (M.Khovanov, A. L. 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 Categorfcaton Unversty) of quantum groups June 29th, / 33

28 Inectvty of γ Inectvty of the map γ: U + K 0(R) uses that U q + 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 Gronowsk and Vazran. M.Khovanov, A. L. (arxv: ) Ths theorem has an extenson to the non-smply laced case. The bass of ndecomposable gves a new bass for U + where structure constants are necessarly postve. Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

29 Conecture (Proven n smply-laced case) U + Lusztg-Kashwara canoncal bass K 0 (R) ndecomposable proectve [P] arxv: Brundan and Kleshchev gave an algebrac proof when Γ s a chan or a cycle. arxv: The general case (over ) was proven by Varagnolo and Vasserot who showed that rngs R(ν) n the smply-laced case were somorphc to certan Ext-algebras of Perverse sheaves on Lusztg quver varetes. Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

30 Cyclotomc quotents For a gven weght λ = I λ Λ defne the cyclotomc quotent R λ ν of R(ν) by mposng the addtonal relatons: for any sequence 1 2 m of vertces of Γ λ 1 dots on the frst strand of any sequence s zero λ m = 0 Ths s analogous to takng the Ark-Koke cyclotomc quotent of the affne Hecke algebra: / Hd λ := H d (X 1 q ) λ I Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

31 Cyclotomc quotent conecture The category of fntely-generated graded modules over the rng R λ = ν Æ[I] categorfes the ntegrable verson of the representaton V λ of U q (g) of hghest weght λ. R λ ν V(λ) K 0 (R λ ) Lusztg-Kashwara canoncal bass ndecomposable proectve [P] Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

32 Theorem (Brundan-Kleshchev, arxv: ) There s an somorphsm Rν λ Hλ ν where Hλ ν s a sngle block of the cyclotomc Hecke algebra Hd λ. Usng ths somorphsm they proved the cyclotomc quotent conecture n type A and affne type A. For level 2 quotents the result follows from earler work of Brundan and Stroppel. Corollary There s a -gradng on blocks H λ ν of affne Hecke algebras. Imples there s a new -gradng on blocks of the symmetrc group. Leads to graded Specht module theory, see Brundan-Kleshchev-Wang, arxv: Leads to a graded verson of the generalzed LLT-conecture. Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33

33 Generalzatons A.L (arxv: ) There s a graphcal 2-category categorfyng the ntegral form of the dempotent completon of the entre quantum group U q (sl 2 ) U = K0 ( U) the Grothendeck rng/category of ths 2-category Indecomposable 1-morphsms Lusztg canoncal bass element The 2-category U acts on cohomology of terated flag varetes, categorfyng the rreducble N-dmensonal rep of U q (sl 2 ) M. Khovanov, A.L. (arxv: ) 2-category U has an extenson to a categorfcaton of U(sl n ). Conectural categorfcaton of the ntegral form of U(g) for any Kac-Moody algebra. arxv: Closely related 2-categores were recently studed by Rouquer. Aaron Lauda Jont wth Mkhal Khovanov (Columba Categorfcaton Unversty) of quantum groups June 29th, / 33