Quantum algebras and symplectic reflection algebras for wreath products Nicolas Guay

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1 Quantum agebras and sympectc refecton agebras for wreath products Ncoas Guay Abstract To a fnte subgroup Γ of SL C, we assocate a new famy of quantum agebras whch are reated to sympectc refecton agebras for wreath products S Γ va a functor of Schur-Wey type. We expan that they are deformatons of matrx agebras over ran-one sympectc refecton agebras for Γ and construct for them a PBW bass. When Γ s a cycc group, we are abe to gve more nformaton about ther structure and to reate them to Yangans. 1 Introducton The theory of sympectc refecton agebras was ntroduced a few years ago n the semna paper [EtG] of P. Etngof and V. Gnzburg. Snce then, appcatons have been found n representaton theory and n agebrac geometry, see e.g. [Bo, EGGO, GoSt, GoSm]. One mportant exampe of such agebras s gven by the ratona Cheredn agebras [Ch],[GGOR],[BEG] assocated to a compex refecton group W actng on the sympectc vector space h h, h beng ts refecton representaton. A arge cass of sympectc refecton agebras are those assocated to wreath products Γ S Γ Γ S for Γ a fnte subgroup of SL C. In ths paper, we ntroduce a new famy of quantum agebras that we ca Γ-deformed doube current agebras Γ-DDCA. They are fat deformatons of the enveopng agebra of an enargement of ŝ n C[u, v] Γ, the unversa centra extenson of s n C[u, v] Γ. They can aso be vewed as fat deformatons of Ug n A 1 Γ where A 1 s the frst Wey agebra. We construct a PBW bass for Γ-DDCA by usng a Schur-Wey functor whch reates them to sympectc refecton agebras for Γ. When Γ Z/rZ, we are abe to gve a second defnton of Γ-DDCA by reazng them as certan subagebras of a cycc verson of affne Yangans. One can consder the genera probem of studyng spaces of maps X g from an agebrac varety to a semsmpe Le agebra g. When X s smooth and of dmenson one, ths eads to current Le agebras g C C[u], oop agebras g C C[u, u 1 ] and ther unversa centra extensons, the affne Le agebras. When X has dmenson two, the most natura case to consder s the two-dmensona torus X C C, but two smper cases are X C C and X C. Quantzatons of the correspondng enveopng agebras are nown as quantum toroda ageras [GKV], affne Yangans and deformed doube current agebras [Gu], respectvey. We may aso consder snguar varetes and one of the smpest exampe s a Kenan snguarty C /Γ. We are thus ed to consder the Le agebra g C C[u, v] Γ and ts unversa centra extenson. We can aso foow one of the man deas n [EtG] and repace the rng of nvarants C[u, v] Γ by the smash-product C[u, v] Γ snce t s beeved by rng theorsts that the atter encodes more the geometry of the quotent C /Γ and of ts mnma resouton of snguartes than the former. Ths s another motvaton for studyng Γ-DDCA. The representaton theory of quantum toroda agebras was studed n [He1, He, VaVa1, VaVa] and, va geometrc methods, n [VaVa3]. We hope that understandng the representatons of Γ-DDCA w eventuay ead to a better understandng of quantum toroda agebras and affne Yangans, but w aso exhbt new phenomena whch do not occur for these two types of agebras. After recang the defnton of sympectc refecton agebras for wreath products, we devote two sectons to the Le agebras s n C[u, v] and ŝ nc[u, v] Γ, gvng presentatons n terms of fames of generators and reatons whch are usefu ater on. The man dea s to obtan presentatons wth ony fntey many generators and reatons of ow degree. The prncpa resuts here are emma.5 and 1

2 emma 5.1. The atter s modfed verson of a theorem of C. Kasse and J.L. Loday [KaLo] whch s usefu for our purpose. We aso menton some resuts about the frst cycc homoogy group of the smash product C[u, v] Γ snce ths space gves the center of the unversa centra extenson ŝ nc[u, v] Γ. The foowng secton s smpy devoted to defnng the Γ-deformed doube current agebras D n β,b. Secton 7 contans one of the man resuts of ths paper: we expan how to extend the cassca Schur-Wey functor to the doube affne setup and when t yeds an equvaence of certan categores of modues - see theorem 7.1. When λ 0, the Γ-deformed doube current agebras are enveopng agebras of Le agebras cosey reated to s n wth entres n a ran-one sympectc refecton agebra for Γ: ths s the content of secton 8. Secton 9 contans our second man theorem: we prove that the assocated graded rng of D n β,b s somorphc to the undeformed rng Dn β0,b0, whence the name PBW property by anaogy wth the cassca Poncaré-Brhoff-Wtt theorem. The second haf of the paper a of secton 10 s devoted to the speca case Γ Z/dZ. The sympectc refecton agebras for the wreath product Z/dZ S are ratona Cheredn agebras, so they afford a Z-gradng. Ths expans n part why we can obtan more resuts n ths specfc case. We start by studyng certan degenerate affne Hece agebras assocated to Γ and then extend the resuts of [Gu1] to the doube affne trgonometrc settng where we have a functor of Schur-Wey type see theorem 10.1 between modues for a ocazaton of a ratona Cheredn agebra for Γ and a certan agebra whch turns out to be somorphc to a Yangan for s nd see coroary The man goa of secton 10 s reached n subsecton 10.3 where we prove that deformed doube current agebras for Z/dZ can be reazed as subagebras of certan oop Yangans: see theorem Ths provdes another set of generators, whch mght be convenent n the study of representatons. Throughout ths paper, we w assume that n : anaogous resuts most probaby hod for n, 3, but some defntons may nvove more compcated reatons and certan proofs woud have to be modfed accordngy. Acnowedgments The author gratefuy acnowedges the fnanca support of the Mnstère franças de Ensegnement supéreur et de a Recherche and woud e to than Davd Hernandez and the Laboratore de Mathématques de Unversté de Versaes-St-Quentn-en-Yvenes for ther hosptaty whe ths paper was wrtten. The author s now supported by an NSERC Dscovery Grant. 3 Sympectc refecton agebras for wreath products Isomorphsm casses of fnte subgroups of SL C are nown to be n bectve correspondence wth affne Dynn dagrams of type A,D,E va the McKay correspondence. We w denote by Γ such a subgroup. For nstance, the group correspondng to the Dynn dagram of type Âr 1 s the cycc group Γ Z/rZ. In ths secton, we reca a few defntons and facts about sympectc refecton agebras for the wreath product Γ S Γ of Γ wth the symmetrc group S. Let ω be a non degenerate sympectc form on U C and choose a bass {x, y} of U such that ωx, y 1. We w denote by {x, y } the same bass of C, ths tme vewed as the th drect summand of U. Note that Γ acts on U. The defnton of a sympectc refecton agebra depends on two parameters: t C and c κ d + \{d} c γγ ZΓ, whch s an eement n the center ZΓ of C[Γ]. We have adapted the defnton of the sympectc refecton agebra H t,c Γ from [GaG]. For γ Γ, we wrte γ for d,..., d, γ, d,..., d Γ where γ s n the th poston. Defnton 3.1. The sympectc refecton agebra H t,c Γ s defned as the agebra generated by the two

3 sets of parwse commutng eements x 1,..., x, y 1,..., y and by τ Γ wth the reatons: τ x τ 1 τx, τ y τ 1 τy, 1,...,, τ Γ 1 [x, y ] t + κ σ γ γ \{d} c γ γ, 1,..., where σ S s the permutaton. For and any w 1, w U, w span{x, y }: [w 1, w ] κ ωγw 1, w σ γ γ 1 3 To smpfy the notaton, we w wrte ω x γ ωγx, x, ω y γ ωγy, y, ω x,y γ ωγx, y. It s possbe to fter the agebra H t,c Γ by gvng degree 1 to the generators x, y, 1, and degree 0 to the eements of Γ. Ths ftraton w be denoted F H t,c Γ and the correspondng assocated graded rng gr H t,c Γ. Theorem 3.1 PBW Property, [EtG]. The canonca map H t0,c0 Γ gr H t,c Γ s an somorphsm. Doube current agebras Before defnng Γ-deformed doube current agebras n secton 6, we need to prove a seres of emmas for the Le agebra ŝ nc[u, v] Γ, the unversa centra extenson of s n C[u, v] Γ. In ths secton, we threat the case Γ {d}. We w need to assume that n n ths secton and, a fortor, for the rest of the paper. The frst emma s smar to proposton 3.5 n [MRY] and shoud admt an anaog for other semsmpe Le agebras; however, we doubt that emma. admts such a generazaton, except perhaps by addng a few reatons. We start wth a theorem whch gves a descrpton of ŝ n[u, v]. We w denote by Ω 1 C the space of poynoma 1-forms on the affne pane C and by dc[u, v] the space of exact 1-forms on C. Theorem.1. [Ka] The Le agebra ŝ n[u, v] s somorphc to the Le agebra s n [u, v] Ω1 C dc[u,v] the foowng bracet where, s the Kng form on s n : [m 1 p 1, m p ] [m 1, m ] p 1 p + m 1, m p 1 dp, m 1, m s n, p 1, p C[u, v] and the eements of Ω 1 C dc[u,v] are centra. We denote by Ĉn 1 c 0, n 1 the n n Cartan matrx of affne type Ân 1: Ĉ n The set of roots of s n w be denoted {α 1 n} wth choce of postve roots + {α 1 < n}. The ongest postve root θ equas α 1n. The eementary matrces w be wrtten E, H E E +1,+1 for 1 n 1 and H E E. We set E + E,+1, E E +1,, 1 n 1. The foowng s emma.5 n [Gu]. wth 3

4 Lemma.1. The Le agebra ŝ n[u, v] s somorphc to the Le agebra whch s generated by the eements X ±,r, H,r, X 0,r +, 1 n 1, r 0 whch satsfy the foowng reatons: [H,r, H,s ] 0 1, n 1, r, s 0, [H,0, X ±,r ] ±c X ±,r 1 n 1, 0 n 1, r 0 [X ±,r, X±,s ] 0, [X±,r+1, X±,s ] [X±,r, X±,s+1 ], [H,r+1, X ±,s ] [H,r, X ±,s+1 ] 1, n 1, r, s 0 5 [X 0,r +, X+ 0,s ] 0, [X+,r+1, X+ 0,s ] [X+,r, X+ 0,s+1 ], [H,r+1, X 0,s + ] [H,r, X 0,s+1 + ] 1 n 1, r, s 0 6 [X +,r, X,s ] δ H,r+s, 0 n 1, 1 n 1, r, s 0 7 [ X ±,0, [X±,0, X±,r ]] 0 r 0 f c 1, [X ±,0, X±,r ] 0 r 0 f c 0, 8 Remar.1. In 8, when 0 or 0, we have defnng reatons ony n the + case. We w need a smper set of generators and reatons for the Le agebra ŝ n[u, v], whence the mportance of the next emma, whch s emma.7 n [Gu], except for a mnor dfference. Lemma.. The Le agebra ŝ n[u, v] s somorphc to the Le agebra whch s generated by the eements X ±,r, H,r, X 0,r +, 1 n 1, r 0, 1 satsfyng the foowng reatons: [H,r, H,s ] 0, 1, n 1, r, s 0, 1, [H,0, X ±,r ] ±c X ±,r, 1 n 1, 0 n 1, r 0, 1 9 [X ±,0, X±,1 ] 0, [X±,1, X±,0 ] [X±,0, X±,1 ], [H,1, X ±,0 ] [H,0, X ±,1 ], 1, n 1, 10 [X 0,0 +, X+ 0,1 ] 0, [X+,1, X+ 0,0 ] [X+,0, X+ 0,1 ], [H,1, X 0,0 + ] [H,0, X 0,1 + ], 1 n 1, 11 An somorphsm [X +,r, X,0 ] δ H,r [X +,0, X,r ], 0 n 1, 1 n 1, r 0, 1 1 [ X ±,0, [X±,0, X±,0 ]] 0 f c 1, [X ±,0, X±,0 ] 0 f c 0, 0, n 1 13 ŝ n[u, v] s gven by X ±,r E± v r, H,r H v r for 1 n 1, X + 0,r E n1 uv r, r 0, 1. We w need a coroary of the prevous emma whch gves a fourth presentaton of ŝ n[u, v]. It s an mmedate consequence of emma 3. snce we are ony emnatng X 0,1 + from a the reatons, so we w use the same etter. Lemma.3. The Le agebra ŝ n[u, v] s somorphc to the Le agebra whch s generated by the eements X ±,r, H,r, X 0,0 +, 1 n 1, r 0, 1 satsfyng the foowng reatons: [H,r, H,s ] 0, 1, n 1, r, s 0 or 1, [H,0, X ±,r ] ±c X ±,r, 1, n 1, r 0, 1 1 [X ±,0, X±,1 ] 0, [X±,1, X±,0 ] [X±,0, X±,1 ], [H,1, X ±,0 ] [H,0, X ±,1 ], 1, n 1 15 [X +,1, X+ 0,0 ] [ H,1, [X +,0, X+ 0,0 ]], [H 1,1, X 0,0 + ] [H n 1,1, X 0,0 + ], 1, n 1, 16 [ X + 0,0, [H n 1,1, X 0,0 + ]] 0, [H,0, X 0,0 + ] c 0X 0,0 +, [H,1, X 0,0 + ] 0 f 1, n 1, 17 [X +,r, X,s ] δ H,r+s, [ [H n 1,r, X 0,0 + ],,s] X 0, 0 n 1, 1 n 1, 0 r + s 1 r 0 f 0 18 [ X ±,0, [X±,0, X±,0 ]] 0 f c 1, [X ±,0, X±,0 ] 0 f c 0, 0, n 1 19

5 In the prevous emmas, the eements X ±,r, H,r wth 0 generate a Le subagebra whch s somorphc to s n [v], whereas those wth r 0 aong wth X + 0,0 generate an somorphc mage of s n[u]. We woud e to end ths secton by gvng one ast defnton of ŝ n[u, v] n whch these two agebras pay a more symmetrc roe, but before that we need to ntroduce one more emma whch s probaby nown to other peope. Lemma.. The Le agebra s n [v] s somorphc to the Le agebra f generated by eements E ab s n, E ab v for 1 a b n whch satsfy the foowng reatons: [E ab, E bc v] E ac v [E ad v, E dc ], [E ab v, E bc v] [E ad v, E dc v] f a b c a d c [E ab, E cd v] 0 [E ab v, E cd v] f a b c d a Proof. We want to defne eements E ab v, 1 a b n, by settng nductvey E ab v +1 [E ac v, E cb v ] for some c a, b. Ths does not depend on the choce of c, for f d a, b, c and 1 s satsfed for nstead of + 1, then: [E ad v, E db v ] [ E ad v, [E dc, E cb v ] ] [ [E ad v, E dc ], E cb v ] [E ac v, E cb v ]. We have to show that [E ab v, E bc v ] E ac v + f + + 1, a b c a 0 and [E ab v, E cd v ] 0 f or 1, + 1, when a b c d a. 1 We proceed by nducton on, the case 0 beng true by the defnton of f. Assume that Suppose that a b c a and choose d a, b, c. Frst, suppose that 1. [E ab v, E bc v ] [ [E ad v, E db v 1 ], E bc v ] [E ad v, E dc v + 1 ] E ac v + If 0, + 1, then [E ab, E bc v +1 ] [ E ab, [E bd v, E dc v ] ] [E ad v, E dc v ] E ac v +1. We have estabshed 0, so et us turn to 1. If a b, c a, choose d a, b, c. Then, f + +1 and, wthout oss of generaty,, [E ab v, E ac v ] [ E ab v, [E ad v, E dc v 1 ] ] 0 by nducton. Smary, [E ab v, E cb v ] 0 f + + 1, and, f 1, + 1, we can show that [E ab v, E ac v +1 ] 0 [E ab v, E cb v +1 ]. If a, b, c, d are a dstnct and + + 1, 1,, then [E ab v, E cd v ] [ E ab v, [E cb v, E bd v 1 ] ] [E cb v, E ad v + 1 ] [ E cb v, [E ac v 1, E cd v ] ] [E ab v, E cd v ]. Comparng the frst and ast terms, we see that [E ab v, E cd v ] 0. If 0, + 1, then [E ab, E cd v +1 ] [ E ab, [E cb v, E bd v ] ] [E cb v, E ad v ] 0 by the prevous case. The same argument wors f + 1, 0. Fnay, f agan a, b, c, d are a dstnct, we have [E ab v, E cd v +1 ] [ E ab v, [E ca v, E ad v] ] [E cb v +1, E ad v] [ [E cd v +1, E db ], E ad v] ] [E cd v +1, E ab v] Comparng the frst and ast terms shows that [E ab v, E cd v +1 ] 0. 5

6 Lemma.5. The Le agebra ŝ n[u, v] s somorphc to the Le agebra t that s generated by the eements E ab s n, E ab u, E ab v for 1 a b n wth the foowng reatons: For any w 1 a 1 u + b 1 v, w a u + b v, a, b C, [E ab, E bc w 1 ] E ac w 1 [E ab w 1, E bc ] [E ab w 1, E bc w ] [E ad w 1, E dc w ], [E ab w 1, E bc w ] [E ab w, E bc w 1 ] f a b c a d c, 3 [E ab w 1, E cd w ] 0 [E ab, E cd w 1 ] f a b c d a Proof. We can defne an epmorphsm t by the formuas X +,1 E,+1v, X,1 E +1,v, H,1 H,+1 v for 1 n 1, X + 0,0 E n1u We have to chec that ths respects the reatons We w expan why ths s ndeed the case for the frst equaton n 1, the frst and second one n 16, the frst one n 17 and the second one n 18, but before we do ths, we need to deduce a few consequences of the reatons n ths emma. For a b, we defne H ab w by H ab w [E ab w, E ba ]. Choose c a, b, so H ab w [ [E ac, E cb w], E ba ] [Ebc, E cb w] + [E ac, E ca w] [E bc, E cb w] + [ [E ab, E bc ], E ca w ] [E ab, E ba w] H ba w Startng from [E ab w 1, E bc w ] [E ad w 1, E dc w ] wth a, b, c, d a dstnct and appyng [, E ba ] gves the reaton [H ab w 1, E bc w ] [E bd w 1, E dc w ]. 5 Athough we needed to assume that a, b, c, d were dstnct to deduce ths equaty, t s aso true that [H ab w 1, E bc w ] [E ba w 1, E ac w ] f a, b, c are a dstnct, due to reaton 3. Smary, [H ab w, E bc w] [E bd w, E dc w] and commutng both sdes wth E ca yeds [H ab w, E ba w] [E ca w, E bc w] [E bd w, E da w] [E bd w, E da w]. We now appy [E ab, ] to both sdes of ths equaton to get [E ab w, E ba w] + [E bd w, E db w] + [E da w, E ad w] 0 6 Ths s a usefu equaton snce t heps us deduce the foowng for a, b, c a dstnct: [H ab w, H bc w] [ [E ab w, E ba ], [E bc, E cb w] ] [ [E ac w, E ba ], E cb w] [E bc, [E ab w, E ca w] ] [E bc w, E cb w] + [E ac w, E ca w] [E ab w, E ba w] 0 7 The frst equaton n 1 s now an mmedate consequence of 7. Appyng [E 1, ] to 0 [E n 1,1 u, E 1 v] gves 0 [E n 1, u, E 1 v] + [E n 1,1 u, H 1 v]. Therefore, [H 1 v, E n 1,1 u] [E n 1, u, E 1 v] [E n 1,n v, E n1 u] [H n 1,n v, E n 1,1 u], the ast equaty beng a consequence of appyng [, E n,n 1 ] to 0 [E n 1,n v, E n 1,1 u]. We now use [E n,n 1, ] agan to obtan [H 1 v, E n1 u] [H n 1,n v, E n1 u] + [E n,n 1 v, E n 1,1 u] [H n 1,n v, E n1 u] + [H n,n 1 v, E n1 u] [H n 1,n v, E n1 u]. Ths mpes that the second reaton n 16 s respected. As for the frst reaton n 16 when n 1 or 1, t s a consequence of 5 wth w 1 v, w u. 6

7 The frst reaton n 17 and the second one n 18 foow aso from 5 and from. To prove that t s an somorphsm, we woud e to construct an nverse ξ. We do ths by usng emma.3 whch dentfes wth the Le agebra n theorem.1. We set ξe ab u E ab u and ξe ab v E ab v. Ceary, ths defnes a Le agebra map t ŝ n[u, v]. Tang the composte wth the map t above yeds a homomorphsm ŝ n[u, v] whch s the somorphsm gven n emma.3 see the formuas after emma.. Therefore, t. 5 The unversa centra extenson of s n C[u, v] Γ For an arbtrary assocatve agebra A, s n A s defned as the space of matrces n g n A wth trace n [A, A]. Ths Le agebra s perfect, so t admts a unversa centra extenson whose erne s somorphc to the frst cycc homoogy group HC 1 A [KaLo]. When A s the group rng A C[Γ], HC 1 A 0 see chapter 9 n [We], so we concude that s n C[Γ] s unversay cosed. Therefore, theorem 5.1 gves a descrpton of s n C[Γ]. We w need to use the foowng theorem of C. Kasse and J.L. Loday n the case A C[u, v] Γ. We w compute ater n ths secton HC 1 C[u, v] Γ. Theorem 5.1. [KaLo] Let A be an assocatve agebra over C. The unversa centra extenson ŝ na of s n A s the Le agebra generated by eements F ab p, 1 a b n, p A, satsfyng the foowng reatons: F ab t 1 p 1 + t p t 1 F ab p 1 + t F ab p t 1, t C, p 1, p A 8 [F ab p 1, F bc p ] F ac p 1 p f a b c a 9 [F ab p 1, F cd p ] 0 f a b c d a 30 We w need to smpfy theorem 5.1 when A C[u, v] Γ. A generazaton of the foowng emma, under the extra condton that n 5, s gven by proposton 3.3 n [Gu3]. Lemma 5.1. The unversa centra extenson ŝ nc[u, v] Γ s somorphc to the Le agebra e generated by eements E ab w, w tu + sv, E ab γ, 1 a b n, γ Γ, s, t C such that the foowng reatons hod: If a b c a d c and w t u + s v, 1, : If a b c d a: E ab w te ab u + se ab v, [E ab w 1, E bc w ] [E ad w, E dc w 1 ], 31 [E ab γ, E bc w] [E ad γw, E dc γ], [E ab γ 1, E bc γ ] E ac γ 1 γ 3 [E ab w 1, E cd w ] 0 [E ab γ 1, E cd γ ] [E ab γ, E cd w] 33 Proof. We w ntroduce eements E ab q for any q C[u, v] Γ and show that they satsfy the reatons n theorem 5.1. When q C[u, v], the eements E ab q can be defned usng the map ŝ n[u, v] e gven by emma.5 and theorem 5.1 n the case A C[u, v]. Suppose q pγ, p C[u, v], γ Γ. We can assume that p u e1 v e. Set e e 1 + e ; we w use nducton on e. Choose a b and c a, b; set E ab q [E ac p, E cb γ]. We cam that ths defnton of E ab q does not depend on the choce of c. Ths s true when the degree of p s one accordng to 3. Indeed, suppose that d a, b, c and e 1, e 1 the cases e 1 0 or e 0 are smar and wrte E ac p [E ad v, E dc u e1 v e 1 ]. Argung by nducton, we can assume that [E db u e1 v e 1, E db γ] 0. Then E ab q [ [E ad v, E dc u e1 v e 1 ], E cb γ ] [ E ad v, [E dc u e1 v e 1, E cb γ] ] [ E ad v, [ E dc u e1 v e 1, [E cd, E db γ] ]] [E ad v, [ [E dc u e1 v e 1, E cd ], E db γ ]] [E ad u e1 v e, E db γ] + [ [E dc u e1 v e 1, E cd ], E ab vγ ] [E ad u e1 v e, E db γ] 7

8 snce [E dc u e1 v e 1, E ab vγ] 1 [ [Hdc u, E dc u e1 1 v e 1 ], E ab vγ ] 0 by nducton. Let us assume that [E ab p 1 γ 1, E bc p γ ] E ac p 1 γ 1 p γ 1 γ for any a b c a and aso that [E ab p 1 γ 1, E cd p γ ] 0 for any a b c d a, any γ 1, γ Γ and any p 1, p C[u, v] of tota degree < e. We want to prove that the same reaton hod when the tota degree of p 1, p s e. Step 1: Suppose that a, b, c, d are a dstnct and e 1 1. Set p u e1 1 v e, so p u p. Usng nducton, we get [E ab γ, E bc p] [ E ab γ, [E bd u, E dc p] ] [ [E ab γ, E bd u], E dc p ] [ [E ab γu, E bd γ], E dc p ] [ E ab γu, [E bd γ, E dc p] ] [E ab γu, E bc γ pγ] [ E ab γu, [E bd γ p, E dc γ] ] [ [E ab γu, E bd γ p], E dc γ ] [E ad γu p, E dc γ] E ac γpγ Step : Assume that a b c d a and e 1 1. There are three subcases to consder: a, b, c, d are a dstnct, a c, b d. In the frst subcase, [E ab p, E cd γ] 1 [ [Hab u, E ab p], E cd γ ] 0 by nducton. In the second subcase, choose e a, c, d; then [E ab p, E ad γ] [[E ae u, E eb p], E ad γ] 0 by nducton snce deg p < e. The thrd subcase s smar to the second one. Step 3: Choose a, b, c, d a dstnct. We now from step that [E ab p 1, E dc γ ] 0, so [E ab p 1, E bc p γ ] [ E ab p 1, [E bd p, E dc γ ] ] [ [E ab p 1, E bd p ], E dc γ ] [E ad p 1 p, E dc γ ] E ac p 1 p γ Step : Agan, suppose that a, b, c, d are a dstnct. [E ab p 1 γ 1, E bc p ] [ [E ad p 1, E db γ 1 ], E bc p ] [ E ad p 1, [E db γ 1, E bc p ] ] The ast equaty s a consequence of step 3. [E ad p 1, E dc γ 1 p γ 1 ] E ac p 1 γ 1 p γ 1 Step 5 : Assume that a b c d a. As n step, there are three subcases to consder. In the frst subcase, usng step twce, we get [E ab p 1 γ 1, E cd γ ] [ [E ad p 1, E db γ 1 ], E cd γ ] [E ad p 1, E cb γ γ 1 ] 0. In the second subcase, choosng e a, b, d, we get [E ab p 1 γ 1, E ac γ ] [ [E ae p 1, E eb γ 1 ], E ad γ ] 0. The thrd subcase s smar to the second one. Step 6: Suppose that a, b, c, d are a dstnct. [E ab p 1 γ 1, E bc p γ ] [ E ab p 1 γ 1, [E bd p, E dc γ ] ] [ [E ab p 1 γ 1, E bd p ], E dc γ ] [E ad p 1 γ 1 p γ 1, E dc γ ] [ [E ab p 1 γ 1 p, E bd γ 1 ], E dc γ ] [ E ab p 1 γ 1 p, [E bd γ 1, E dc γ ] ] [E ab p 1 γ 1 p, E bc γ 1 γ ] E ac p 1 γ 1 p γ 1 γ Step 7: Fnay, suppose that a b c d a and q 1 p 1 γ 1, q p γ. As n step and 5, there are three subcases. In the frst case, [E ab q 1, E cd q ] [ [E ac q 1, E cb ], E cd q ] [E ad q 1 q, E cb ] [ [E ab q 1, E bd q ], E cb ] [Eab q 1, E cd q ] Comparng that frst and ast terms, we concude that [E ab q 1, E cd q ] 0. In the second case, suppose that a, b, d are a dstnct and choose e a, b, d. Then [E ab q 1, E ad q ] [[E ae, E eb q 1 ], E ad q ] 0 by the prevous subcase and step. The thrd case can be handed as the second one. 8

9 As recaed earer, the center of ŝ nc[u, v] Γ s nown to be somorphc to HC 1 C[u, v] Γ, see [KaLo]. Consequenty, the foowng proposton w be usefu: Proposton 5.1. The frst cycc homoogy group of C[u, v] Γ s somorphc to Ω 1 C[u, v] Γ /dc[u, v] Γ, the quotent of the space of Γ-nvarant 1-forms on the compex affne pane by the space of exact forms comng from Γ-nvarant poynomas. Proof. It s proved n [Fa] that the Hochschd homoogy of C[u, v] Γ s gven by: HH 0 C[u, v] Γ C[u, v] Γ C cγ 1, HH 1 C[u, v] Γ C[u, v] C U Γ where U span{u, v} C HH C[u, v] Γ C[u, v] Γ, HH C[u, v] Γ 0 for 3. Here, cγ s the number of conugacy casses of Γ. There exsts an exact sequence HH 0 C[u, v] Γ HH 1 C[u, v] Γ HC 1 C[u, v] Γ 0 and the frst map s gven by the dfferenta d, the space C[u, v] U beng dentfed wth the space of reguar 1-forms on C U. Proposton 5.. The frst cycc homoogy group of C[u, v] Γ can be dentfed as a vector space wth C[u, v] Γ. Proof. The form ω see secton 3 aows us to dentfy Ω 1 C[u, v] wth the space V of poynoma vector feds on C. We assume here that ωu, v 1. There s a contracton map V C[u, v], so we can defne a near map Ω 1 C[u, v] C[u, v] gven expcty by u s v r du ru s v r 1, u s v r dv su s 1 v r. Snce ω s Γ-nvarant, ths restrcts to a surectve map Ω 1 C[u, v] Γ C[u, v] Γ and the erne of ths ast map s dc[u, v] Γ, whch equas dc[u, v] Γ. Thus, HC 1 C[u, v] Γ C[u, v] Γ. These two propostons suggest that t may be possbe to reate the enveopng agebra of ŝ nc[u, v] Γ to Ug n A 1 Γ where A 1 s the frst Wey agebra: ths s expaned n secton 8. In the ast secton, we w consder deformatons of an agebra reated to Uŝ nc[u, v] Γ, namey Uŝ nc[u ±1, w] Γ when Γ Z/dZ s cycc, acts trvay on w and on u by ξu ζu, ξ beng a generator of Z/dZ and ζ a prmtve d th -root of unty. It s expaned n [GHL] that s n C[u ±1, w] Γ s nd C[s ±1, w]: ths foows from the somorphsm of assocatve agebras gven n oc. ct. C[u ±1 ] Γ M d C[s ±1 ] where s u d. It foows that HC 0 C[u ±1, w] Γ C[s ±1, w] C[u ±1, w] Γ and HC 1 C[u ±1, w] Γ Ω1 C[s ±1, w] dc[s ±1, w] C s 1 ds C[s ±1, w]wds. Let g be the Le agebra defned by the reatons n defnton 10.5 when λ 0, β 0. The next proposton w be usefu to understand the agebras n secton 10. Proposton 5.3. The Le agebra g s somorphc to ŝ nc[u ±1, w] Γ. Proof. See proposton. n [GHL] and aso [MRY]. An somorphsm s gven by For 1 n 1, X ±1,r, E± w r e, H,r, H w r e X + 0,r, E n1 w r ue, X 0,r, E 1n w r u 1 e, H 0,r, E nn w r e E 11 w r e +1 + δ 0 s 1 w r ds Here, we dentfy the center of s n C[u ±1, w] Γ wth HC 1 C[u ±1, w] Γ as above. Let a be the Le subagebra of g generated by X ±,r,, H,r,, X + 0,r,, X 0,r+1, for 1 n 1, r 0, 0 d 1. Va the somorphsm g ŝ nc[u ±1, w] Γ, we see that a contans ŝ nc[u, v] Γ wth v u 1 w. As we have mentoned earer, we are nterested n deformatons of the enveopng agebra 9

10 of a Le agebra sghty bgger than ŝ nc[u, v] Γ. The proecton g g/ n 1 nectve on a note that n d 1 0 H,0, g s d 1 0 H,0, s a centra eement of g, so we can vew ŝ nc[u, v] Γ as contaned n the Le subagebra ã of g whch s the mage of a under the prevous proecton. The Le subagebra of g generated by X ±,0,, H,0,, H 0,0,, X + 0,0, for 1 n 1, 0 d 1 s arger than s n C[u] Γ. Here, denotes the mage under the proecton g g. For nstance, n 1 H 0,0, E 11 e e +1 s n C[u] Γ. Note that span{e e +1 0 d 1} d 1 1 C ξ C[Γ]. In the ast secton, we w expan how Γ-DDCA n the cycotomc case are deformatons of Uã. 6 Γ-deformed doube current agebras We ntroduce n ths secton a new famy of quantum agebras whch are deformatons of the enveopng agebra of an enargement of ŝ nc[u, v] Γ and whch are reated to sympectc refecton agebras for wreath products of Γ va a functor of Schur-Wey type. By enargement of a Le agebra g, we mean a Le agebra ǧ whch contans a. Before defnng them, we need to ntroduce the Le agebra g n C[Γ] whch s the Le subagebra of g n C[Γ] spanned by s n C[Γ] and by the eements E aa γ γ Γ\{d}, 1 a n. The necessty to consder g n C[Γ] nstead of ust s n C[Γ] w become cear n sectons 7 and 8. We w use the foowng notaton: gven an agebra A and eements a 1, a A, we w set Sa 1, a a 1 a + a a 1. Defnton 6.1. The Γ-deformed doube current agebra D n β,b wth parameters β C, b ZΓ, b λ d + \{d} b γγ s the agebra generated by the eements of g n C[Γ], E ab t 1 w 1 + t w for 1 a b n, t 1, t C, w, w 1, w U whch satsfy E ab t 1 w 1 + t w t 1 E ab w 1 + t E ab w and the foowng reatons: If a b c a d c, [E ab γ, E bc w] [E ad γw, E dc γ], [E aa γ, E ac w] [E ab γ, E bc w] [E ac γw, E cc γ] 3 [E ab w, E bc w 1 ] [E ad w 1, E dc w ] + ωw 1, w E ac b + β + λ 8 ωw 1, w,1 S [E ab γ 1, E ], [E, E bc γ] + S [E ad γ, E ], [E, E dc γ 1 ] λ ωγw 1, w ωw 1, w E bb γ 1 E ac γ + E dd γe ac γ 1 35 If a, b, c are a dstnct, [E cc γ, E ab w] 0, and f a b c d a, then [E ab γ, E cd w] 0 and [E ab w 1, E cd w ] λ ωγw 1, w S E ad γ 1, E cb γ 36 n Set b \{d} b γγ, so b bλ 0. Let D n be the subagebra of β, b Dn β,bλ0 generated by the eements E ab w 1, E ab w, E ab γ for a b. Lemma 5.1 says that D n β0,b0 s somorphc to the enveopng agebra of ŝ nc[u, v] Γ. When β 1, λ 0 b γ for γ d, D n β1,b0 s exacty the enveopng agebra of g n A 1 Γ where A 1 s the frst Wey agebra. D n β0,b0 s the enveopng agebra of a Le agebra that we denote š nc[u, v] Γ and we have š nc[u, v] Γ ŝ nc[u, v] Γ. See secton 8 for more detas. When Γ s the trva group, D n β,b s somorphc to the agebra Dn λ,β n [Gu] see aso [Gu1]. The man dfference n the defntons of D n β,b and Dn λ,β s that the former does not nvove any Yangan. 10

11 Actuay, there does not seem to be any sensbe noton of Yangan assocated to Γ n genera, whch, from a heurstc pont of vew, s not surprsng snce, when Γ s not cycc, C s not the drect sum of two one-dmensona Γ-nvarant subspaces. When Γ s a fnte cycc group, we can gve another defnton of D n β,b whch nvoves Yangans: see secton 10. By gvng degree 0 to the eements of g n C[Γ] and degree one to E ab w, w U, we can defne a ftraton F on D n β,b such that Dn β0,b0 gr F D n β,b. The PBW theorem n secton 9 says that ths canonca map s an somorphsm. 7 Schur-Wey functor and equvaence of categores Gven a rght modue M over H t,c Γ, we set SWM M C[S ] C n. We woud e to gve SWM a structure of eft modue over D n β,b. The acton of E ab s smpy va the s n -modue structure on C n. Let us assume that u, v, x, y U are such that the map u x, v y s a Γ-equvarant automorphsm of the sympectc vector space U, so that ωγu, v ωγx, y. In partcuar, {u, v} s a sympectc bass of C. We woud e to et E ab w, E ab γ D n λ,β act on SWM n the foowng way: E ab wm v mw E ab v, E abγm v Here, w t 1 x + t y f w t 1 x + t y, v v 1 v C n and mγ 1 E ab v v 1 v 1 E ab v v +1 v. E ab v These operators defne a representaton of D n β,b on SWM f and ony f the foowng reatons hod between t, c, λ, b: λ κ, b γ c γ 1 for γ d and β t nκ Γ κ. To prove our cam, we have to verfy that the operators above satsfy the defnng reatons of D n β,b. We start by computng that, for a b c a d c, [Eab u, E bc u] [E ad u, E dc u] m v, κ κ κ mx x x x E ab E bc E ad E dc v ωγmσ x γ γ 1,, ωγmγ x γ 1, ωγmγ x γ 1 E ab E bc E ad E dc v E bb E ac E dd E ac v H bd E ac v κ ωγh x bd γ 1 E ac γm v The computatons are the same when u s repaced by v and ω x γ s repaced by ω y γ. Under the same assumpton on a, b, c, d, we now compute: 11

12 [Eab v, E bc u] [E ad u, E dc v] m v mx y y x E ac v + κ m t + κ, E ac t + κ, ω x,y γ, \{d}, γ γ 1 σ + m[x, y ] E ab E bc v \{d} mγ γ 1 σ E ab E bc v κ c γ γ 1 m v + κ ωγ x,y mγ γ 1 E bb E ac v κ c γ γ E ac v, n e1, ω x,y γ mγ γ 1, m[y, x ] E ad E dc v mγ γ 1 σ E ad E dc v E ae E ec v ωγ x,y mγ γ 1 E dd E ac v E ac t + c γ γ 1 κn Γ m v + κ n S E ae γ 1, E ec γ m v \{d} e1,e a,c + κ S H ab γ 1, E ac γ + S H cd γ, E ac γ 1 + S H ad γ, E ac γ S H cb γ 1, E ac γ m v κ ω x,y We now chec that [E ab γ, E bc u] [E ad γu, E dc γ] m v 0, m x γ 1 γ 1 γ x E ac v + m[γ 1, γ x ] E ad E dc v γ 1 E bb γ 1 E ac γ + E dd γe ac γ 1 m v, m[x, γ 1 ] E ab E bc v snce γ 1 γ x x γ 1 n H t,c Γ and γ x x γ, γ Γ f. Exacty the same computatons wor wth v nstead of u and n the case a b or c d wth γ d. Now, we w assume nstead that a b c d a: [E ab u, E cd v]m v κ m[y, x ] E ab E cd v κ ωγ x,y mγ γ 1 E cb E ad v κ ωγ x,y mσ γ γ 1 E ab E cd v ω x,y γ E cb γe ad γ 1 m v The computatons are the same for [E ab u, E cd u]m v and [E ab v, E cd v]m v wth ω x γ resp. ω y γ nstead of ω x,y γ. We can state what we have proved so far n ths secton, but before that we need a defnton. 1

13 Defnton 7.1. A modue over D n β,b s caed ntegrabe f t s a drect sum of ntegra weght spaces under the acton of h and s ocay npotent under the acton of E ab w for any 1 a b n and any w U. Defnton 7.. A modue over D n β,b s sad to be of eve f, as a modue over s n, t decomposes as a drect sum of rreducbe s n -submodues of C n. Proposton 7.1. Suppose that λ κ, β t κn Γ κ and b γ c γ 1 for γ d. Then there exsts a functor SW : mod R H t,c Γ mod nt, L D n β,b gven by SWM M C[S ] C n. Here, mod nt, L s the category of ntegrabe eft modues of eve. Ths proposton can be strengthened to yed a new generazaton of the cassca Schur-Wey duaty theorem between s n and S. Theorem 7.1. Suppose that λ κ, β t κn Γ κ and b γ c γ 1 for γ d. If + < n, then the functor SW yeds an equvaence between the category of rght H t,c Γ -modues and the category of eft modues over D n β,b whch are ntegrabe of eve. The proof of ths theorem w foow the same nes as the anaogous resut n [Gu1],[Gu] see aso [ChPr1], [VaVa1]. However, before provng t, we have to estabsh a smar resut for Γ and s n C[Γ]. When Γ s the cycc group Z/dZ, a more genera resut was estabshed n [ATY] n the context of cycotomc Hece agebras and where s n C[Γ] s d n s repaced by g m1 g md. We w need the foowng emma. Lemma 7.1. If v v 1 v s a generator of C n as a modue over s n e.g. f for any, then m v 0 m 0. Proposton 7.. The functor SW : mod R C[Γ ] mod L s n C[Γ] gven by M M C[S ] C n whch s an equvaence of categores of fnte dmensona modues when + 1 < n. Proof. Gven a rght Γ -modue M, we can put on M C[S ]C n a structure of eft modue over s n C[Γ] by settng E ab γm v mγ 1 E ab v. Ths extends the cassca Schur-Wey functor to mod R C[Γ ] and mod L s n C[Γ]. The second part of the proposton requres more wor; to prove t, we w foow the approach and deas n [ChPr1]. Suppose that + 1 < n. Let N be a eft s n C[Γ]-modue whch s of eve as s n -modue. Then N M C[S ] C n as eft s n -modue for some rght S -modue M by the cassca case. We want to show that M s a rght modue over the group Γ. For 1, set v v v v n v +1 v where {v 1, v,..., v n } s the standard bass of C n. Let w be the same eement of C n as v except that v n s repaced by v 1. As n [ChPr1], we wrte w τ for the eement obtaned by permutng the factors of w by τ S. The set {w τ τ S } s a bass for the subspace of C n of weght λ ɛ ɛ where ɛ s the weght on dagona matrces gven by E δ, so we can wrte E 1n γm v τ S m τ w τ for some m τ M. Ths can be rewrtten as E 1n γm v m w for some m M. By emma 7.1 above, m s unque, so there exsts a near endomorphsm ζ γ, 1n of M such that m ζ γ, 1n m for a m M. 13

14 One can show, exacty as n emma.5 n [ChPr1], that E 1n γm v n ζγ, 1n m E 1n v for [ any v C n. Instead of the quantzed Serre reaton that they use, one shoud consder the reaton En,n 1, [E n,n 1, E 1n γ] ] 0, whch s a consequence of [E 1n γ, E n,n 1 ] [E 1,n γ, E n,n 1 ]. Smary, t s possbe to show aso that, for any 1 a b n, there exsts an endomorphsm ζ γ, ab End C M such that E ab γm v ζγ, ab m E ab v. We cam that, for any choce of a b, c d, ζ γ, ab ζ γ, cd. Suppose, for nstance, that b d a, Then E adγ [E ab γ, E bd ], so E ad γm v [E ab γ, E bd ]m v ζ γ, ab m E ad v. Snce ths s true for any v C n, m M, ζ γ, ab ζ γ, ad. The other cases can be treated smary. Therefore, we can defne ζ γ, End C M unambguousy by settng ζ γ, ζ γ, ab for any choce of a b. We can now show that settng mγ ζ γ 1, m gves M a structure of rght modue over Γ. We w prove the foowng reatons: 1. mγ γ mγ γ, γ, γ Γ.. mγ γ m γ γ γ, γ Γ f. 3. mσ γ mγ σ, 1, γ Γ. 1: Set v v v v n v +1 v and ṽ v v v 1 v +1 v Snce [E 1,n 1 γ 1, E n 1,n γ 1 ] E 1n γ 1 γ 1, we obtan [E 1,n 1 γ 1, E n 1,n γ 1 ]m v mγ γ ṽ E1n γ 1 γ 1 m v. Ths equaty, aong wth emma 7.1, mpy that mγ γ mγ γ, whch s what we wanted. : Suppose that 1 <. Set v v 3 v +1 v n v + v v n 1 v +1 v and ṽ v 3 v +1 v 1 v + v v v +1 v. Snce [E 1n γ 1, E,n 1 γ 1 ] 0, we get 0 [E 1n γ 1, E,n 1 γ 1 ]m v mγ γ ṽ m γ γ ṽ, so, by emma 7.1, mγ γ m γ γ. 3: Set v v v v n v +1 v 1 v n 1 v +1 v and ṽ σ v; et v be the same as v except that v n s repaced by v 1 and set v σ v. Then mσ γ v E n1 γ 1 mσ v E n1 γ 1 m v mγ ṽ mγ σ v. Agan, emma 7.1 aows us to concude that mσ γ mγ σ. Fnay, one can chec that the functor F s bectve on sets of morphsms. Proof of theorem 7.1. Let N be a eft modue over D n β,b whch s ntegrabe and of eve. Proposton 7. says that N M C[S ] C n for some rght Γ -modue M. We have to extend ths to a rght modue structure over H t,c Γ. We can proceed exacty as n the proof of proposton 7. mmcng the arguments n [ChPr1] to show that there exst endomorphsms ζ w End CM such that E ab wm v ζw m E ab v. We proceed as n [Gu] to show that settng mx ζ um, my ζ v m turns M nto a modue over H t,c Γ. 1

15 Fx 1,,. Choose v v 1 v such that, n 1, r r + f r <, r, r r + 1 f r >, r. Set ṽ E n E 1,n 1 v. On the one hand, E1,n 1 w E n w 1 E n w 1 E 1,n 1 w m v so s1 r1 mw 1 rw s E s 1,n 1 Er n v s1 r1 mw sw 1 r E r n Es 1,n 1 v m w 1 w w w 1 ṽ Usng reaton 36 for E ab E n and E cd E 1,n 1, we fnd that [E 1,n 1 w, E n w 1 ] λ ωγw 1, w E n,n 1 γ 1 E 1 γ, [E 1,n 1 w, E n w 1 ]m v λ λ ωγw 1, w mγ 1 γ E n,n 1 E 1 v ωγw, w 1 mγ γ 1 σ ṽ. Therefore, m w 1w w w1 + λ ωγw 1, w σ γ γ 1 ṽ 0. From emma 7.1 and our assumpton that λ κ, we deduce that m w 1w w w1 + κ ωγx, yσ γ γ 1 0. We use equaton 35 n the case a, b n, 1, c, d n 1, 1. [E n1 v, E 1,n 1 u] [E n1 u, E 1,n 1 v] s equa to E n,n 1 b + β + λ n SE n γ, E,n 1 γ 1 + λ S H n1 γ, E n,n 1 γ 1 1 It mpes that the dfference + S E n,n 1 γ, H n 1,1 γ 1 λ ωγu, v 1 E 11 γ 1 E n,n 1 γ + E 11 γe n,n 1 γ 1 Now fx and et v be determned by n 1, + 1 f. Set v E n,n 1 v. Appyng both sdes of the prevous equaty to m v, we deduce that κ mx y y x v m β + λ + b γ γ 1 v + λn Γ m v + λ γ d n, n, mσ γ γ 1 Lemma 7.1 and our assumpton that λ κ, β t κn Γ κ, b γ c γ 1 mpy that [x, y ] t + σ γ γ 1 + c γ γ. γ d That the functor SW s bectve on sets of morphsms foows from the cassca Schur-Wey duaty and emma 7.1. v 8 Specazaton at λ 0 In [Gu], we proved that, when the parameters λ 0 and β 0, the deformed doube current agebra s somorphc to the enveopng agebra of the Le agebra g n over the frst Wey agebra, whch s a sympectc refecton agebra of ran one for the trva group Γ {1}. Therefore, t s natura to conecture that, for an arbtrary fnte subgroup Γ of SL C, a smar resut s true, the frst Wey agebra beng repaced by a sympectc refecton agebra of ran one for Γ. Theorem 8.1 confrms ths. 15

16 Defnton 8.1. [CBHo] Set c \{d} c γγ. Let A t, c be the agebra generated by the eements x, y, γ Γ and satsfyng the reatons γ x γ 1 γx, γ y γ 1 γy and xy yx t + c. Here, span C {x, y} C and Γ thus acts on span C {x, y}. We reca that the Le agebra s n A t, c s defned as the Le subagebra of g n A t, c of matrces wth trace n [A t, c, A t, c ]. As a vector space, s n A t, c s n C A t, c d [A t, c, A t, c ], where d [A t, c, A t, c ] s the space of scaar matrces n g n A t, c wth entres n [A t, c, A t, c ]. Lemma 8.1. For a c and a t C except a countabe set, the Le agebra s n A t, c s unversay cosed, that s, t s ts own unversa centra extenson. Proof. Theorem 1.7 n [KaLo] states the the center of the unversa centra extenson of s n A, where A s an arbtrary assocatve agebra, s somorphc to the frst cycc homoogy group HC 1 A. It shown n [EtG] that the frst Hochschd homoogy group HH 1 A t, c vanshes for a c and a t C outsde a countabe set. The groupe HC 1 A t, c s a quotent of HH 1 A t, c, so t vanshes aso. Theorem 8.1. Suppose that β t and b γ c γ 1 for γ d. Then the agebra D n s somorphc to the β, b enveopng agebra of the Le agebra ŝ na t, c, the unversa centra extenson of s n A t, c. Proof. It foows from the defnton of D n β, b and theorem 5.1 that Uŝ na t, c s a quotent of D n β,b. To prove that the quotent map s an somorphsm, we construct eements E ab p D n β, b for 1 a b n and any p A t, c and show that they satsfy the reatons n theorem 5.1. We w gve a proof when n 5; t ustrates how the cacuatons are sometmes smper when n 5. Let g be the Le agebra defned by the reatons n defnton 6.1 wth λ 0. Lemma. gves us homomorphsms s n [v] g, s n [u] g. Defne E ab v γ [E ac v, E cb γ] for γ d, a, b, c a dstnct, and set nductvey E ab u v γ [E ac u, E cb u 1 v γ] for some c a, b and for, 1. We defne E ab p by nearty when p s a sum of monomas. We have to show that [E ab p 1, E bc p ] E ac p 1 p f a b c a and [E ab p 1, E cd p ] 0 f a b c d a for any p 1 u 1 v 1 γ 1, p u v γ. The frst step, however, s to show that the defnton of E ab u v γ does not depend on the choce of c. Snce we are assumng that n 5, choose d, e such that a, b, c, d, e are a dstnct and assume that. The case 1, 1 s smar. Then [E ac u, E cb u 1 v γ] [E ac u, [ E cd, [E de u, E eb u v γ] ]] [ [E ac u, E cd ], [E de u, E eb u v γ] ] [E ad u, E db u v γ] The arguments used are smar to those n the proofs of emma.,.5 and 5.1. We proceed agan by nducton on degp 1 + degp to prove the two equates above, whch hod when degp 1 + degp 1. If a b c d a, choose e a, b, c, d. Wthout oss of generaty, we can suppose that p 1 u p 1 and degp 1. Then [E ab p 1, E cd p ] [ [E ae u, E eb p 1 ], E cd p ] 0 by nducton. If p 1 u p 1 wth deg p 1 1, choose a, b, c, d a dstnct, so that, by nducton, [E ab p 1, E bc p ] [ [E ad u, E db p 1 ], E bc p ] [ E ad u, [E db p 1 ], E bc p ] ] [E ad u, E dc p 1 p ] E ac u p 1 p E ac p 1 p If p 1 v r wth r 1 and p u p, then choose a, b, c, d, e a dstnct, so that [E ab p 1, E bc p ] [ [E ad v, E db v r 1 ], E bc p ] [ E ad v, [E db v r 1, E bc p ] ] [E ad v, E dc v r 1 p ] [E ad v, E dc uv r 1 p + E dc [v r 1, u] p ] 16

17 [ E ad v, [E de u, E ec v r 1 p ] ] + [E ad v, E dc [v r 1, u] p ] [ [E ad v, E de u], E ec v r 1 p ] + E ac v[v r 1, u] p [ [E ad u, E de v], E ec v r 1 p ] [E ae c, E ec v r 1 p ] + E ac v[v r 1, u] p E ac uv r p E ac cv r 1 p + E ac v[v r 1, u] p E ac p 1 p If p 1 p 1 γ wth p 1 a monoma n u, v, p u p and a, b, c, d, e are a dstnct, then [E ab p 1, E bc p ] [ [E ae p 1, E eb γ], [E bd u, E dc p ] ] [E ae p 1, [ [E eb γ, E bd u], E dc p ]] [ E ae p 1, [ [E eb γu, E bd γ], E dc p ]] [E ae p 1, E ec γu p γ] [E ae p 1, E ec γp γ] E ac p 1 p The ast ne foows from the prevous cases snce p 1 s assumed to be a monoma n u, v. Coroary 8.1. For a bλ 0 and a β C outsde a countabe set, the agebra D n β,bλ0 s somorphc to the enveopng agebra of the Le agebra g n A t, c wth t β, c γ b γ 1 for γ d. Ths s true, n partcuar, when λ 0 b γ for γ d and β 0. Proof. Lemma 8.1 and theorem 8.1 mpy that D n s somorphc to s β, b na t, c wth t β, c γ b γ 1 for γ d. The somorphsm gven n theorem 8.1 can be extended to D n β,bλ0 and g na t, c by sendng E aa γ to E aa γ for γ d. Note that gna t, c s na t, c A t, c [A t, c,a t, c ] HC 0A t, c and t s proved n [EtG] that dm C A t, c cγ 1 for generc vaues of the parameters n the same sense as above. 9 PBW bases We foow the same approach as n [Gu] to prove that a Γ-deformed doube current agebra admts a vector space bass of PBW type. Ths can be formuated by sayng that the map D n β0,b0 gr F D n β,b s an somorphsm. We w construct nductvey a vector space bass of D n β,b whch yeds the natura PBW bass on gr F D n β,b Uš nc[u, v] Γ. We mae the same assumpton on u, v, x, y as n secton 7. We need to assume that β + λ λn Γ 0 n ths secton. We need to choose n C[u, v] a Γ-nvarant space E compementary to C[u, v] Γ, so that C[u, v] C[u, v] Γ E as Γ-modues. We can suppose that E m 1 E[m] s graded by the degree of the monomas and E1 U. Let us assume that we have constructed eements F ab p D n β,b for a p E span{ pγ p C[u, v], γ d} of degree N 1 a, b n and aso p C[u, v] Γ + C[Γ] of degree N 1 a, b n, such that F ab ph v hpx, y, γ E ab v f h H t,cγ, v C n. Ths s aready nown to hod for N 1. We use the notaton F ab p nstead of E ab p because we must set F ab γ E ab γ 1, F ab t 1 u + t v E ab t 1 u + t v, t 1, t C and F ab wγ [E ac γ 1, E cb w] for some c a, b. We want to construct by nducton such eements F ab p for any p C[u, v] Γ, 1 a, b n. Set H ab p [F ab p, F ba ] for 1 a b n f F ab p has aready been defned. Let pu, v C[u, v] Γ, pu, v 0 be a poynoma of degree N 1 we can assume that pu, v s homogeneous. In the computatons beow for P h v, we w not need to use that pu, v s Γ-nvarant. However, we have to start wth ths case n the nducton step. Suppose that pu, v r,s 0 cp r,sru s v r 1 c p r,ssu s 1 v r where c p r,s, c p r,s C and c p r,s 0 cp, r, s f r + s f. The proof of proposton 5. suggests that we consder the foowng eements: 17

18 P c p r,s [H u s v r, H u] + c p r,s[h u s v r, H v] and P 1 r,s 0 n P. 1 P h v n 1 r,s 0 n 1 r,s 0, r,s 0 n 1 r,s 0 d0 n c p r,s h[x, x s y] r c p r,sh[x s y, r y ] E + E +1,+1 v c p r,s d0 s 1 1 r,s 0, t + r,s 0 c p r,s r,s 0 d0 c p r,s h[x, x s y r ] c p r,sh[x s y r, y ] H r 1 c p r,s, r 1 d0 hx s y d [x, y ]y r d 1 c p r,s d0 hxd [x, x ]x s d 1 y r cp r,s H v s 1 hx d [x, y ]x s d 1 y r hxd [x, y ]x s d 1 c p r,s hxs y d [x, y ]y r d 1 cp r,s hxs y d [y, y ]y r d 1 c p r,s rhx s y r 1 r 1 κ s 1 κ c p r,s r,s 0 d0 κ H κ H t,, 1 r,s 0 d0 c p r,sshx s 1 y r v hx s yσ d γ γ 1 y r d 1 + \{d} hx d σ γ γ 1 x s d 1 y r + \{d} n s 1 hωγc x p r,sx d σ γ γ 1 v n r 1 H 1 r,s 0 d0 + H v hpx, y v r,s 0 \{d} + κ c p r,s r,s 0, hω x,y γ d0 x s d 1 c p r,sx s y d σ γ γ 1 y r d 1 y r c p r,sω x,y d0 y r v H H H v H v c γ hx s yγ d y r d 1 v c γ hx d γ x s d 1 γ y r v x d σ γ γ 1 xs d 1 y r ωγ c y p r,sx s y d σ γ γ 1 yr d 1 r 1 s 1 c γ c p r,s hx s yγy d r d 1 γ c p r,s hx d γx s d 1 yγ r n e,1 r 1 c p r,s d0 s 1 hx d γx s d 1 y r γ γ 1 d0 hx s yγy d r d 1 γ γ 1 E e E e v 18 v

19 t κ κ r,s 0 s 1, d0 1 n c p r,sωγ x c p r,sω x,y γ hx d 1 γ 1 x s d 1 yγ r γ 1 E E 1 E,+1 E +1, 1 E +1, E,+1 r,s 0 r 1, n d0 1 c p r,sω x,y γ E E 1 E,+1 E +1, 1 E +1, E,+1 hpx, y v κn + κ + + κ κ r 1 r,s 0 d0 e,1 e r 1 r,s 0 d0 \{d} 1 + κn κ κ + κ r 1 r,s 0 d0 \{d} 1 s 1 r,s 0 d0 e,1 e s 1 r,s 0 d0 s 1 r 1 v c p r,sω y γ hx s y d 1 γ 1 y r d 1 γ γ 1 r,s 0 d0 v c p r,shx s y r 1 v n c p r,sf e γv r d 1 γf e u s v d γ 1 h v 37 n c γ c p r,s F u s v d γv r d 1 γ + c p r,sf u d γu s d 1 v r γ h v 38 n c p r,s1 ω x,y γ 1 F γv r d 1 γ, F u s v d γ 1 h v 39 n c p r,sf e γu s d 1 v r γf e u d γ 1 h v 0 n r,s 0 d0 \{d} 1 s 1 c p r,shx s 1 y r v n r,s 0 d0 \{d} 1 s 1 r,s 0 d0 1 c p r,s1 ω x,y γ F γu s d 1 v r γf u d γ 1 h v 1 n c p r,sωγ F γu s d 1 v r γf u d γ 1 h v n c p r,sωγ x 1 cp r,sωγ x,y F +1, γu s d 1 v r γf,+1 u d γ 1 +F,+1 γu s d 1 v r γf +1, u d γ 1 h v 3 + κ r 1 n c p r,sωγf y γv r d 1 γf u s v d γ 1 h v + κ r,s 0 d0 \{d} 1 r 1 r,s 0 d0 1 n c p r,sω x,y γ c p r,sωγ y F 1 +1, γv r d 1 γf,+1 u s v d γ 1 +F,+1 γv r d 1 γf +1, u s v d γ 1 h v 5 Set Ĩp P where 37 s the expresson on ne 37 but wthout h v, 1 and Ip β + λ nλ Γ Ĩp. Then Iph v hpx, y v. 19

20 Fnay, set F 11 p 1 n Ip + n n H,+1p, so F 11 ph v hpx, y E 11 v. We can then obtan F p wth the same property. We et F ab p [F aa p, F ab ] f 1 a b n. We must now construct eements F ab p p E span{ pγ p C[u, v], γ d} of degree N + 1 and 1 a, b n. Suppose frst that 1 a b n and put p u s v r γ wth r + s N + 1 and, wthout oss of generaty, r 1. Choose c a, b and set F ab u s v r [F ac v, F cb u s v r 1 ]. We compute that F ab u s v r h v equas + κ hx s y r E ab v +,, hx s y r E ab v + d0 h[x s y r 1, y ] E ac E cb v s 1 hx d [x, y ]x s 1 d y r 1 E ac E cb v, r hx s y[y d, y ]y r d hx s y r E ab v κ, d0 d0 E ac E cb v s 1, d0 ω x,y γ hx d σ γ γ 1 r ωγhx y s yσ d γ γ 1 y r d E ac E cb v x s 1 d y r 1 E ac E cb v κ hx s y r E ab v κ, d0 s 1, d0 ω x,y γ hx d γx s 1 d r ωγhx y s yγy d r d γ γ 1 E cc E ab v hx s y r E ab v s 1 γy r 1 γ γ 1 E cc E ab v κ ωγ x,y F cc γu s 1 d γv r 1 γf ab u d γ 1 6 d0 r + ωγf y ab u s v d γ 1, F cc γv r d γ h v 7 d0 Settng F ab u s v r F ab u s v r 6 7 where 6 s the expresson on ne 6 but wthout h v, we have obtaned an eement wth the requred property. For γ d, we can set F ab u s v r γ [E aa γ 1, F ab u s v r ] and H ab u s v r γ [F ab u s v r γ, E ba ]. If p pγ, γ d, then p s a sum of monomas u s v r of degree N + 1, so we can aso defne F ab p and H ab p. Suppose that γ Γ \ {d} and et uγ, vγ be a bass of U C consstng of egenvectors of γ necessary for non-trva egenvaues µγ, νγ, respectvey. The vector xγ, yγ are defned smary. For s 1, we set Duγ s vγ r 1 n γ 1 µγ 1 [H uγ s 1 vγ r γ, H uγ], whereas f 0

21 s 0 and r 1, we set Dvγ r γ 1 1 νγ n 1 [H vγ r 1 γ, H vγ]. Let us assume that s 1; then Duγ s vγ r γh v equas 1 1 µγ h[xγ, xγ s yγ r γ ] v + 1 hxγ s+1 yγ r µγ γ v + 1 µγ µγ 1 1 µγ n 1 n 1,, d0 n r 1 d0 n 1, s 1 hxγ d [xγ, xγ ]xγ s 1 d r 1 hxγ s yγ d [xγ, yγ ]yγ r 1 d d0 hxγ s+1 yγ r γ v + µγ 1 µγ κ 1 µγ κ 1 µγ n r 1 hxγ s yγ d t + κ d0 n 1 n 1 s 1, d0 r 1, d0 ω xγ γ ω xγ,yγ γ hxγ s+1 yγ r γ v + trµγ 1 µγ + κµγ 1 µγ + µγ 1 µγ κ 1 µγ r 1 d0 ω xγ,yγ γ r 1, r 1 d0 d0 \{d} s 1 n h 1 1 h[xγ, xγ s yγ r γ ] H H hxγ s yγ d [xγ, yγ ]yγ r 1 d γ v yγ r γ H γ H σ γ γ 1 + hxγ d σ γ γ 1 xγ s 1 d hxγ s yγ d σ γ γ 1 n hxγ s yγ d γyγ r 1 d hxγ s yγ r 1 γ v \{d} H v H v c γ γ yγ r 1 d γ v yγ r γ H yγ r 1 d γ 1 γ γ σ v c γ hxγ s yγ d γyγ r 1 d γ γ v d0 ω xγ γ xγ s yγ d γ 1 yγ r 1 d hxγ s+1 yγ r γ v γ H H v xγ d γ 1 xγ s 1 d γ 1 yγ r γ γ 1 γ + γ γ 1 γ H v v E E E,+1 E +1, E +1,,+1 E v + trµγ 1 µγ Iuγs vγ r 1 γh v n rf aa uγ s vγ r 1 γh v 8 a1 1

22 + κµγ 1 µγ + µγ r 1 1 µγ κ 1 µγ r 1 n d0 a,b1 d0 \{d} 1 d0 F ab γvγ r 1 d γγf ba uγ s vγ d γ 1 h v 9 c γ I uγ s vγ d γvγ r 1 d γγ h v 50 n s 1 ω xγ γ F γ 1 uγ s 1 d γ 1 vγ r γ 1 γf uγ d γ 51 F +1, γ 1 uγ s 1 d γ 1 vγ r γ 1 γf,+1 uγ d γ 5 F,+1 γ 1 uγ s 1 d γ 1 vγ r γ 1 γf +1, uγ d γ h v 53 κ 1 µγ n r 1 ω xγ,yγ γ F γ 1 vγ r 1 d γ 1 γf uγ s vγ d γ 5 1 d0 F +1, γvγ r 1 d γ 1 γf,+1 uγ s vγ d γ 55 F,+1 γvγ r 1 d γ 1 γf +1, uγ s vγ d γ h v 56 Set Duγ s vγ r γ Duγ s vγ r γ where 8 denotes the expresson on ne 8 but wthout h v. We defne F uγ s vγ r γ n the foowng way: F nn uγ s vγ r γ 1 n Duγ s vγ r γ n 1 1 H uγ s vγ r γ and, recursvey, F uγ s vγ r γ H uγ s vγ r γ + F +1,+1 uγ s vγ r γ for 1 n 1. If p pγ, γ d, then p s a sum of monomas u s v r of degree N + 1, and u s v r can be expressed unquey as a sum of monomas n uγ, vγ, so we can aso defne F p. Fnay, we shoud expan how to construct eements F p when p E[N + 1]. It s enough to consder the case when p s a monoma uγ s vγr on whch some eement γ d acts by the non-zero egenvaue µγ s r. We have ust seen how to defne F uγ s vγ r γ, and we set F uγ s vγ r 1 µ r s 1 [F uγ s vγ r γ, F γ 1 ]. We have thus constructed eements F ab p p C[u, v] Γ, 1 a, b n. Let B {F ab u s v r γ 1 a, b n, r, s 0, γ Γ} Let us fx an order on B. We can assume that F a1b 1 u s1 v r1 γ 1 < F ab u s v r γ f a 1 a and a b. Theorem 9.1. The canonca map D n β0,b0 gr F D n β,b s an somorphsm. Proof. We foow the same deas as n [Gu]. We w prove that, when β + λ nλ Γ 0, the set of ordered monomas n the eements of B s a vector space bass of D n β,b. Suppose that we have a non-trva reaton of the type cd, A, B, R, S, γ, EMd, A, B, R, S, γ, E 0 57 d S 1 A,B,R,S,γ,E S d where S 1, S d are fnte sets, S 1 Z 0, S d [1, n] d [1, n] d Z 0 d Z 0 d Γ d Z >0 d, A a 1,..., a d, B b 1,..., b d, R r 1,..., r d, S s 1,..., s d, γ γ 1,..., γ d, E e 1,..., e d, [1, n] {1,..., n} and Md, A, B, R, S, γ, E F a1b 1 u s1 v r1 γ 1 e1 F ad b d u s d v r d γ d e d s an ordered monomas n the eements of B. In partcuar, F ab u s v r γ F ab u s v r γ f. Let us choose a specfc ď, Ǎ, ˇB, Ř, Š, ˇγ, Ě such that

23 1. d g1 r g + s g e g has maxmum vaue for whch cd, A, B, R, S, γ, E 0;. and, among these, t has maxmum vaue for d g1 1 δ a gb g e g ; 3. and, among these, t has maxmum vaue for d g1 δ a gb g e g. Ths choce may not be unque. Set ˇM Mď, Ǎ, ˇB, Ř, Š, ˇγ, Ě Set δ d g1 e g. Now suppose that, 1,..., n Z 0 are such that n δ. We et v v 1 v, v v 1 v be the foowng eements of C n for δ: for e e g e e, set v g v b, v g v a, m g x s g yg r, γ g γg Γ, and set v g v g v f δ g δ Because of our assumpton that equaty 57 hods, cd, A, B, R, S, γ, EMd, A, B, R, S, γ, E1 v d S 1 A,B,R,S,γ,E S d Consder the vector space bass of H t,c Γ C[S ] C n space s a modue for D n β,b gven by the monomas xqx wth t, c, β, b as n secton 7, so that ths y qy x qx 1 1 yqy 1 1 γ v where qx, qy Z 0, γ Γ, v v 1 v wth v g {v 1,..., v n }. We can decompose the eft-hand sde of 58 as a sum of vectors n that bass and do the same for ˇM1 v. The coeffcent of m δ m δ 1 m 1 γ δ γ 1 v n ˇM1 v s equa to ccď, Ǎ, ˇB, Ř, Š, ˇγ, Ě for some c 0 whch depends on the mutpctes e,. Furthermore, the ony other monomas n the eft-hand sde of 57 whch can produce a non-zero mutpe of m δ m δ 1 m 1 γ δ γ 1 v when apped to 1 v dffer from ˇM ony by the vaue of e g for g such that a g b g, m g 1, γ g d. Because of our assumpton on the order on B, these eements aways appear at the end of each monomas n 57. Therefore, the coeffcent of m δ m δ 1 m 1 γ γ γ 1 v n the eft-hand sde of 58 can be vewed as a poynoma n 1,..., n. Snce ths poynoma vanshes for nfntey many vaues of these varabes, whch can be gven arbtrary arge ndependent vaues, t must vansh dentcay, so ts coeffcents are zero and cď, Ǎ, ˇB, Ř, Š, ˇγ, Ě 0. Repeatng ths argument, we concude that a the coeffcents cd, A, B, R, S, γ, E n 57 equa zero. Ths competes the proof of the near ndependance when β + λ λn Γ 0. Ths means that the map D n β0,b0 gr F D n β,b s an somorphsm f β + λ λn Γ 0. By upper-semcontnuty, t must be an somorphsm for any β, b. Let Φ be the agebra homomorphsm D n β,b End CV comng from the D n β,b-modue structure on H t,c Γ C[S ] C n. Coroary 9.1. of the proof of theorem 9.1 Suppose that β + λ λn Γ 0. Gven M D n β,b, M 0, there exsts an 0 such that Φ M s not dentcay zero. 10 Cycotomc case It s possbe to generaze most of the resuts of [Gu1] and [Gu] to the case when Γ s a cycc group of order d. In order to do ths, we frst have to consder a famy of graded Hece agebras for the compex refecton groups S Z/dZ whch were frst ntroduced n [RaSh] and studed when d n [De1] and, n genera, n [De]. We w then prove an equvaence of Schur-Wey type between a ocazaton of a ratona Cheredn agebra and an affne Yangan, generazng the wor n [Gu1]. Afterwards, we w expan how to reaze the Z/dZ-deformed doube current agebra D n β,b as a subagebra of ths affne Yangan. 3