ON THE ROSENBERG-ZELINSKY SEQUENCE IN ABELIAN MONOIDAL CATEGORIES

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1 KCL-MTH ZMP-HH/07-13 Hambug itäg zu Matmatik N. 294 ON THE ROSENERG-ZELINSKY SEUENCE IN AELIAN MONOIDAL CATEGORIES Till ami a,b, Jügn Fucs c, Ingo Runkl b, Cistop Scwigt a a Oganisationsinit Matmatik, Univsität Hambug Scwpunkt Algba und Zalntoi undsstaß 55, D Hambug b Dpatmnt of Matmatics, King s Collg London Stand, London WC2R 2LS, Unitd Kingdom c Totisk fysik, Kalstads Univsitt Univsittsgatan 5, S Kalstad Dcmb 2007 Abstact W consid Fobnius algbas and ti bimoduls in ctain ablian monoidal catgois. In paticula w study t Picad goup of t catgoy of bimoduls ov a Fobnius algba, i.. t goup of isomopism classs of invtibl bimoduls. T Rosnbg-Zlinsky squnc dscibs a omomopism fom t goup of algba automopisms to t Picad goup, wic owv is typically not sujctiv. W invstigat und wic conditions t xists a Moita quivalnt Fobnius algba fo wic t cosponding omomopism is sujctiv. On motivation fo ou considations is t obifold constuction in confomal fild toy. addsss: bami@mat.uni-ambug.d, jfucs@fucs.tkn.kau.s, ingo.unkl@kcl.ac.uk, scwigt@mat.uni-ambug.d

2 1 Intoduction In t study of associativ algbas it is oftn advantagous to collct algbas into a catgoy wos mopisms a not algba omomopisms, but instad bimoduls. On motivation fo tis is povidd by t following obsvation. Lt k b a fild and consid finit-dimnsional unital associativ k-algbas. T condition on a k-lina map to b an algba mopism is obviously not lina. As a consqunc t catgoy of algbas and algba omomopisms as t unplasant fatu of not bing additiv. On t ot and, instad of an algba omomopism ϕ: A on can quivalntly consid t -A-bimodul ϕ wic as a k-vcto spac coincids wit and wos lft action is givn by t multiplication of wil t igt action is application of ϕ composd wit multiplication in. Tis is consistnt wit composition in t sns tat givn anot algba omomopism ψ: C t is an isomopism C ψ ϕ Cψ ϕ of C-A-bimoduls. It is tn natual not to stict on s attntion to suc spcial bimoduls, but to allow all -A-bimoduls as mopisms fom A to [, sct. 5.7]. Of cous, as bimoduls com wit ti own mopisms, on tn actually dals wit t stuctu of a bicatgoy. T advantag is tat t 1-mopism catgoy A, i.. t catgoy of -A-bimoduls, is additiv and vn ablian. Taking bimoduls as mopisms as fut intsting consquncs. Fist of all, t concpt of isomopy of two algbas A and is now placd by Moita quivalnc, wic quis t xistnc of an invtibl A--bimodul. Indd, in applications involving associativ algbas on oftn finds tat not only isomopic but also Moita quivalnt algbas can b usd fo a givn pupos. T classical xampl is t quivalnc of t catgoy of lft (o igt) moduls ov Moita quivalnt algbas. Anot illustation is t Moita quivalnc btwn invaiant subalgbas and cossd poducts, s.g. [Ri]. Exampls in t alm of matmatical pysics includ t obsvations tat matix tois on Moita quivalnt noncommutativ toi a pysically quivalnt [Sc], and tat Moita quivalnt symmtic spcial Fobnius algbas in modula tnso catgois dscib quivalnt ational confomal fild tois [FFRS1, FFRS3]. As a scond consqunc, instad of t automopism goup Aut(A) on now dals wit t invtibl A-bimoduls. T isomopism classs of ts paticula bimoduls fom t Picad goup Pic(A-imod) of A-bimoduls. Wil Moita quivalnt algbas may av diffnt automopism goups, t cosponding Picad goups a isomopic. On finds tat fo any algba A t goups Aut(A) and Pic(A-imod) a latd by t xact squnc 0 Inn(A) Aut(A) Ψ A Pic(A-imod), (1.1) wic is a vaiant of t Rosnbg-Zlinsky [RZ, KO] squnc. H Inn(A) dnots t inn automopisms of A, and t goup omomopism Ψ A is givn by assigning to an automopism ω of A t bimodul A ω obtaind fom A by twisting t igt action of A on itslf by ω. In ot wods, Pic(A-imod) is t om fo t obstuction to a Skolm-Not tom. It sould b noticd tat t goup omomopism Ψ A in (1.1) is not ncssaily a sujction. ut fo pactical puposs in conct applications it can b of intst to av an xplicit alisa- 2

3 tion of t Picad goup in tms of automopisms of t algba availabl. Tis lads natually to t following qustions: Dos t xist anot algba A, Moita quivalnt to A, suc tat t goup omomopism Ψ A : Aut(A ) Pic(A -imod) in (1.1) is sujctiv? And, onc suc an algba A as bn constuctd: Dos tis sujction admit a sction, i.. can t goup Pic(A-imod) b idntifid wit a subgoup of t automopism goup of t Moita quivalnt algba A? W will invstigat ts qustions in a mo gnal stting, namly w consid algbas in k-lina monoidal catgois mo gnal tan t on of k-vcto spacs. Lik many ot sults valid fo vcto spacs, also t squnc (1.1) continus to old in tis stting, s [VZ, pop. 3.14] and [FRS3, pop. 7]. W stat in sction 2 by collcting som aspcts of algbas and Moita quivalnc in monoidal catgois and viw t dfinition of invtibl objcts and of t Picad catgoy. Sction 3 collcts infomation about fixd algbas und som subgoup of algba automopisms. In sction 4 w answ t qustions aisd abov fo t spcial cas tat t algba A is t tnso unit of t monoidal catgoy D und considation. As calld in sction 2, t catgoical dimnsion povids a caact on t Picad goup wit valus in k. T main sult of sction 4, Poposition 4.3, supplis, fo any finit subgoup H of t Picad goup on wic tis caact is tivial, an algba A tat is Moita quivalnt to t tnso unit suc tat t lmnts of H can b idntifid wit automopisms of A. Tom 4.12, in tun, givs a caactisation of goup omomopisms H Aut(A) in tms of cocains on H. In tis cas t subgoup H is not only quid to av tivial caact, but in addition a t-cocycl on Pic(A-imod) must b tivial wn stictd to H. T lvant t-cocycl is obtaind fom t associativity constaint of D, s q. (4.23) blow. W also comput t fixd algba und t cosponding subgoup of automopisms. In sction 5 ts sults a gnalisd to algbas not ncssaily Moita quivalnt to t tnso unit, poviding an affimativ answ to t abov qustions also in t gnal cas. Howv, simila to t A 1 cas, on nds to stict onslf to a finit subgoup H of Pic(A-imod) suc tat t cosponding invtibl bimoduls av catgoical dimnsion qual to 1 in A-imod and fo wic t associativity constaint of A-imod is tivial. Tis is statd in Tom 5.6, wic is t main sult of tis pap. Lt us also bifly mntion a motivation of ou considations wic coms fom confomal fild toy. A consistnt ational confomal fild toy (on ointd sufacs wit possibly non-mpty bounday) is dtmind by a modul catgoy M ov a modula tnso catgoy C [FRS1]. T Picad goup of t catgoy of modul ndofunctos of M dscibs t symmtis of tis CFT [FFRS3]. T xplicit constuction of tis CFT quis not just t abstact modul catgoy, but at a conct alization as catgoy of moduls ov a Fobnius algba A, as tis povids a natual fogtful functo fom M to C wic nts cucially in t constuction. T modul ndofunctos a alisd as t catgoy of A-A-bimoduls. Fo pactical puposs it can b usful 3

4 to coos t algba A suc tat a givn subgoup H of t symmtis Pic(A-imod) of t CFT is alisd as automopisms of A. Tom 5.6 povids us wit conditions fo wn suc a psntativ xists. Finally, t fixd algba und tis subgoup of automopisms is latd to t CFT obtaind by obifolding t oiginal CFT by t symmty H. Acknowldgmnts: T is suppotd by t Euopan Supsting Toy Ntwok (MCFH ) and tanks King s Collg London fo ospitality. JF is patially suppotd by VR und pojct no IR is patially suppotd by t EPSRC Fist Gant EP/E005047/1, t PPARC olling gant PP/C507145/1 and t Mai Cui ntwok Supsting Toy (MRTN-CT ). CS is patially suppotd by t Collaboativ Rsac Cnt 676 Paticls, Stings and t Ealy Univs - t Stuctu of Matt and Spac-Tim. 2 Algbas in monoidal catgois In tis sction w collct infomation about a fw basic stuctus tat will b ndd blow. Lt D b an ablian catgoy nicd ov t catgoy Vct k of finit-dimnsional vcto spacs ov a fild k. An objct X of D is calld simpl iff it as no pop subobjcts. An ndomopism of a simpl objct X is it zo o an isomopism (Scu s lmma), and nc t ndomopism spac Hom(X, X) is a finit-dimnsional division algba ov k. An objct X of D is calld absolutly simpl iff Hom(X, X) k id X. If k is algbaically closd, tn vy simpl objct is absolutly simpl; t convs olds.g. if D is smisimpl. Wn D is monoidal, tn witout loss of gnality w assum it to b stict. Mo spcifically, fo t st of tis pap w mak t following assumption. Convntion 2.1. (D,, 1) is an ablian stict monoidal catgoy wit simpl and absolutly simpl tnso unit 1, and nicd ov Vct k fo a fild k of caactistic zo. In paticula, Hom(1, 1) k id 1, wic w idntify wit k. Dfinition 2.2. A igt duality on D assigns to ac objct X of D an objct X, calld t igt dual objct of X, and mopisms b X Hom(1, X X ) and d X Hom(X X, 1) suc tat (id X d X ) (b X id X ) id X and (d X id X ) (id X b X ) id X. (2.1) A lft duality on D assigns to ac objct X of D a lft dual objct X togt wit mopisms bx Hom(1, X X) and d X Hom(X X, 1) suc tat ( d X id X ) (id X b X ) id X and (id X d X ) ( b X id X) id X. (2.2) Not tat d 1 1 is nonzo; sinc by assumption 1 is simpl, w tus av 1 1. In t sam way on ss tat 1 1. Fut, givn a igt duality, t igt dual mopism to a mopism f Hom(X, Y ) is t mopism f : (d Y id X ) (id Y f id X ) (id Y b X ) Hom(Y, X ). (2.3) 4

5 Lft dual mopisms a dfind analogously. Hby ac duality funiss a functo fom D to D op. Fut, t objcts (X Y ) and Y X a isomopic. Dfinition 2.3. A sovign 1 catgoy is a monoidal catgoy tat is quippd wit a lft and a igt duality wic coincid as functos, i.. X X fo vy objct X and f f fo vy mopism f. In a sovign catgoy t lft and igt tacs of an ndomopism f Hom(X, X) a t scalas (mmb tat w idntify End(1) wit k) t l (f) : d X (id X f) b X and t (f) : d X (f id X ) b X, (2.4) spctivly, and t lft and igt dimnsions of an objct X a t scalas dim l (X) : t l (id X ), dim (X) : t (id X ). (2.5) ot tacs a cyclic, and dimnsions a constant on isomopism classs, multiplicativ und t tnso poduct and additiv und dict sums. Fut, on as t l (f) t (f ), and using t fact tat in a sovign catgoy ac objct X is isomopic to its doubl dual X it follows tat t igt dimnsion of t dual objct quals t lft dimnsion of t objct itslf, dim l (X) dim (X ), (2.6) and vic vsa. In paticula, any objct tat is isomopic to its dual, X X, as qual lft and igt dimnsion, wic w tn dnot by dim(x). T tnso unit 1 is isomopic to its dual and as dimnsion dim(1) 1. Nxt w collct som infomation about algba objcts in monoidal catgois. Rcall tat a (unital, associativ) algba in D is a tipl (A, m, η) consisting of an objct A of D and mopisms m Hom(A A, A) and η Hom(1, A), suc tat m (id A m) m (m id A ) and m (id A η) id A m (η id A ). (2.7) Dually, a (counital, coassociativ) coalgba is a tipl (C,, ε) wit C an objct of D and mopisms Hom(C, C C) and ε Hom(C, 1), suc tat ( id C ) (id C ) and (id C ε) id C (ε id C ). (2.8) T following concpts a also wll known, s.g. [Mü, FRS1]. Dfinition 2.4. (i) A Fobnius algba in D is a quintupl (A, m, η,, ε), suc tat (A, m, η) is an algba in D, (A,, ε) is a coalgba and t compatibility lation (id A m) ( id A ) m (m id A ) (id A ) (2.9) btwn t algba and coalgba stuctus is satisfid. 1 Wat w call sovign is somtims fd to as stictly sovign, compa [i, ]. 5

6 (ii) A Fobnius algba A is calld spcial iff m β A id A and ε η β 1 id 1 wit β 1, β A k. A is calld nomalisd spcial iff A is spcial wit β A 1. (iii) If D is in addition sovign, an algba A in D is calld symmtic iff t two mopisms Φ 1 : ((ε m) id A ) (id A b A ) and Φ 2 : (id A (ε m)) ( b A id A ) (2.10) in Hom(A, A ) a qual. Fo (A, m A, η A ) and (, m, η ) algbas in D, a mopism f: A is calld a (unital) mopism of algbas iff f m A m (f f) in Hom(A A, ) and f η A η. Similaly on dfins (counital) mopisms of coalgbas and mopisms of Fobnius algbas. An algba S is calld a subalgba of A iff t is a monic i: S A tat is a mopism of algbas. A (unital) lft A-modul is a pai (M, ρ) consisting of an objct M in D and a mopism ρ Hom(A M, M), suc tat ρ (id A ρ) ρ (m id M ) and ρ (η id M ) id M. (2.11) Similaly on dfins igt A-moduls. An A-A-bimodul (o A-bimodul fo sot) is a tipl (M, ρ, ϱ) suc tat (M, ρ) is a lft A-modul, (M, ϱ) a igt A-modul, and t lft and igt actions of A on M commut. Analogously, A--bimoduls cay a lft action of t algba A and a commuting igt action of t algba. Fo (M, ρ M ) and (N, ρ N ) lft A-moduls, a mopism f Hom(M, N) is said to b a mopism of lft A-moduls (o bifly, a modul mopism) iff f ρ M ρ N (id A f). Analogously on dfins mopisms of A--bimoduls. Tby on obtains a catgoy, wit objcts t A--bimoduls and mopisms t A--bimodul mopisms. W dnot tis catgoy by D A and t st of bimodul mopisms fom M to N by Hom A (M, N). T Fobnius popty (2.9) mans tat t copoduct is a mopism of A-bimoduls. Dfinition 2.5. An algba is calld (absolutly) simpl iff it is (absolutly) simpl as a bimodul ov itslf. Tus A is absolutly simpl iff Hom A A (A, A) k id A. Rmak 2.6. Sinc D is ablian, on can dfin a tnso poduct of A-bimoduls. Tis tuns t bimodul catgoy D A A into a monoidal catgoy. Fo xampl, D D 1 1 as monoidal catgois. S t appndix fo mo dtails on tis and spcially on tnso poducts ov spcial Fobnius algbas. Rmak 2.7. If A is a (not ncssaily symmtic) Fobnius algba in a sovign catgoy, tn t mopisms Φ 1 and Φ 2 in (2.10) a invtibl, wit invss Φ 1 1 (d A id A ) (id A ( η)) and Φ 1 2 (id A d A ) (( η) id A ), (2.12) spctivly. So if A is Fobnius, A and A a isomopic, nc t lft and igt dimnsion of A a qual. Accodingly w will wit dim(a) fo t dimnsion of a Fobnius algba in t 6

7 squl. Fut on can sow (s [FRS1], sction 3) tat fo any symmtic spcial Fobnius algba A t lation β A β 1 dim(a) olds. In paticula, dim(a) 0. Futmo, witout loss of gnality on can assum tat t copoduct is nomalisd suc tat β 1 dim(a) and β A 1, i.. A is nomalisd spcial. Lmma 2.8. Lt (A, m, η) b an algba wit dim k Hom(1, A) 1. Tn A is an absolutly simpl algba. Poof. y Poposition 4.7 of [FS] on as Hom(1, A) Hom A (A, A). T sult tus follows fom 1 dim k Hom A A (A, A) dim k Hom A (A, A). Rmak 2.9. Obviously t tnso unit 1 is a symmtic spcial Fobnius algba. On also asily vifis tat fo any objct X in a sovign catgoy t objct X X wit stuctual mopisms m : id X d X id X, η : b X, : id X b X id X, ε : d X (2.13) povids an xampl of a symmtic Fobnius algba. If t objct X as nonzo lft and igt dimnsions, tn tis algba is also spcial, wit β X X dim l (X), β 1 dim (X). (2.14) T objct X is natually a lft modul ov X X, wit psntation mopism ρ id X d X, wil t objct X is a igt modul ov X X wit ϱ d X id X. Nxt w call t concpt of Moita quivalnc of algbas (fo dtails s.g. [Pa, VZ]). Dfinition A Moita contxt in D is a sxtupl (A,, P,, f, g), w A and a algbas in D, P is an A--bimodul and A is a -A-bimodul, suc tat f: P A and g: a isomopisms of A- and -bimoduls, spctivly, and t two diagams (P ) f id A ( ) g id P ( ) A (P ) (2.15) id g P P id f A A commut. If suc a Moita contxt xists, w call t algbas A and Moita quivalnt. In t squl w will suppss t isomopisms f and g and wit a Moita contxt as A P,. Lmma Lt D b in addition sovign and lt U b an objct of D wit nonzo lft and igt dimnsion. Tn t symmtic spcial Fobnius algba U U is Moita quivalnt to t tnso unit, wit Moita contxt 1 U,U U U. 7

8 Poof. W only nd to sow tat U U U U 1. Sinc U U is symmtic spcial Fobnius, t idmpotnt P U,U fo t tnso poduct ov U U, as dscibd in appndix A, is wll dfind. On calculats tat P U,U (dim l (U)) 1 bu d U. Tis implis tat t tnso unit is indd isomopic to t imag of P U,U. Finally, commutativity of t diagams (2.15) follows using t tcniqus of pojctos as psntd in t appndix. Dfinition An objct X in a monoidal catgoy is calld invtibl iff t xists an objct X suc tat X X 1 X X. If t catgoy D as small sklton, tn t st of isomopism classs of invtibl objcts foms a goup und t tnso poduct. Tis goup is calld t Picad goup Pic(D) of D. Lmma Lt D b in addition sovign. (i) Evy invtibl objct of D is simpl. (ii) An objct X in D is invtibl iff X is invtibl. (iii) An objct X in D is invtibl iff t mopisms b X and b X a invtibl. (iv) Evy invtibl objct of D is absolutly simpl. Poof. (i) Lt X X X X 1. Assum tat : U X is monic fo som objct U. Tn id X : X U X X is monic. Indd, if (id X ) f (id X ) g fo som mopisms f and g, tn by applying t duality mopism d X w obtain (d X id U ) (id X f) (d X id U ) (id X g). As is monic tis amounts to (d X id U ) (id X f) (d X id U ) (id X g), wic by applying b X and using t duality popty of d X and b X sows tat f g. Tus id X : X U X X 1 is monic. As 1 is quid to b simpl, it is tus an isomopism. Tn id X id X is an isomopism as wll. y assumption t xists an isomopism b: 1 X X. Wit t lp of b w can wit (b 1 id X ) (id X id X ) (b id U ). Tus is a composition of isomopisms, and nc an isomopism. In summay, :U X bing monic implis tat is an isomopism. Hnc X is simpl. (ii) Not tat X X (X X) 1 1, and similaly X X 1. (iii) Sinc by pat (ii) X is invtibl, so is X X. y pat (i), X X is tfo simpl and b X : 1 X X is a nonzo mopism btwn simpl objcts. y Scu s lmma it is an isomopism. T agumnt fo b X pocds along t sam lins, and t convs statmnt follows by dfinition. (iv) T duality mopisms giv an isomopism Hom(X, X) Hom(X X, 1). Fom pats (i) and (ii) w know tat X X 1, and so Hom(X, X) Hom(1, 1). Tat X is absolutly simpl now follows bcaus 1 is absolutly simpl by assumption. Lmma 2.13 implis tat fo an invtibl objct X on as dim l (X) dim (X) dim l (X) dim l (X ) dim l (X X ) dim l (1) 1. (2.16) 8

9 Wit t lp of tis quality on ccks tat t invs of b X is givn by dim l (X) d X, dim l (X) d X b X dim l (X) dim (X) id 1 id 1. (2.17) Analogously w av dim (X) d X b X id 1 ; tus in paticula t lft and igt dimnsions of an invtibl objct X a nonzo. Fut w av dim l (X) b X d X id X X and dim (X) b X d X id X X. (2.18) W dnot t objct psnting an isomopism class g in Pic(D) by L g, i.. [L g ] g Pic(D). Tn L g L Lg. As t psntativ of t unit class 1 w tak t tnso unit, L 1 1. Lmma Lt D b in addition sovign and H a subgoup of Pic(D). (i) T mappings dim l (L ) and dim (L ) a caacts on H. (ii) If H is finit, tn dim l ( H L ) is it 0 o H. It is qual to H iff dim l (L ) 1 fo all H. Poof. Claim (i) follows dictly fom t multiplicativity of t lft and igt dimnsion und t tnso poduct and fom t fact tat t dimnsion only dpnds on t isomopism class of an objct. caus of dim l ( H L ) H dim l (L ), pat (ii) is a consqunc of t otogonality of caacts. Dfinition T Picad catgoy Pic(D) of D is t full subcatgoy of D wos objcts a dict sums of invtibl objcts of D. 3 Fixd algbas W intoduc t notion of a fixd algba und a goup of algba automopisms and stablis som basic sults on fixd algbas. Dfinition 3.1. Lt (A, m, η) b an algba in D and H Aut(A) a goup of (unital) automopisms of A. Tn a fixd algba und t action of H is a pai (A H, j), w A H is an objct of D and j: A H A is a monic wit α j j fo all α H, suc tat t following univsal popty is fulfilld: Fo vy objct in D and mopism f: A wit α f f fo all α H, t is a uniqu mopism f: A H suc tat j f f. T objct A H dfind tis way is uniqu up to isomopism. T following sult justifis using t tm fixd algba, at tan fixd objct. Lmma 3.2. Givn A, H and (A H, j) as in dfinition 3.1, t xists a uniqu algba stuctu on t objct A H suc tat t inclusion j: A H A is a mopism of algbas. 9

10 Poof. Fo abitay α H consid t diagams A H j A α A A H j A α A m (j j) and η (3.1) A H A H 1 Sinc α is a mopism of algbas, w av α m (j j) m ((α j) (α j)) m (j j); as tis olds fo all α H, t univsal popty of t fixd algba yilds a uniqu poduct mopism µ: A H A H A H suc tat j µ m (j j). y associativity of A w av j µ (µ id A H) m (m id A ) (j j j) m (id A m) (j j j) j µ (id A H µ). (3.2) Sinc j is monic, tis implis associativity of t poduct mopism µ. Similaly, applying t univsal popty of (A H, j) on η givs a mopism η : 1 A H tat as t poptis of a unit fo t poduct µ. So (A H, µ, η ) is an associativ algba wit unit. W pocd to sow tat fixd algbas und finit goups of automopisms always xist in t situation studid. Lt H Aut(A) b a finit subgoup of t goup of algba automopisms of A. St P P H : 1 α End(A). (3.3) H α H Tn P P 1 H 2 α,β H α β 1 H 2 α,β H β 1 H β H β P, i.. P is an idmpotnt. Analogously on sows tat α P P fo vy α H. Fut, sinc D is ablian, w can wit P wit monic and pi. Dnot t imag of P P H by, so tat : A and id AP. Lmma 3.3. T pai (, ) satisfis t univsal popty of t fixd algba. Poof. Fom id AP w s tat α α α P P fo all α H. Fo an objct of D and f: A a mopism wit α f f fo all α H, st f : f. Tn f f P f 1 H α H α f 1 H α H f f. Fut, if f is anot mopism satisfying f f, tn f f and, sinc is monic, f f, so f is uniqu. Hnc t objct satisfis t univsal popty of t fixd algba A H. W would lik to xpss t stuctual mopisms of t fixd algba toug and. To tis nd w intoduc a candidat poduct m P and candidat unit η P on : w st m P : m ( ) and η P : η. (3.4) 10

11 Not tat is a mopism of unital algbas: m P m ( ) P m ( ) 1 H α H α m ( ) 1 1 H α H m ((α ) (α )) H α H m ( ) m ( ), η P η P η 1 H α H α η 1 H α H η η. (3.5) Lmma 3.4. T algba (, m P, η P ) is isomopic to t algba stuctu tat inits as a fixd algba. Poof. An asy calculation sows tat m P is associativ and η P is a unit fo m P ; tus (, m P, η P ) is an algba. Moov, sinc accoding to lmma 3.2 t is a uniqu algba stuctu on A H suc tat t inclusion into A is a mopism of algbas, it follows tat and A H a isomopic as algbas. In t following discussion t tm fixd algba will always f to t algba. Lmma 3.5. Wit P 1 H α H α as in (3.3), w av t following qualitis of mopisms: m ( P ) m ( id A ), m (P ) m (id A ) and P m ( ) m ( ). Poof. Indd, making us of α and α fo all α H, w av m ( P ) 1 m ( α) 1 α m ((α 1 ) id A ) H H α H α H 1 m ( id A ) m ( id A ). H α H (3.6) (3.7) T ot two qualitis a stablisd analogously. Rmak 3.6. Wit t lp of t gapical calculus fo mopisms in stict monoidal catgois (s [JS, Ka, Ma, K], and.g. Appndix A of [FFRS2] fo t gapical psntation of t stuctual mopisms of Fobnius algbas in suc catgois), t qualitis in Lmma 3.5 can b visualisd as follows: A A P (3.8) A P A P A A If A is a Fobnius algba, it is undstood tat Aut(A) consists of all algba automopisms of A wic a at t sam tim also coalgba automopisms. Tn fo a Fobnius algba A t 11

12 idmpotnt P can also b omittd in t following situations, wic w dscib again pictoially: A A P P (3.9) P A Poposition 3.7. Lt A b a Fobnius algba in D and H Aut(A) a finit goup of automopisms of A. (i) is a Fobnius algba, and t mbdding : A is a mopism of algbas wil t stiction : A is a mopism of coalgbas. (ii) If t catgoy D is sovign and A is symmtic, tn is symmtic, too. (iii) If t catgoy D is sovign, A is symmtic spcial and is an absolutly simpl algba and as nonzo lft (quivalntly igt, cf. mak 2.7) dimnsion, tn is spcial. Poof. (i) T algba stuctu on as alady bn dfind in (3.4), and accoding to (3.5) is a mopism of algbas. Dnoting t copoduct on A by and t counit by ε, w fut st P : ( ) and ε P : ε. Similaly to t calculation in (3.5) on vifis tat is a mopism of coalgbas, and tat P is coassociativ and ε P is a counit. Rgading t Fobnius popty, w giv a gapical poof of on of t qualitis tat must b satisfid: P m P P P (m P id AP ) (id AP P ). (3.10) H it is usd tat accoding to mak 3.6 w a allowd to mov and inst idmpotnts P, and tn t Fobnius popty of A is invokd. T ot alf of t Fobnius popty is sn analogously. (ii) T following cain of qualitis sows tat is symmtic: A P A P A P A P (3.11) 12

13 H t notations b X X X and bx X X (3.12) a usd fo t duality mopisms b X and b X, spctivly, of an objct X. T mopisms d X and d X a dawn in a simila way. (iii) W av ε P η P ε η ε η, wic is nonzo by spcialnss of A. As is associativ, m P is a mopism of -bimoduls. T Fobnius popty nsus tat P is also a mopism of bimoduls. Hnc m P P is a mopism of bimoduls, and by absolut simplicity of it is a multipl of t idntity. Moov, m P P is not zo: w av ε P m P P η P ε m (id A P ) η (3.13) wic, as A is symmtic, is qual to t l (P ) dim l ( ) 0. W conclud tat m P P 0. Hnc is spcial. Rmak 3.8. In t abov discussion t catgoy D is assumd to b ablian, but tis assumption can b laxd. Of t poptis of an ablian catgoy w only usd tat t mopism sts a ablian goups, tat composition is bilina, and tat t lvant idmpotnts factois in a monic and an pi, i.. tat D is idmpotnt complt. In addition w assumd tat mopisms sts a finit-dimnsional k-vcto spacs. Fom q. (3.3) onwads, and in paticula in poposition 3.7, it is in addition usd tat D is nicd ov Vct k. If tis is not t cas, on can no long, in gnal, dfin an idmpotnt P toug 1 H α H α, and t nd not xist a copoduct on t fixd algba AH, vn if t is on on A. 4 Algbas in t Moita class of t tnso unit Rcall tat accoding to ou convntion 2.1 (D,, 1) is ablian stict monoidal, wit simpl and absolutly simpl tnso unit and nicd ov Vct k wit k of caactistic zo. Fom now on w fut assum tat D is skltally small and sovign. W now associat to an algba (A, m, η) in D a spcific subgoup of its automopism goup t inn automopisms wic a dfind as follows. T spac Hom(1, A) bcoms a k-algba by dfining t poduct as f g : m (f g) fo f, g Hom(1, A). T mopism η Hom(1, A) is a unit fo tis poduct. W call a mopism f in Hom(1, A) invtibl iff t xists a mopism f Hom(1, A) suc tat f f η f f. Now t mopism ω f : m (m f ) (f id A ) Hom(A, A) (4.1) is asily sn to b an algba automopism. T automopisms of tis fom a calld inn automopisms; ty fom a nomal subgoup Inn(A) Aut(A) as is sn blow. 13

14 Dfinition 4.1. Fo A an algba in D and α, β Aut(A), t A-bimodul α A β (A, ρ α, ϱ β ) is t bimodul wic as A as undlying objct and lft and igt actions of A givn by ρ α : m (α id A ) and ϱ β : m (id A β), (4.2) spctivly. Ts lft and igt actions of A a said to b twistd by α and β, spctivly, and αa β is calld a twistd bimodul. Tat tis indd dfins an A-bimodul stuctu on t objct A is asily cckd wit t lp of t multiplicativity and unitality of α and β. Fut, as sown in [VZ, FRS3], t bimoduls α A β a invtibl. Dnot t isomopism class of a bimodul X by [X]. y stting on obtains an xact squnc Ψ A (α) : [ id A α ] (4.3) 0 Inn(A) Aut(A) Ψ A Pic(D A A ) (4.4) of goups. In paticula on ss tat t subgoup Inn(A) is in fact a nomal subgoup, as it is t knl of t omomopism Ψ A. T poof of xactnss of tis squnc in [VZ, FRS3] is not only valid in baidd monoidal catgois, but also in t psnt mo gnal situation. Lt now A and b Moita quivalnt algbas in D, wit, a Moita contxt (P, A ). Tn t mapping Π,P : Pic(D A A ) Pic(D ) [X] [ A X ] (4.5) constituts an isomopism btwn t Picad goups Pic(D A A ) and Pic(D ). In paticula, if A is an algba tat is Moita quivalnt to t tnso unit 1, tn w av an isomopism Pic(D A A ) Pic(D). As Moita quivalnt algbas nd not av isomopic automopism goups, t imags of t goup omomopisms Ψ A : Aut(A) Pic(D A A ) and Ψ : Aut() Pic(D ) will in gnal b non-isomopic. In t following w will consid subgoups of t goup Pic(D). Fo a subgoup H Pic(D) w put (H) : L. (4.6) H Rmak 4.2. Sinc t objct is t dict sum ov a wol subgoup of Pic(D) and L g L g 1, it follows tat. As a consqunc, lft and igt dimnsions of a qual, and accodingly in t squl w us t notation dim() fo bot of tm. Poposition 4.3. Lt H Pic(D) b a finit subgoup suc tat dim() 0 fo (H). Tn wit t algba A A(H) : (4.7) 14

15 and t Moita contxt 1, A intoducd in lmma 2.11, w av H im(π, Ψ A ), i.. t subgoup H is covd as t imag of t composit map Aut(A) Ψ A Pic(D A A ) Π, Pic(D). (4.8) Poof. T isomopism Π, : Pic(D) Pic(D A A ) is givn by [L g ] [ L g ]. Fo H w want to find automopisms α of A suc tat id A α L as A-bimoduls. W fist obsv t isomopisms L g H L g L g H L g. W mak a (in gnal noncanonical) coic of isomopisms f : L, wit t mopism f 1 cosn to b t idntity id. Tn fo ac H w dfin t ndomopism α of by α : f 1 L f (4.9) Ts a algba mopisms: f 1 f m (α α ) f 1 f f 1 L f f f 1 f 1 f L f 1 f f 1 f α m. (4.10) H w av usd tat by lmma 2.14(ii) w av dim l (L ) 1 fo H. T tid quality is tn a consqunc of id L L b L d L, s quation (2.18); in t fout quality f is canclld against f 1 by using poptis of t duality. (Also, fo btt adability, and blow w fain fom lablling som of t L -lins.) Fut, t mopisms α a also unital: α η f 1 f f 1 f L η, (4.11) w again by lmma 2.14 w av dim (L ) 1. T invs of α is givn by α 1 f f (4.12) L 15

16 as is sn in t following calculations: α α 1 f 1 f f f L id, α 1 α f f f f id. (4.13) f 1 f f 1 f w in paticula (2.18) and dim l (L ) 1 is usd. A bimodul isomopism L id A α Fist w s tat F is invtibl wit invs F 1 f is now givn by F : id (( d L id ) (id L f )). (4.14) stands fo t dual of t invs of f. Tat F 1 id ((id L f ) (b L id )), w is indd invs to F is sn as follows: F F 1 f f id, L L F 1 F f f id L. (4.15) f f L L Moov, F claly inttwins t lft actions of A on L and on id A α. inttwins t igt actions as wll is vifid as follows: Tat it f f f (4.16) L f f 1 f L f f 1 f L f 1 L H simila stps a pfomd as in t poof tat α spcts t poduct of A. W conclud tat w av [ L ] im(ψ A ) fo all H, and tus Π, (H) is a subgoup of im(ψ A ). 16

17 On t ot and, fo g H, L g, H L g is not isomopic to, not vn as an objct, so tat Π, (g) im(ψ A ). Togt it follows tat im(π, Ψ A ) H. Rmak 4.4. Similaly to t calculation tat t mopisms α in (4.9) a mopisms of algbas, on sows tat ty also spct t copoduct and t counit of A(H). So in fact w av found automopisms of Fobnius algbas. W dnot t inclusion mopisms L g H L g by and t pojctions L by, suc tat g 0 fo g and g g id Lg. Tn g g P g is a nonzo idmpotnt in End(), and w av H P id. Lmma 4.5. Lt H Pic(D) b a finit subgoup suc tat dim() 0 fo (H). Givn g H and an automopism α of A suc tat Ψ A (α) [ L g ], t xists a uniqu isomopism f g Hom( L g, ) suc tat g f g ( 1 id Lg ) id Lg as in (4.9). and α α g wit α g Poof. W stat by poving xistnc. Lt ϕ g : L g id A α b an isomopism of bimoduls. As a fist stp w sow tat ϕ g id fo som mopism : L g : ϕ g 1 dim() ϕ g 1 dim() ϕ g (4.17) L g L g L g H in t fist stp w just instd t dimnsion of, using tat it is nonzo, and t scond stp is t statmnt tat ϕ g inttwins t lft action of A on L g and on id A α. Not tat upon stting f g : 1 dim() ϕ g (4.18) L g tis amounts to ϕ g id (( d Lg id ) (id Lg fg )), as in poposition 4.3. Similaly t condition tat ϕ g inttwins t igt action of A on L g and id A α mans tat f g α f g (4.19) L g L g and tis is quivalnt to quality of t sam pictus wit t idntity mopisms at t lft sids 17

18 movd. Applying duality mopisms to bot sids of t sulting quality w obtain L g L g f g α f g (4.20) Nxt w apply t mopism (id d Lg id ) (f 1 g id L g id ), lading to α f 1 g f g (4.21) Using also tat by lmma 2.14 (ii) w av dim (L g ) 1, tis sows tat α α g wit α g as in (4.9). Sinc L g is absolutly simpl (s lmma 2.13 (iv)), w av g f g ( 1 id Lg ) ξ id Lg fo som ξ k. As g is injctiv and f g is an isomopism, g f g : L g L g is nonzo. ut tis mopism can only b nonvanising on t dict summand 1 of, and nc also g f g ( 1 id Lg ) 0, i.. ξ k. Finally not tat α dos not cang if w plac f g by a nonzo multipl of f g ; nc aft suitabl scaling f g obys bot α α g and g f g ( 1 id Lg ) id Lg, tus poving xistnc. To sow uniqunss, suppos tat t is anot isomopism f g: L g suc tat α α g (w α g is givn by (4.9) wit f g instad of f g ) and g f g ( 1 id Lg ) id Lg. Composing bot sids of t quality α g α g wit f g id t sulting mopism L g givs fom t igt and taking a patial tac ov of (d id ) (id α g ) (id f g id ) ( b id Lg id ) (d id ) (id α g) (id f g id ) ( b id Lg id ) (4.22) Substituting t xplicit fom of α g and α g, and using tat dim() 0 and tat L g is absolutly simpl, on finds tat f g λ f g fo som λ k. T nomalisation conditions g f g ( 1 id Lg ) id Lg and g f g ( 1 id Lg ) id Lg tn foc λ 1, poving uniqunss. T constuction of t automopisms α psntd in poposition 4.3 and lmma 4.5 still dpnds on t coic of isomopisms f o ϕ. As ac suc automopism gts mappd to [ L ], du to xactnss of t squnc (4.4) diffnt coics of f lad to automopisms wic diff only by inn automopisms. On t ot and, t mapping α nd not b a omomopism of goups fo any coic of t isomopisms f. In t following w will fomulat ncssay and sufficint conditions tat H Pic(D) must satisfy fo t assignmnt α to yild a goup omomopism fom H to Aut(A). Rcall tat L g L Lg. Tus by lmma 2.13 (iv) t spacs Hom(L g L, L g ) a on-dimnsional (but t is no canonical coic of an isomopism to t gound fild k). Fo ac pai 18

19 g, Pic(D) w slct a basis isomopism g b Hom(L g L, L g ). W dnot ti invss by g b Hom(L g, L g L ), i.. g b g b id Lg L and g b g b id Lg. Fo g 1 w tak 1 b g and gb 1 to b t idntity, wic is possibl by t assumd stictnss of D. Fo any tipl g 1, g 2, g 3 Pic(D) t collction { g b } of mopisms povids us wit two bass of t on-dimnsional spac Hom(L g1 L g2 L g3, L g1 g 2 g 3 ), namly wit g1 g 2 b g3 ( g1 b g2 id Lg3 ) as wll as g1 b g2 g 3 (id Lg1 g2 b g3 ). Ts diff by a nonzo scala ψ(g 1, g 2, g 3 ) k: g 1 g 2 b g3 ( g1 b g2 id Lg3 ) ψ(g 1, g 2, g 3 ) g1 b g2 g 3 (id Lg1 g2 b g3 ). (4.23) T pntagon axiom fo t associativity constaints of D implis tat ψ is a t-cocycl on t goup Pic(D) wit valus in k (s.g. appndix E of [MS], capt 7.5 of [FK], o [Ya]). Any ot coic of bass lads to a coomologous t-cocycl. Obsv tat by taking 1 b and b 1 to b t idntity on L t cocycl ψ is nomalisd, i.. satisfis ψ(g 1, g 2, g 3 ) 1 as soon as on of t g i quals 1. Lmma 4.6. Fo g,, k Pic(D), t bass intoducd abov oby t lation L k L k 1 g L k L k 1 g k b k 1 g b 1 g 1 ψ(, 1 k, k 1 g) b 1 g 1k b k 1 g (4.24) Poof. In pictus: L L 1 g L L 1 g L k L k 1 g L k L k 1 g L k L k 1 g L k L k 1 g k b k 1 g k b k 1 g k b k 1 g b 1 g b 1 g 1 k b k 1 g 1k b k 1 g 1 ψ b 1 k k b k 1 g 1k b k 1 g 1 ψ b 1 k 1k b k 1 g (4.25) L L 1 g L L 1 g L L 1 g L L 1 g w in t scond stp w instd t dfinition of ψ and abbviatd ψ ψ(, 1 k, k 1 g). Dfinition 4.7. Givn t nomalisd cocycl ψ on Pic(D), a nomalisd two-cocain ω on H wit valus in k is calld a tivialisation of ψ on H iff it satisfis dω ψ H. Poposition 4.8. Givn a finit subgoup H of Pic(D) and a function ω: H H k, dfin : H L and m m(h, ω) : g, H ω(g, ) g g b ( g ) Hom(, ), η η(h, ω) : 1 Hom(1, ), (H, ω) : H 1 g, H ω(g, ) 1 ( g ) g b (4.26) g Hom(, ), ε ε(h, ω) : H 1 Hom(, 1). Tn t following statmnts a quivalnt: (i) ω is a tivialisation of ψ on H. 19

20 (ii) (, m, η) is an associativ unital algba. (iii) (,, ε) is a coassociativ counital coalgba. Moov, if any of ts quivalnt conditions olds, tn (H, ω) (, m,, η, ε) is a spcial Fobnius algba wit m id and ε η H id 1. Poof. T quivalnc of conditions (i) (iii) follows by dict computation using only t dfinitions; w fain fom giving t dtails. T Fobnius popty tn follows wit t lp of lmma 4.6. Lmma 4.9. Lt ω b a tivialisation of ψ on H Pic(D) and lt (H, ω) b t spcial Fobnius algba dfind in poposition 4.8. Tn is symmtic iff dim() 0. Poof. Rcall t mopisms Φ 1 and Φ 2 fom (2.10). Sinc H L, t condition Φ 1 Φ 2 is quivalnt to Φ 1 g Φ 2 g fo all g H. y t dfinition of t multiplication on, tis amounts to 1 L g 1 L g 1 1 ω(g, g 1 ) g b g 1 ω(g 1, g) g 1bg (4.27) L g L g wic in tun, by applying duality mopisms and composing wit t mopisms g b g 1, is quivalnt to 1 ω(g, g 1 ) gb g 1 g b g 1 ω(g 1, g) g 1bg (4.28) g b g 1 T igt and sid of (4.28) is valuatd to 1 1 ω(g 1, g) g 1b g ω(g 1, g) ψ(g 1, g, g 1 ) g b g 1 1 b g 1 g 1b 1 1 ω(g 1, g)ψ(g 1, g, g 1 ) 1 L g ω(g 1, g) ψ(g 1, g, g 1 ) dim l (L g ) id 1. (4.29) T fist stp is an application of lmma 4.6, t scond is du to ou convntion tat t mopisms 1 b g a cosn to b idntity mopisms. So t condition tat is a symmtic Fobnius algba is quivalnt to t condition ω(g, g 1 ) ω(g 1, g) ψ(g 1, g, g 1 ) dim l (L g ) fo all g H. As ω is a tivialisation of ψ and dω(g 1, g, g 1 ) ω(g, g 1 )/ω(g 1, g), tis is quivalnt to dim l (L g ) 1 fo all g H. y lmma 2.14 t latt condition olds iff dim() 0. Dfinition An admissibl subgoup of Pic(D) is a finit subgoup H Pic(D) suc tat dim l (L ) 1 fo all H and suc tat t xists a tivialisation ω of ψ on H. 20

21 Rmak W s tat (H, ω) is a symmtic spcial Fobnius algba if and only if H is an admissibl subgoup of Pic(D) and ω is a tivialisation of ψ on H. On can sow tat vy stuctu of a spcial Fobnius algba on t objct H L is of t typ (H, ω) dscibd in poposition 4.8 fo a suitabl tivialisation ω of ψ (s [FRS2], poposition 3.14). So giving poduct and copoduct mopisms on is quivalnt to giving a tivialisation ω of ψ. Also obsv tat multiplying a tivialisation ω of ψ wit a two-cocycl γ on H givs anot tivialisation ω of ψ. On can sow tat t Fobnius algbas (H, ω) and (H, ω ) a isomopic as Fobnius algbas if and only if ω and ω diff by multiplication wit an xact two-cocycl. Accodingly, in t squl w call two tivialisations ω and ω fo ψ quivalnt iff ω/ω dη fo som on-cocain η. Tus if ψ is tivialisabl on H, tn t quivalnc classs of tivialisations fom a toso ov H 2 (H, k ). Tom Lt H b an admissibl subgoup of Pic(D), put (H) as in (4.6), and lt A A(H) b t algba dfind in (4.7). Tn t is a bijction btwn tivialisations ω of ψ on H and goup omomopisms α: H Aut(A) wit Π, Ψ A α id H. Poof. Dnot by T t st of all tivialisations ω of ψ on H, and by H t st of all goup omomopisms α: H Aut(A) satisfying Π, Ψ A α id H. T poof tat T H as sts is oganisd in t stps: dfining maps F : T H and G: H T, and sowing tat ty a ac ot s invs. (i) Lt ω T. Fo ac H dfin g f : g H ω(g, ) gb (4.30) g L Sinc ω taks valus in k, ts a in fact isomopisms, wit invs givn by f 1 g H ω(g, ) 1 ( g id L ) g b g. (4.31) Dfin t function F (ω) fom H to Aut(A) by F (ω): α fo α givn by (4.9) wit f as in (4.30). W pocd to sow tat F (ω) H. Abbviat α F (ω). Tat Π, Ψ A α id H follows fom t poof of poposition 4.3. To s 21

22 tat α(g) α() α(g), fist wit α(g) α() using (4.30) and (4.31): α(g) α() k,l,m,n ω(k, g) ω(l, ) ω(m, g) ω(n, ) m m b g mg n n b n L g k kb g kg l L lb l k,m ω(k, g) ω(kg, ) ω(m, g) ω(mg, ) m m b g mg b mg L g L k kb g kgb kg m k m k k,m ξ km g b m b g gb kb g k,m ξ km m b g L g kb g mg kg mg kg wit ξ km (4.32) ω(k, g) ω(kg, ) ψ(k, g, ) ω(m, g) ω(mg, ) ψ(m, g, ). (4.33) H t scond stp uss tat t a no nonzo mopisms L n L mg unlss n mg; by t sam agumnt w conclud tat l kg. In t tid stp on applis lation (4.23). Now t condition α(g) α() α(g) is quivalnt to ω(k, g) ω(kg, ) ψ(k, g, ) ω(m, g) ω(mg, ) ψ(m, g, ) ω(k, g) ω(m, g) fo all m, k H, (4.34) wic in tun can b wittn as dω(m, g, ) dω(k, g, ) ψ(m, g, ) ψ(k, g, ) fo all m, k H. (4.35) T last condition is satisfid bcaus by assumption dω ψ H. So indd w av F (ω) H. (ii) Givn α H, fo ac H t automopism α() satisfis t conditions of lmma 4.5. As a consqunc w obtain a uniqu isomopism f : L suc tat α() α and f ( 1 id L ) id L. Dfin a function ω: H H k via L g L g ω(g, ) gb g f (4.36) g L g L L g L Tn dfin t map G fom H to functions H H k by G(α) : ω, wit ω obtaind as in (4.36). W will sow tat G(α) T. 22

23 Givn α H, abbviat ω G(α). Fist not tat ω taks valus in k, as f is an isomopism. Nxt, t nomalisation condition f ( 1 id L ) id L implis ω(1, ) 1 fo all H. Sinc α is a goup omomopism w av α(1) id. y t uniqunss sult of lmma 4.5 tis implis tat f 1 id, and so ω(g, 1) 1 fo all g H. Altogt it follows tat ω is a nomalisd two-cocain wit valus in k. y following again t stps (4.32) to (4.35) on sows tat, sinc α is a goup omomopism, ω must satisfy (4.35). Stting k 1 and using tat ω and ψ a nomalisd finally dmonstats tat dω ψ H. Tus indd G(α) T. (iii) Tat F (G(α)) α is immdiat by constuction, and tat G(F (ω)) ω follows fom t uniqunss sult of lmma 4.5. W av sn tat if H is an admissibl subgoup of Pic(D) and ω a tivialisation of ψ, tn (H, ω) is a symmtic spcial Fobnius algba. Using t poduct and copoduct mopisms of (H, ω), t automopisms α inducd by ω as dscibd in tom 4.12 can b wittn as α H P (4.37) Not tat in tis pictu t cicl on t lft stands fo t copoduct of, wil t cicl on t igt stands fo t dual of t poduct. Givn a tivialisation ω of ψ, tom 4.12 allows us to alis H as a subgoup of Aut(A). In paticula t fixd algba und t action of H is wll dfind; w dnot it by A H. W alady know fom poposition 3.7 tat A H is a symmtic Fobnius algba. It will tun out tat it is isomopic to (H, ω). In paticula, by mak 4.11 any quivalnt coic of a tivialisation ω of ψ will giv an isomopic fixd algba. Tom Lt H b an admissibl subgoup of Pic(D) wit tivialisation ω, put A A(H) as in (4.7), and mbd H Aut(A) as in tom Tn t fixd algba A H is wll dfind and it is isomopic to (H, ω). Poof. Lt (H, ω). y lmma 2.14 w av dim() H. Idntify H wit its imag in Aut(A) via t mbdding H Aut(A) dtmind by ω as in tom Now dfin mopisms i: A and s: A by i : (m id ) (id b ), s : (id d ) ( id ). (4.38) W av s i id, implying tat i is monic and is a tact of A. W claim tat i is t inclusion mopism of t fixd algba. Rcall fom (3.3) t dfinition P H 1 H α of t idmpotnt cosponding to t fixd algba objct. In t cas und considation, P 23

24 taks t fom P (4.39) as follows fom (4.37) togt wit id H P. W calculat tat i s P : i s (2) (3) (4) (5) (6) (7) P. (4.40) H in t scond stp a counit mopism is intoducd and in t tid stp t Fobnius popty is applid. T nxt stp uss coassociativity, wil t fift stp follows bcaus (H, ω) is symmtic. Tn on uss t Fobnius popty and duality. So satisfis t univsal popty of t imag of P and nc t univsal popty of t fixd algba. W now calculat t poduct mopism tat t fixd algba inits fom, stating fom (3.4): (4.41) 24

25 T fist stp uss duality, t scond on olds by associativity of m. In t last stp on uss tat s i id. T initd unit mopism is givn by (4.42) wic du to dim k Hom(1, ) 1 is qual to ζη fo som ζ k. Applying ε to bot ts mopisms and using tat dim() H, w s tat ζ 1. Similaly on sows tat t copoduct and counit mopisms tat inits as a fixd algba qual tos dfind in poposition 4.8. So (H, ω) is isomopic to t fixd algba A H as a Fobnius algba. Rmak T algba stuctu on t objct (H, ω) is a kind of twistd goup algba of t goup H wic is not twistd by a closd two-cocain, but at by a tivialisation of t associato of D. Algbas of tis typ av appad in applications in confomal fild toy [FRS2]. 5 Algbas in gnal Moita classs In tis sction w solv t poblm discussd in t pvious sction fo algbas tat a not Moita quivalnt to t tnso unit. Tougout tis sction w will assum t following. Convntion 5.1. (C,, 1) as t poptis listd in convntion 2.1 and is in addition skltally small and sovign. (A, m, η,, ε) is a simpl and absolutly simpl symmtic nomalisd spcial Fobnius algba in C, and H Pic(C A A ) is a finit subgoup. Rcall fom mak 2.7 tat t conditions abov imply dim(a) 0. In t squl w will find a symmtic spcial Fobnius algba A A (H) and a Moita contxt,p A in C, suc tat H is a subgoup of im(π P,P Ψ A ), w Π P,P gnaliss t sults of poposition 4.3. is t isomopism intoducd in (4.5). Tis W will apply som of t sults of t pvious sction to t stictification D of t catgoy C A A. Not tat D as t poptis statd in convntion 2.1 and is in addition sovign, as can b sn by staigtfowad calculations wic a paalll to tos of [FS] fo t catgoy C A of lft A-moduls. y applying t invs quivalnc functo D C A A tis will tn yild a symmtic spcial Fobnius algba in C A A tat as t dsid poptis. Not tat t gapical psntations of mopisms usd blow a mant to psnt mopisms in C. Ptinnt facts about t stuctu of t catgoy C A A a collctd in appndix A; in t squl w will fly us t tminology psntd t. It is wot mpasising tat fo stablising vaious of t sults blow, it is ssntial tat A is not just an algba in C, but vn a simpl and absolutly simpl symmtic spcial Fobnius algba. 25

26 As a fist stp w study ow concpts lik algbas and moduls ov algbas can b tanspotd fom C A A to C. If X is an objct of C A A, i.. an A-bimodul in C, w dnot t cosponding objct of C by Ẋ. Poposition 5.2. (i) Lt (, m, η ) b an algba in C A A. Tn (Ḃ, m,, η η) is an algba in C. (ii) If (C, C, ε C ) is a coalgba in C A A, tn (Ċ, C,C C, ε ε C ) is a coalgba in C. (iii) A mopism γ: of algbas in C A A is also a mopism of algbas in C. (iv) Lt (, m, η ) b an algba in C A A and (M, ρ) b a lft -modul in C A A. Tn (Ṁ, ρ,m) is a lft Ḃ-modul in C. Similaly igt -moduls and -bimoduls in C A A can b tanspotd to C. Fut, if f: (M, ρ M ) (N, ρ N ) is a mopism of lft -moduls in C A A, tn f is also a mopism of lft Ḃ-moduls in C, and an analogous statmnt olds fo mopisms of igt- and bimoduls. (v) If (, m,, η, ε ) is a Fobnius algba in C A A, tn (Ḃ, m,,,, η η, ε ε ) is a Fobnius algba in C. If is spcial in C A A, tn Ḃ is spcial in C. If is symmtic in C A A, tn Ḃ is symmtic in C. Poof. (i) Tat m is an associativ poduct fo in C A A mans tat m (id A m ) α,, m (m A id ), (5.1) w α,, is t associato dfind in (A.4). Aft insting t dfinitions of t tnso poduct of mopisms and t associato α,, tis ads m,, A, A m, m m, (5.2), A, A, ( A ) A ( A ) A Aft composing bot sids of tis quality wit t mopism A, (, id ) t sulting idmpotnts P, A, P A, and P, can b doppd fom t lft and sid, and P A, fom t igt and sid. W s tat m, is indd an associativ poduct fo Ḃ in C. Nxt 26

27 consid t mopism η ; w av m, m, m, m η, η η,a η,a m (id A η ) ρ A () 1 id, (5.3) wit ρ A t unit constaint as givn by (A.7) in t appndix. H in t fist stp t idmpotnt P, is intoducd and tn movd downwads, and likwis in t tid stp. T fift stp is t unit popty of η in C A A. So w s tat η η is indd a igt unit fo Ḃ in C, similaly on sows tat it is also a lft unit. (ii) is povd analogously to t pcding statmnt. (iii) Lt m and m dnot t poducts of and in C A A. Tn γ m, m (γ A γ), m, (γ γ) P, m, P, (γ γ) m, (γ γ), (5.4) w t tid quality uss tat γ is a mopism in C A A. Fut w av γ η η η η, as γ spcts t unit η of in C A A. So γ is also a mopism of algbas in C. (iv) T statmnt tat M is a lft -modul in C A A ads ρ (id A ρ) α,,m ρ (m A id M ), (5.5) wic is an quality in Hom A A (( A ) A M, M). Similaly to t poof in i), on sows tat tis indd implis tat (M, ρ,m ) is a lft Ḃ-modul in C. Now fo any mopism f: (M, ρ M ) (N, ρ N ) w av f ρ M,M ρ N (id A f),m ρ N,N (id f) P,M ρ N,N P,N (id f) ρ N,N (id f), (5.6) sowing tat f is a mopism of lft igt- and bimoduls. Ḃ-moduls in C. Similaly on vifis t conditions fo (v) To s tat t poduct and copoduct mopisms fo Ḃ satisfy t Fobnius popty in C 27

28 consid t following calculation:, m,, A, m m A,, A, α 1,,, (5.7) m, A,,,, A, A,, T fist quality is t asstion tat is a Fobnius algba in C A A, t scond on implmnts t dfinition of α 1,,. In t last stp t sulting idmpotnts a movd up o down, upon wic ty can b doppd. A paalll agumnt stabliss t scond idntity. Similaly on ccks tat spcialnss and symmty of Ḃ a tanspotd to C as wll. To apply t sults of t pvious sction to t algba A w also nd to dal wit Moita quivalnc in C A A. W stat wit t following obsvation. Lmma 5.3. Lt b a symmtic spcial Fobnius algba in C A A, and C and D b algbas in C A A and lt ( C M, ρ C, ϱ ) b a C--bimodul and ( N D, ρ, ϱ D ) a -D-bimodul. Tn t tnso poduct M N in C A A is isomopic, as a Ṁ Ḃ Ṅ ov t algba Ḃ in C. Ċ-Ḋ-bimodul in C, to t tnso poduct Poof. Sinc is symmtic spcial Fobnius in C A A, t idmpotnt PM,N cosponding to t tnso poduct of M and N ov is wll dfind in C A A. Explicitly it ads (ϱ A ρ ) α 1 M,, A N (id M A α,,n ) (id M A ( η ) A id N )) (id M A λ A (N) 1 ). (5.8) On calculats tat tis quals t mopism givn by t following mopism in C: M A N M,N M, ϱ ρ,n, η (5.9) M,N M A N Composing wit t mopisms M,N and M,N fo t tnso poduct ov A, w s tat P ḂṀ,Ṅ M,N P M,N M,N M,N M,N M,N M,N. Tis funiss a diffnt dcomposition 28

29 of P ḂṀ,Ṅ into a monic and an pi, nc t is an isomopism f: M N Ṁ Ḃ Ṅ of t imags of PM,N and P ḂṀ,Ṅ, suc tat f M,N M,N ḂṀ,Ṅ and M,N M,N ḂṀ,Ṅ f. Now t lft action of C on M N is givn by M N M N ρ C M,N M,N M,N M,N C A M,N ρ C M,N (ρ C A id N ) α 1 C,M,N (id C A M,N) C,M C A M,N M,N C,M M,N C,M A N C,M A N M,N M,N C,M N C,M N C A (M N) C A (M N) (5.10) Compos tis mopism fom t igt wit C,M N and dop t sulting idmpotnt to gt t tanspotd lft action of Ċ. Now composing wit f fom t lft and placing f M,N M,N by ḂṀ,Ṅ and M,N M,N by ḂṀ,Ṅ f sows tat f also inttwins t lft actions of Ċ. Similaly on sows tat f is also an isomopism of Ḋ-igt moduls. Coollay 5.4. Assum tat and C a symmtic spcial Fobnius algbas in C A A, and tat P,P C is a Moita contxt in C A A. Tn Ḃ P, P Ċ is a Moita contxt in C. Poof. Follows fom lmma 5.3 abov. T commutativity of t diagams (2.15) in C follows fom t commutativity of ti countpats in C A A. W a now in a position to gnalis t sults of t pvious sction to t cas of algbas in abitay Moita classs. Poposition 5.5. Lt H Pic(C A A ) b a finit subgoup of t Picad goup of C A A and assum tat dim A ( H L ) 0 fo psntativs L of H in C A A. Tn t xists an algba A in C and a Moita contxt,p A in C suc tat Π P,P maps H into t imag of Ψ A in Pic(C A A ). In ot wods, fo any H t is an algba automopism β of A suc tat t twistd bimodul id A β is isomopic to (P A L ). Poof. Applying poposition 4.3 to t tnso unit of D yilds, by t quivalnc D C A A, a symmtic spcial Fobnius algba in C A A and a Moita contxt A, in C A A, suc tat t a automopisms β of and -bimodul isomopisms F : ( A L ) A id β fo all H. y poposition 5.2 and coollay 5.4 tis givs is to a Moita contxt Ȧ P,P Ḃ in C, w P and P. tanspotd to C, s poposition 5.2. It follows tat (P A L ) id Ḃ β as Ḃ-bimoduls in C. T mopisms F main isomopisms of bimoduls wn 29

30 Sinc t algba Ḃ migt av a lag automopism goup in C, its imag und Ψ Ḃ migt b lag in Pic(CḂ Ḃ ). So w cannot conclud tat H im(ψḃ) in tis cas. ut as w av Aut A A () Aut C (Ḃ) as subgoups, w can still gnalis tom Tis funiss t main sult of tis pap: Tom 5.6. Lt C b a skltally small sovign ablian monoidal catgoy wit simpl and absolutly simpl tnso unit tat is nicd ov Vct k, wit k a fild of caactistic zo. Lt A b a simpl and absolutly simpl symmtic spcial Fobnius algba in C, and lt ψ b a nomalisd t-cocycl dscibing t associato of t Picad catgoy of C A A. Lt H b an admissibl subgoup of Pic(C A A ) (cf. dfinition 4.10). Tn t xist a symmtic spcial Fobnius algba A in C and a Moita contxt,p A suc tat fo ac tivialisation ω of ψ on H (cf. dfinition 4.7) t following olds. (i) T is an injctiv omomopism α ω : H Aut(A ) suc tat Π P,P Ψ A α ω id H. T assignmnt ω α ω is injctiv. (ii) T fixd algba of im(α ω ) Aut(A ) is isomopic to (H, ω), w (H, ω) is t algba (H, ω) in C A A as dscibd in poposition 4.8, tanspotd to C. Poof. (i) Dnot again by D t stictification of t bimodul catgoy C A A. y popositions 4.3 and 4.8 and tom 4.12 w find a symmtic spcial Fobnius algba in C A A and a Moita contxt A, in C A A suc tat t is a omomopism φ ω : H Aut A A () wit Π, Ψ : Aut A A () Pic(C A A ) as on-sidd invs. Accoding to poposition 5.2, Ḃ is a symmtic spcial Fobnius algba in C and,p Ḃ is a Moita contxt in C wit P and P. Fut, w can xtnd Ψ to Aut C (Ḃ) by putting Ψ Ḃ (γ) [ idḃγ] fo γ Aut C (Ḃ). Sinc w av Aut A A () Aut C (Ḃ) as a subgoup, φ ω tn givs a omomopism α ω : H Aut C (Ḃ) tat as Π P,P ΨḂ as on-sidd invs. As was sn in tom 4.12, t assignmnt ω φ ω is a bijction, and nc t assignmnt ω α ω is still injctiv. (ii) Put β : α ω (). Fom tom 4.13 w know tat t algba (H, ω) is isomopic to t fixd algba H in C A A. It coms togt wit an algba mopism i: (H, ω) and a coalgba mopism s: (H, ω) suc tat s i id (H,ω) and i s H 1 H β. Now lt f Hom C (X, Ḃ) b a mopism obying β f f fo all H. Tn fo f : s f: X (H, ω) w find i f i s f 1 H H β f f, i.. (H, ω) satisfis t univsal popty of t fixd algba in C. So ḂH (H, ω) as objcts in C. y poposition 5.2, i is still a mopism of Fobnius algbas wn tanspotd to C. It follows tat ḂH (H, ω) as Fobnius algbas in C. 30