Bodo Pareigis. Abstract. category. Because of their noncommutativity quantum groups do not have this

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1 Quantum Goups { Th Functoial Sid Bodo aigis Sptmb 21, 2000 Abstact Quantum goups can b intoducd in vaious ways. W us thi functoial constuction as automophism goups of noncommutativ spacs. This constuction shows in paticula why quantum goups should not b considd as goups in th catgoical sns. Th lina psntations of an odinay goup fom a symmtic monoidal catgoy. Bcaus of thi noncommutativity quantum goups do not hav this symmty of th tnso poduct of psntations. In many cass, howv, it can b placd by a baiding, fo xampl in th catgoy of YttDinfld moduls o of moduls ov a quasitiangula opf algba. A gnalization of th dnition of YttDinfld moduls lads to a catgoical iddl an xampl of a univsalcounivsal poblm, that is dnd by a simultanous unit and counit. Intoduction In this suvy pap w want togivanintoduction to quantum goups using a functoial constuction. It is wll known to th spcialists that quantum goups a not goups in a catgoy. W discuss how clos thy a to catgoical goups. This slight gnalization lads to an unusual bhavio of thi psntations. To gt a simila psntation thoy as fo goups on imposs a baid stuctu on th catgoy of psntations. W discuss what kind of lmnts of a quantum goup a sponsibl fo such baid stuctus and wh such baid stuctu occu natually. 1 Quantum Automophism Goups of Noncommutativ Spacs Quantum goups aos fom th dfomation of function algbas of (Li)goups. Function algbas of (Li)goups and mo gnally of manifolds a automatically commutativ, bcaus th functions hav valus in th ld of complx numbs o an abitay ld). But quantum physics quis noncommutativ function algbas. Dfomation tchniqus hav bn succssfully applid to classical Ligoups and to univsal nvloping algbas of Lialgbas to constuct quantum goups. In this suvy pap w want to pusu a dint appoach. W will constuct quantum goups as automophism goups of noncommutativ spacs. It is a classic tchniqu to dscib gomtic spacs X though thi function algba O(X )Fun(X C). Fo xampl stat spacs in physics a dscibd by thi algba of obsvabls. Und suitabl assumptions this lads to a duality btwn th catgoy 1

2 2 Bodo aigis of gomtic spacs and of commutativ algbas. Famous xampls a th duality of th GlfandNaimak Thom and th duality btwn an algbaic manifolds and nitly gnatd commutativ algbas. Quantum thoy focs us to consid noncommutativ algbas as \function algbas" of noncommutativ gomtic spacs o quantum spacs. Thus w dn Dnition 1.1 (Manin [6]) Th catgoy of quantum spacs o of noncommutativ spacs is th dual of th catgoy of noncommutativ (not ncssaily commutativ) algbas. This dnition should b sucint fo a catgoical mindd ad. But somhow on is missing th cosponding gomtic stuctus. Th hav bn sval attmpts to mak gomtic stuctus visibl. On of th simplst is th following. Th catgoy of quantum spacs can b idntid with th catgoy of covaiant psntabl functos on th catgoy Alg of not ncssaily commutativ algbas. Thus a quantum spac X with givn function algba A can b viwd as th psntabl functo X (B) Alg(A B). Th lmnts of X (B) a calld th Bpoints of X. Ths sts X (B) can b considd as a placmnt fo a gomtic spac associatd with th function algba A. Indd if A is psntd as A khx 1 x 2 i(p 1 (x i ) p 2 (x i ) ) a sidu class algba of th noncommutativ polynomial ing (f algba) on th vaiabls x 1 x 2, thn ach Bpoint f A ;! B is dscibd by th valus b 1 b 2 (with b i f(x i )) that a zos of th polynomials p 1 (x i ) p 2 (x i ).SoX (B) can b considd as th st of zos of ctain noncommutativ polynomials with coodinats in B. W want to constuct goups in th catgoy of quantum spacs. Fo this pupos w nd to know th \poduct" of two quantum spacs. Th a vaious asons basd in physics as wll as in algba not to us th catgoical poduct. W intoduc a poduct that is much \small" than th catgoical poduct. Quantum goups will b dnd with spct to this small poduct. Dnition 1.2 Lt X and Y b quantum spacs. Thn thi othogonal poduct is dnd by (X Y)(B) f( ) 2X(B) Y(B) j8a 2O(X ) a 0 2O(Y) (a)(a 0 )(a 0 )(a)g Th Bpoints and a calld commuting points. (Th imags of th functions and in B commut.) Th functo X Y is a psntabl functo and thus a quantum spac with psnting algba O(X ) O(Y). Th othogonal poduct dns th stuctu of a monoidal catgoy on th catgoy of quantum spacs. W want to dn an automophism goup fo a quantum spac X, in th catgoy of quantum spacs. That mans th goup to b constuctd should liv in th catgoy of quantum spacs and should act in a suitabl way on an oth quantum spac. Fo all considations w us th othogonal poduct instad of th catgoical poduct. W bgin with th notion of an action of on quantum spac M on anoth quantum spac X.

3 Quantum Goups { Th Functoial Sid 3 Dnition 1.3 Lt X and M b quantum spacs. A (lft) action of M on X is a mophism MX;!X. An action MX ;!X is calld a univsal action if fo vy action ZX ;! X th is a uniqu factoization f Z;!Msuch that th following diagam commuts ZX f 1 MX Q QQ QQs It is an opn qustion und which gnal conditions on X such univsal actions xist. But w hav th following Thom 1.4 (Tambaa [13]) Lt X b a quantum spac with nit dimnsional psnting algba A. Thn th xists a univsal action of a quantum spac on X. A univsal action pfoms a supising littl magic. It tuns out that a univsal action automatically cais a natual stuctu of a monoid and that this stuctu coms in such away that th action on th quantum spac X is a monoid action i.. it is associativ with unit. Futhmo th monoid stuctu and th monoid action a dnd with spct to th othogonal poduct, not with spct to th catgoical poduct. oposition 1.5 Lt MX ;!X b a univsal action. Thn M is in a uniqu way a monoid in th monoidal catgoy of quantum spacs with tnso poduct such that it acts on X by a monoid action (associativ with unit). Sktch of poof Th poof is actually quit simpl. W giv only th dnition of th multiplication of th monoid. Th commutativ diagam MMX id MX id X MX dns by th univsal popty of th action a uniqu mophism MM;!M. Fo this multiplication of M th action bcoms associativ by th sam diagam. W lav th st of th poof to th ad. A complt poof can b found in [8]. Thus M can b considd as th ndomophism monoid of X. It is calld th quantum ndomophism monoid of X. It tuns out to satisfy an additional univsal popty, it is th univsal monoid acting on X by a monoid action. Rmak 1.6 Th univsal action and th monoid stuctu of M tanslats back into th psnting algbas as a univsal algba homomophism A ;! B A, th stuctu of a bialgba on B (algba plus coalgba plus compatibility) and th stuctu of a Bcomodul algba on A though th algba homomophism A ;! B A. In paticula vy nit dimnsional algba A has a univsal bialgba B that maks A into a Bcomodul algba. X

4 4 Bodo aigis Thus w hav achivd pat of ou task to nd a quantum automophism goup of a quantum spac. In th simplst st thotic situation wwould nowhav to collct th invtibl ndomophisms of X and w would thn obtain th automophism goup with all its univsal poptis. But picking lmnts in M and chcking fo invtibility is not quit th ight way in th situation of quantum spacs. So w will b looking fo submonoids A of M that hav somthing lik an\invt" function S A;!Aand hopfully pick th bst such submonoid. Sinc a asonabl invs function S A;!Ados not xist fo most quantum monoids (it cannot xist if th quantum monoid is a pop noncommutativ spac, i.. th function algba is noncommutativ and if th multiplication of th quantum monoid is also noncommutativ) w hav to go to th cosponding function algbas. And th, indd, w may nd an invs function S ;! that bhavs vy much lik foming invss in a goup. Th poblm is that this invs function, usually calld th antipod, is only a lina map and not an algba homomophism. W dn th invs o antipod by Dnition 1.7 A bialgba is calld a opf algba if th is a (uniqu) (lina) map S ;!, th antipod, such that th diagam commuts. " C S1 1S 6 Th antipod of a opf algba is uniqu and it is an algba antihomomophism and a coalgba antihomomophism. This dnition was usd by Dinfld Dnition 1.8 (Dinfld [3]) A quantum goup is (th dual of) a (noncommutativ and noncocommutativ) opf algba. Th nxt poblm w fac is to nd a univsal quantum subgoup of a givn quantum monoid. This would thn psnt th st of invtibl lmnts of th monoid. This poblm has alady bn solvd many yas ago by M.Takuchi on th function algba sid. Thom 1.9 (Takuchi [14]) Lt B b a bialgba. Thn th xists a univsal opf algba (B) togth with a bialgba homomophism B ;! (B). This guaants that w may adjoin an antipod to any givn bialgba. Tanslatd back into th languag of quantum spacs th thom says that fo vy quantum monoid M th is a univsal quantum goup G(M) togth with a homomophism of quantum monoids G(M) ;!M. In th languag of st thotic monoids and goups this amounts to th constuction of th goup of invtibl lmnts G(M) of a monoid M. W hav obtaind th coct functo fo th \quantum automophism goup" of a quantum spac X. Indd whav Coollay 1.10 Lt X b a quantum spac that posssss a univsal action. Thn th is a univsal quantum goup G acting on X by a monoid action (o as an automophism goup).

5 Quantum Goups { Th Functoial Sid 5 2 Baidings and Rpsntations of Quantum Goups Lt G b a goup (in th catgoy of sts). A (complx vcto spac) psntation of G is a complx vcto spac togth with a modul action C[G] ;!. Th psntations of a goup hav two poptis that w would lik to s fo psntations of quantum goups as wll. Givn two psntations and Q of G thn Q is also a psntation of G with th action g (pq) gp gq. With this tnso poduct th catgoy C[G]Mod bcoms a monoidal catgoy. This monoidal catgoy is symmtic with th symmty map Q 3 p q 7! q p 2 Q. On asily chcks that this is a natual isomophism of C[G]moduls. Th function algba of G is C G, th commutativ algba of maps fom G to C. If G is a nit goup thn th isomophism om([g] ) om( C G ) tansfoms vy modul action of C[G] on to a comodul action of C G C[G] on.thus th catgoy of C G comoduls is quivalnt to th catgoy of C[G]moduls o psntations of G. Sinc th bialgba B fom th pvious sction is a function algba and th opf algba (B) is to b considd as a function algba as wll, th psntations of ths bialgbas should b comoduls ov B sp. ov (B). Comoduls ov B also fom a monoidal catgoy with (pq) p (0) q (0) p (1) q (1) 2 QB (Swdl notation). Sinc th axioms fo bialgbas B o opf algbas a slf dual, psntations could b dnd to b moduls o comoduls ov B sp.. Both dnitions giv a monoidal catgoy. Th pincipal qustion is now if th monoidal catgoy of moduls o of comoduls is symmtic o has at last a good intchang map Q ;! Q. If is cocommutativ thn th catgoy Mod is symmtic by th usual intchang map (p q) q p. If is commutativ thn th catgoy Comod is also symmtic by th usual intchang map (p q) q p. In gnal, howv, this is not th cas. To intoduc th appopiat tminology w dn Dnition 2.1 Lt C b a monoidal catgoy. A natual isomophism Q ;! Q in C is calld a baiding if th following diagams commut Q R 1(Q R) j Q R ( Q)1 j ( Q R) R Q ( QR) Q R R Q ( R)1 * Q R 1( R) * commut. (W assum that C is stict, so that w may omit th associativity mophisms.)

6 6 Bodo aigis W now stict ou discussion to ight moduls ov a opf algba with bijctiv antipod S and dvlop ncssay and sucint conditions fo Mod to hav a baiding. Simila considations hold fo comoduls. Bfo w intoduc a suitabl intchang map Q ;! Q, lt us study a simpl cas of a natual tansfomation. W consid th following univsal poblm. Givn an lmnt h 2. Thn this lmnt inducs a natual tansfomation h! ;!! fo th undlying functo! Mod ;! Vc by th commutativ diagam 1h!( ) Q QQ h Q Qs!( )!( ) Instad of ( )!( ) ;!!( )w simply wit ;!. univsal poblm This lads to th Thom 2.2 (aigis [9]) Fo vy natual tansfomation ;! (in Vc) th is a uniqu h 2 such that h Actually this thom says that th algba can b constuctd fom th natual ndomophisms of th undlying functo! Mod ;! Vc. Now w study a simila univsal poblm fo th tnso poduct!!. oposition 2.3 Fo vy natual tansfomation Q ;! Q th is a uniqu R 2 such that Q Q R Q Q It tuns out that th natual tansfomation Q ;! Q is a baiding fo th catgoy of ight moduls i R satiss th axioms of a univsal Rmatix, i.. 1. R is invtibl in, 2. ((h)) R(h)R ;1, 3. ( id)r R 13 R 23, 4. (id )R R 13 R 12,

7 Quantum Goups { Th Functoial Sid 7 wh R 12 R 1, R 23 1 R, andr 13 R 1 1 R 2 (with R R 1 R 2 ). A opf algba togth with a univsal Rmatix is calld a quasitiangula opf algba. This is actually what on would lik to call a quantum goup. But th dnition givn in th st sction has now bn gnally accptd, spcially in viw of th fact that th a also coquasitiangula opf algbas, i.. th catgoy of comoduls is baidd. In statistical physics ths univsal Rmatics povid solutions fo th quantum YangBaxt quation. So it is of gat intst to know if quasitiangula opf algbas xist and how to constuct thm. W will com back to this qustion lat on. 3 Baidings and YttDinfld Moduls Th is anoth stup of opf algbas that povids baidings fo f. Lt us consid th catgoy of vcto spacs that a ight moduls and ight comoduls at th sam tim togth with th following compatibility condition X (p(0) h (1) ) p (1) h (2) X (p h (2) ) (0) h (1) (p h (2) ) (1) o Ths moduls fom th catgoy YD of Ytt Dinfld moduls ov. Th compatibility condition has bn known fo long in th cas of cossd Gsts ov a goup G. Th magic of this compatibility condition is oposition 3.1 (Ytt [15]) Th catgoy of YttDinfld moduls YD baidd monoidal catgoy with th baiding is a ( Q) Q ;! Q p q 7! X q (0) p q (1) o Q Q So vy quantum goup givs is to a baidd monoidal catgoy, th catgoy YD. Actually on has an almost univsal popty. Thom 3.2 (Ytt [15]) Lt C b a small stict baidd monoidal catgoy togth with a monoidal functo F C;! Vc f to th catgoy of nit dimnsional

8 8 Bodo aigis vcto spacs. Thn th is a opf algba and a factoization F C Vc f F YD! Vc wh F psvs th baiding. So any study of baidings on nit dimnsional vcto spacs ducs to th study of YttDinfld moduls. This constuction xplains also th famous constuction of th Dinfld doubl Thom 3.3 Lt b a nit dimnsional opf algba. YttDinfld moduls YD Thn th catgoy of is quivalnt as a monoidal catgoy ov Vc to th catgoy of ight moduls ModD() ov th Dinfld doubl D() (with appopiat opf algba stuctu). In paticula ModD() is baidd and thus D() is quasitiangula. Th famous constuction of th Dinfld doubl fo nit dimnsional opf algbas povids a walth of quasitiangula opf algbas and thus of solutions of th quantum YangBaxt quation. 4 Baidings and Doubl Quantum Goups In th catgoy of YttDinfld moduls (and ov th quasitiangula Dinfld doubl) w s moduls and comoduls simultanously. W now study th qustion if a simila stup with two dint opf algbas is possibl. Dnition 4.1 (Bzzinski [1, 2]) Lt b an algba and b a coalgba and lt ;! b a lina map such that th following hold Thn ( ) is calld an ntwining stuctu. Th map is calld an ntwining map.

9 Quantum Goups { Th Functoial Sid 9 Dnition 4.2 Lt M ( ) b th catgoy of objcts that a simultanously comoduls and moduls ( ;! ;! )such that with spct to an ntwining stuctu ( ) o ( )( )( ) holds. Ths objcts will b calld ntwind moduls. Mophisms shall b modul and comodul mophisms. W consid th following biunivsal poblm. Givn a homomophism f ;! in Vc. Thn this homomophism inducs a natual tansfomation f! ;!! fo th undlying functo! M ( ) ;! Vc by th commutativ diagam!( )!( ) 1f q!( )!( ) Instad of f ( )!( ) ;!!( )w wit again ;!. \biunivsal" poblm This lads to th Thom 4.3 (obstaigis [4]) Fo vy natual tansfomation ;! (in Vc) th is a uniqu f ;! such that f If is a bialgba, thn th tnso poduct of two moduls is again an modul by th diagonal multiplication. Similaly, if is a bialgba, thn th tnso poduct of two comoduls is a comodul by th codiagonal comultiplication. Futhmo I( C) is a unit objct fo th tnso poduct if ndowd with th tivial stuctu sp. th tivial stuctu. W want to study conditions und which M ( ) bcoms a monoidal catgoy with th givn multiplication and comultiplication on th tnso poduct of two moduls. Th undlying functo will thn psv th tnso poduct, i.. it will b a monoidal functo. Thom 4.4 [4] Lt and b bialgbas. Th catgoy M ( ) is monoidal i th following additional compatibility conditions fo th ntwining map ;! hold

10 10 Bodo aigis and If ths conditions a satisd wcall ( ) a monoidal ntwining stuctu and a monoidal ntwining map. Th tnso poduct Q of moduls Q 2M ( ) ( ) with th diagonal modul and th codiagonal comodul bcoms an objct in M stuctu. An xtnsion of Thom 4.3 and oposition 2.3 is Thom 4.5 (obstaigis [4]) Fo vy natual tansfomation Q ;! Q th is a uniqu ;! such that Q Q Q Q Now whav all th basic tools to dtmin whn M ( ) bcoms a baidd monoidal catgoy. Thom 4.6 [4] Th natual tansfomation Q ;! Q is a baiding fo M ( ) if and only if th following conditions a satisd 1. is a mophism of moduls o quivalntly 2. is a mophism of comoduls o quivalntly

11 Quantum Goups { Th Functoial Sid is an isomophism o quivalntly th xists a map s ;! such that s s 4. is compatibl with tnso poducts o quivalntly and Rfncs [1] Bzzinski, T. On Moduls associatd to Coalgba Galois Extnsions. pint, qalg/ [2] Bzzinski, T. and ajac.m. Coalgba Extnsions and Algba Coxtnsions of Galois Typ. pint, qalg/ [3] Dinfld, V. G. Quantum goups. In ICM ocdings Bkly, [4] obst, D. and aigis, B. Doubl Quantum Goups. pint (30 p.) [5] Joyal, A. and Stt, R. Th Gomty of Tnso Calculus, I. Adv. Math. 88 (55112) [6] Manin, Y. Quantum Goups and Noncommutativ Gomty. Ls publications CRM, Univsit dmontal, [7] Montgomy, S. opf Algbas and thi Actions on Rings Am. Math. Soc., ovidnc, RI [8] aigis, B. Endomophism Bialgbas of Diagams and of Noncommutativ Algbas and Spacs. In Advancs in opf Algbas. Lctu Nots in u and Applid Mathmatics 158. Macl Dkk, 1994, [9] aigis, B. Nonadditiv Ring and Modul Thoy II. Ccatgois, Cfunctos, and Cmophisms. ubl. Math. Dbcn 24 (351361) 1977.

12 12 Bodo aigis [10] aigis, B. Rconstuction of iddn Symmtis. J. Algba 183 (90154) [11] aigis, B. and Sommhaus, Y. Gaphic Calculus of Baidd Catgois. pint [12] nos, R. Applications of Ngativ Dimnsional Tnsos. In Combinatoial Mathmatics and its Applications. Acadmic ss. (221244) [13] Tambaa, D. Th condomophism bialgba of an algba. J. Fac. Sci. Univ. Tokyo 37, 1990, [14] Takuchi, M. F opf Algbas gnatd by Coalgbas. J. Math. Soc. Japan 23, No.4, [15] Ytt, D. N. Quantum Goups and Rpsntations of Monoidal Catgois. Math. oc. Camb. hil. Soc. 108 No. 2 (261290) 1990.

E F. and H v. or A r and F r are dual of each other.

E F. and H v. or A r and F r are dual of each other. A Duality Thom: Consid th following quations as an xampl = A = F μ ε H A E A = jωa j ωμε A + β A = μ J μ A x y, z = J, y, z 4π E F ( A = jω F j ( F j β H F ωμε F + β F = ε M jβ ε F x, y, z = M, y, z 4π