Generalized Lie Theory and Applications

Transcription

1 Gnralizd Li Thory and pplications ISSN: Journal of Gnralizd Li Thory and pplications lxandr t al., J Gnralizd Li Thory ppl 206, 0: DOI: 0.472/ Rsarch rticl Opn ccss Structur Thory of Rac-Bialgbras lxandr C, Bordmann M 2, Rivièr S 3 and Wagmann F 3 * Dpartmnt of Mathmatics, Univrsité d Strasbourg, 4 Ru Blais Pascal, 6708 Strasbourg, Franc 2 Laboratoir d Mathématiqus, Informatiqu t pplications, Univrsité d Mulhous, Franc 3 Dpartmnt of Mathmatics, Univrsité d Nants, Quai d Tourvill, Nants Cdx, Franc bstract In this papr w focus on a crtain slf-distributiv multiplication on coalgbras, which lads to so-calld rac bialgbra. Inspird by smi-group thory (adapting th Suschwitsch thorm, w do som structur thory for rac bialgbras and cocommutativ Hopf dialgbras. W also construct canonical rac bialgbras (som ind of nvloping algbras for any Libniz algbra and compar to th xisting constructions. W ar motivatd by a diffrntial gomtric procdur which w call th Srr functor: To a pointd diffrntibl manifold with multiplication is ociatd its distribution spac supportd in th chosn point. For Li groups, it is wllnown that this lads to th univrsal nvloping algbra of th Li algbra. For Li racs, w gt rac-bialgbras, for Li digroups, w obtain cocommutativ Hopf dialgbras. Kywords: Coalgbras; Cocommutativ Hopf dialgbras; Canonical rac bialgbras; Manifolds; Drinfld cntr Introduction ll manifolds considrd in this manuscript ar umd to b Hausdorff and scond countabl. Basic Li thory rlis havily on th fundamntal lins btwn ociativ algbras, Li algbras and groups. Som of ths lins ar th pag from an ociativ algbra to its undrlying Li algbra Li which is th vctor spac with th bract [a, b] :=ab ba. On th othr hand, to any Li algbra g on may ociat its univrsal nvloping algbra U(g which is ociativ. Groups aris as groups of units in ociativ algbras. To any group G, on may ociat its group algbra KG which is ociativ. Th thm of th prsnt articl is to invstigat lins of this ind for mor gnral objcts than groups, namly for racs and digroups. Rcall that a pointd rac is a pointd st (X, togthr with a binary opration :X X X such that for all x X, th map y x y is bijctiv and such that for all x, y, z X, th slf-distributivity and unit rlations x(yz=(xy(xz, x=x, and x= ar satisfid. Imitating th notion of a Li group, th smooth vrsion of a pointd rac is calld Li rac. n important cl of xampls of racs ar th so-calld augmntd racs []. n augmntd rac is th data of a group G, a G st X and a map p: X G such that for all x X and all g G, p(g x=gp(xg -. Th st X bcoms thn a rac by stting x y:=p(x y. In fact, augmntd racs ar th Drinfld cntr (or th Yttr-Drinfld moduls in th monoidal catgory of G-sts ovr th (st-thortical Hopf algbra G, s for xampl [2]. ny rac may b augmntd in many ways, for xampl by using th canonical morphism to its ociatd group or to its group of bijctions or to its group of automorphisms. In ordr to formaliz th notion of a rac, on nds th diagonal map diag x : X X X givn by x (x, x. Thn th slf-distributivity rlation rads in trms of maps m (id M m, =m (m m ( id M τ M, M id M (diag M id M id M xiomatizing this ind of structur, on may start with a coalgbra C and loo for rac oprations on this fixd coalgbra [3,4]. natural framwor whr this ind of structur ariss (as w show in Sction 3 is by taing point-distributions (i.. applying th Srr functor ovr (rsp. to th pointd manifold givn by a Li rac. W dub th arising structur as rac bialgbra. Li racs ar intimatly rlatd to Libniz algbra h, i.. a vctor spac h with a bilinar bract [,]:h h h such that for all X, Y, Z h, [X, ] acts as a drivation: [X,[Y,Z]]=[[X,Y],Z] + [Y,[X,Z]]. ( Indd, Kinyon showd that th tangnt spac at H of a Li rac H carris a natural structur of a Libniz algbra, gnralizing th rlation btwn a Li group and its tangnt Li algbra [5]. Convrsly, vry (finit dimnsional ral or complx Libniz algbra h may b intgratd into a Li rac R h (with undrlying manifold h using th rac product. ad X X Y:= ( Y, (2 noting that th xponntial of th innr drivation ad X for ach X h is an automorphism. nothr closly rlatd algbraic structur is that of dialgbras. dialgbra is a vctor spac D with two (bilinar ociativ oprations :D D D and :D D D which satisfy thr compatibility rlations, namly for a, b, c D: (abc=(a bc, a(bc=a(b c,(ab c=a (b c *Corrsponding author: Wagmann F, Dpartmnt of Mathmatics, Univrsité d Nants, Quai d Tourvill, Nants Cdx, Franc, Tl: , Fridrich.Wagmann@univ-nants.fr Rcivd July 7, 206; ccptd Novmbr 9, 206; Publishd Novmbr 30, 206 Citation: lxandr C, Bordmann M, Rivièr S, Wagmann F (206 Structur Thory of Rac-Bialgbras. J Gnralizd Li Thory ppl 0: 244. doi:0.472/ Copyright: 206 lxandr C, t al. This is an opn-accss articl distributd undr th trms of th Crativ Commons ttribution Licns, which prmits unrstrictd us, distribution, and rproduction in any mdium, providd th original author and sourc ar crditd. J Gnralizd Li Thory ppl, an opn accss journal ISSN: Volum 0 Issu

2 Citation: lxandr C, Bordmann M, Rivièr S, Wagmann F (206 Structur Thory of Rac-Bialgbras. J Gnralizd Li Thory ppl 0: 244. doi:0.472/ Pag 2 of 20 dialgbra D bcoms a Libniz algbras via th formula [a,b]=ab b a. In this sns and ar two halvs of a Libniz bract. Loday and Goichot hav dfind an nvloping dialgbra of a Libniz algbra [6,7]. On main point of this papr is th lin btwn rac bialgbras and cocommutativ Hopf dialgbras. In Thorm 2.5, w adapt Suschwitsch s Thorm in smi-group thory to th prsnt contxt. Th clical rsult (s ppndix B trats smi-groups with a lft unit and right invrss (analoguous rsults in th lft-contxt, calld right groups. Suschwitsch shows that such a right group Γ dcomposs as a product Γ=Γ E whr E is th st of all idmpotnt lmnts. Its incarnation hr shows that a cocommutativ right Hopf algbra dcomposs as a tnsor product E whr E is th subspac of gnralizd idmpotnts. Furthrmor, w will show in Thorm 2.6 how to ociat to any augmntd rac bialgbra an augmntd cocommutativ Hopf dialgbra. In Thorm 2.7, w invstigat what Suschwitsch s dcomposition givs for a cocommutativ Hopf dialgbra. It turns out that dcomposs as a tnsor product E H of E with H which may b idntifid to th ociativ quotint of. This rsult prmits to show that th Libniz algbra of primitivs in is a hmismi-dirct product, and thus always split. In this way w arriv onc again at th rsult that Li digroups may srv only to intgrat split Libniz algbras which has alrady bn obsrvd by Covz in his mastr thsis [8]. Lt us commnt on th contnt of th papr: ll our bialgbra notion ar basd on th standard thory of coalgbras, som faturs of which as wll as our notions ar rcalld in ppndix. Rac bialgbras and augmntd rac bialgbras ar studid in Sction 2. Connctd, cocommutativ Hopf algbras giv ris to a spcial cas of rac bialgbras. In Sction 2.2, w ociat to any Libniz algbra h an augmntd rac bialgbra UR (h and study th functorial proprtis of this ociation. This rac bialgbra plays th rol of an nvloping algbra in our contxt. It turns out that a truncatd, non-augmntd vrsion UR(h is a lft adjoint of th functor of primitivs Prim. W also study th group-algbra functor ociating to a rac X its rac bialgbra K[X]. Li in th clical framwor, K[ ] is lft adjoint to th functor Sli ociating to a trac bialgbra its rac of st-li lmnts. Th rlation btwn rac bialgbras and th othr algbraic notion discussd in this papr is summarizd in th diagram (s th nd of Sction 2.2 of catgoris and functors: L i@< x >[ >[ d]h opf@ >[ d] [ x >[ r] G rp[ l] U ( i ( j Prim S li K[ ] ( UR Prim S li K[ >[ d]l ib@< x >[ r] R acbialg[ x >[ r] R acs[ l] In Sction 2.3, w dvlop th structur thory for rac bialgbras and cocommutativ Hopf dialgbras, basd on Suschwitsch s Thorm. Sction 2.3 contains Thorm 2.5, Thorm 2.6 and Thorm 2.7 whos contnt w hav dscribd abov. Rcollcting basic nowldg about th Srr functor F is th subjct of Sction 3. In particular, w show in Sction 3.2 that F is a strong monoidal functor from th catgory of pointd manifolds f * to th catgory of coalgbras, basd on som standard matrial on coalgbras (ppndix. In Sction 3.3, w apply F to Li groups, Li smi-groups, Li digroups, and to Li racs and augmntd Li racs, and study th additional structur which w obtain on th coalgbra. Th cas of Li racs motivats th notion of rac bialgbra. Rcall that for a Libniz algbra h, th vctor spac h bcoms a Li rac R h with th rac product. ad X XY = ( Y. In Thorm 3.8, w show that th rac bialgbra UR (h ociatd to h coincids with th rac bialgbra F(R h. Svral Bialgbras In th following, lt K b an ociativ commutativ unital ring containing all th rational numbrs. Th symbol will always dnot th tnsor product of K-moduls ovr K. For any coalgbra (C, ovr K, w shall us Swdlr s notation (a= (a a ( a (2 for any a. S also ppndix 4 for a survy on dfinitions and notations in coalgbra thory. Th following sctions will all dal with th following typ of nonociativ bialgbra: Lt (B,,ε,,µ b a K-modul such that (B,,ε, is a coociativ counital coaugmntd coalgbra (a C 3 - coalgbra, and such that th linar map µ: B B B (th multiplication is a morphism of C 3 -coalgbras (it satisfis in particular µ( =. W shall call this situation a nonociativ C 3 I-bialgbra (whr I stands for bing an idmpotnt for th multiplication µ. For anothr nonociativ C 3 I bialgbra (B,,,, µ a K-linar map φ: B B will b calld a morphism of nonociativ C 3 I-bialgbras iff it is a morphism of C 3 -coalgbras and is multiplicativ in th usual sns φ(µ(a b=µ (φ(a φ(b for all a, b B. Th nonociativ C 3 I- bialgbra (B,,ε, is calld lft-unital (rsp. right-unital iff for all a B µ( a=a (rsp. µ(a =a. Morovr, considr th ociativ algbra Hom K (B,B quippd with th composition of K-linar maps, and th idntity map id B as th unit lmnt. Thr is an ociativ convolution multiplication * in th K-modul Hom K (B, of all K-linar maps B Hom K (B,B, s ppndix 4, qn (03 for a dfinition with id B ε as th unit lmnt. For a givn nonociativ C 3 I-bialgbra (B,,,,µ w can considr th map µ as a map B Hom K (B,B in two ways: as lft multiplication map L µ : b(bµ(b b or as right multiplication map R µ :b(b µ(b b. W call (B,,ε,,µ a lft-rgular (rsp. right-rgular nonociativ C 3 I- bialgbra iff th map L µ (rsp. th map R µ has a convolution invrs, i.. iff thr is a K-linar map µ : B B B (rsp. µ : B B B such that L µ =L µ =id B ε=l µ L µ (rsp. R µ R µ =id B ε=r µ R µ, or on lmnts a, b B for th lft rgular cas: ( (2 ( (2 µ ( a µ ( a b = ε ( a b= µ ( a µ ( a b. (3 ( a Not that vry ociativ unital Hopf algbra(h,,ε,,µ, S (whr S dnots th antipod, i.. th convolution invrs of th idntity map in Hom K (H,H is right- and lft-rgular by stting µ =µ (S id H and µ =µ ( id H S. Lmma 2.: Lt (B,,ε,,µ b a nonociativ C 3 I-bialgbra.. If B is lft-rgular (rsp. right-rgular, thn th corrsponding K-linar map µ : B B B is uniqu, and in cas is cocommutativ, µ is map of C 3 -coalgbras. 2. If (B,,ε,,µ is lft-unital (rp. right-unital and its undrlying C 3 -coalgbra is connctd, thn (B,,ε,,µ is always lft-rgular (rsp. right-rgular. Proof:. In any monoid (in particular in th convolution monoid two-sidd invrss ar always uniqu. Morovr, as can asily b J Gnralizd Li Thory ppl, an opn accss journal ISSN: Volum 0 Issu

3 Citation: lxandr C, Bordmann M, Rivièr S, Wagmann F (206 Structur Thory of Rac-Bialgbras. J Gnralizd Li Thory ppl 0: 244. doi:0.472/ Pag 3 of 20 chcd, a K-linar map φ: B B B is a morphism of coalgbras iff for ach b B. ( (2 ( L φ ( b L φ ( b = L φ ( b. (4 ( b (and analogously for right multiplications. Both sids of th prcding quation, sn as maps of b, ar in Hom K (B,Hom K (B,B B. Sinc Hom K (B,B B is an obvious right Hom K (B,B-modul, th K-modul Hom K (B,Hom K (B,B B is a right Hom K (B,Hom K (B,B-modul with rspct to th convolution. Dfin th K-linar map F µ : B Hom (B,B B by: K ( (2 F µ ( b := ( L µ ( b L µ ( b L µ ( b. ( b Using qn (4, th fact that L µ is a convolution invrs of L µ, and th cocommutativity of, w gt: F * L = 0, hnc 0 = F * L * L = F *(id ε = F µ µ µ µ µ µ µ B and µ prsrvs comultiplications. similar rasoning whr B B is rplacd by K shows that µ prsrvs counits. Finally, it is obvious that L µ ( is th invrs of th K-linar map L µ (, and sinc th lattr fixs so dos th formr. Th rasoning for right-rgular bialgbras is compltly analogous. 2. For lft-unital bialgbras w gt L µ (=id B, and th gnralizd Tauchi-Swdlr argumnt, s ppndix 4, shows that L µ has a convolution invrs. Right-unital bialgbras ar tratd in an analogous mannr. Not that any C 3 -coalgbra (C,,ε,, bcoms a lft-unital (rsp. right-unital ociativ C 3 I-bialgbra by quipping with th lft-trivial (rsp. right-trivial multiplication. µ 0 (a b ε(ab (rsp.µ 0 (a b ε(ba. (5 W shall call an lmnt c B a gnralizd idmpotnt iff (c c ( c (2 =c. Morovr c B will b calld a gnralizd lft (rsp. right unit lmnt iff for all b B w hav cb=ε(cb (rsp. bc=ε(cb. Rac bialgbras, augmntd rac bialgbras and Libniz algbras Dfinition 2.: rac bialgbra (B,,ε,,µ is a nonociativ C 3 I- bialgbra (whr w writ for all a, b B µ(a b=:ab such that th following idntitis hold for all a, b,c B a=a, (6 a=ε(a, (7 a bc a b a c (8 ( (2 ( = ( (. ( a Th last condition (8 is calld th slf-distributivity condition. Not that w do not dmand that th C 3 -coalgbra B should b cocommutativ nor connctd. Exampl 2.: ny C 3 coalgbra (C,,ε, carris a trivial rac bialgbra structur dfind by th lft-trivial multiplicaton a 0 b ε(ab (9 which in addition is asily sn to b ociativ and lft-unital, but in gnral not unital. nothr mthod of constructing rac bialgbras is gauging: Lt (B,,ε,,µ a rac bialgbra whr w writ µ(a b=:ab for all a, b B, and lt f: B B a morphism of C 3 -coalgbras such that for all a, b B f (ab=a(f (b, (0 i.. f is µ-quivariant. It is a routin chc that (B,,ε,,µ f is a rac bialgbra whr for all a, b B th multiplication is dfind by µ f (a b=:a b:=(f (ab. ( f W shall call (B,,ε,,µ f th f-gaug of (B,,ε,,µ. Exampl 2.2: Lt (H, H,ε H,µ H, H,S b a cocommutativ Hopf algbra ovr K. Thn it is asy to s (cf. also th particular cas B=H and Φ=id H of Proposition 2. that th nw multiplication µ(h H H, writtn µ( h h =hh, dfind by th usual adjoint rprsntation ( (2 h h := ad h( h := h h ( Sh (, (2 ( h quips th C 4 -coalgbra (H, H,ε H, H with a rac bialgbra structur. In gnral, th adjoint rprsntation dos not sm to prsrv th coalgbra structur if no cocommutativity is umd. Exampl 2.3: Rcall that a pointd st (X, is a pointd rac in cas thr is a binary opration :X X X such that for all x X, th map y x y is bijctiv and such that for all x,y,z X, th slfdistributivity and unit rlations: x(yz=(xy(xz, x=x, and x= ar satisfid. Thn thr is a natural rac bialgbra structur on th vctor spac K[X] which has th lmnts of X as a basis. K[X] carris th usual coalgbra structur such that all x X ar st-li: (x=x x for all x X. Th product µ is thn inducd by th rac product. By functoriality, µ is compatibl with and. Obsrv that this construction diffrs slightly from th construction, Sction 3.. Mor gnrally thr is th following structur: Dfinition 2.2: n augmntd rac bialgbra ovr K is a quadrupl (B,Φ,H,l consisting of a C 3 -coalgbra (B,,ε,, of a cocommutativ (! Hopf algbra (H, H,ε H, H, µ H, S, of a morphism of C 3 -coalgbras Φ: B H, and of a lft action l: H B B of H on B which is a morphism of C 3 -coalgbras (i.. B is a H-modul-coalgbra such that for all h H and a B h.=ε H (h (3 (h.a=ad (Φ(a (4 Whr ad dnots th usual adjoint rprsntation for Hopf algbras, s.g. qn (2. W shall dfin a morphism (B,Φ,H, l (B,Φ,H, l of augmntd rac bialgbras to b a pair (φ,ψ of K-linar maps whr φ: (B,,ε, (B,,ε, is a morphism of C 3 -coalgbras, and ψ: H H is a morphism of Hopf algbras such that th obvious diagrams commut: Φ φ=ψ Φ, and l ( φ=φ l (5 n immdiat consqunc of this dfinition is th following: Proposition 2.: Lt (B,Φ,H, l b an augmntd rac bialgbra. Thn th C 3 -coalgbra (B,ε, will bcom a lft-rgular rac bialgbra by mans of th multiplication: ab Φ(a.b (6 for all a,b B. In particular, ach Hopf algbra H bcoms an augmntd rac bialgbra via (H,id H,H,ad. In gnral, for ach augmntd rac bialgbra th map Φ:B H is a morphism of rac bialgbras. Proof: For a proof of this proposition [9]. J Gnralizd Li Thory ppl, an opn accss journal ISSN: Volum 0 Issu

4 Citation: lxandr C, Bordmann M, Rivièr S, Wagmann F (206 Structur Thory of Rac-Bialgbras. J Gnralizd Li Thory ppl 0: 244. doi:0.472/ Pag 4 of 20 Exampl 2.4: Exactly in th sam way as a pointd rac givs ris to a rac bialgbra K[X], an augmntd pointd rac p: X G givs ris to an augmntd rac bialgbra p: K[X] K[G]. Rmar 2.: Motivatd by th fact that th augmntd racs p: X G ar xactly th Yttr-Drinfld moduls ovr th (st-thortical Hopf algbra G, w may as about th rlation of augmntd rac bialgbras to Yttr-Drinfld moduls, and mor gnrally of rac bialgbras to th Yang-Baxtr quation. For ths subjcts. Th lin to Libniz algbras is containd in th following: Proposition 2.2: Lt (B,,ε,, µ b a rac bialgbra ovr K.. Thn its K-submodul of all primitiv lmnts, Prim(B=:h, (s qn (0 of ppndix 4 is a subalgbra with rspct to µ (writtn ab satisfying th (lft Libniz idntity: x(yz=(xyz+y(xz (7 for all x, y, z h=prim(b. Hnc th pair (h,[,] with [x, y] xy for all x, y h is a Libniz algbra ovr K. Morovr, vry morphism of rac bialgbras maps primitiv lmnts to primitiv lmnts and thus inducs a morphism of Libniz algbras. 2. Mor gnrally, h and ach subcoalgbra of ordr, B (, (s qn (02 is stabl by lft -multiplications with vry a B. In particular, ach B ( is a rac subbialgbra of (B,,ε,, µ. Proof: 2. Lt x h and a B. Sinc µ is a morphism of C 3 -coalgbras and x is primitiv, w gt: ( (2 ( (2 ( ax= ( a x ( a + ( a ( a x ( a ( a (7 ( (2 ( (2 ( a ( a = (( a ε( a x + (( ε( a a x =(ax + (ax, whnc ax is primitiv. For th statmnt on th B (, w procd by induction: For =0, this is clar. Suppos th statmnt is tru until, and lt x B (+. Thn: (ax (ax (ax ( ( (2 (2 ( (2 = (( a x ( a x ( a x ( a ( a( x (a ( (a (2 x =( (a( (x x x ( ( (2 (2 = ( a x ( a x ( a( x whr w hav usd th xtndd multiplication (still dnotd :(B B (B B (B B and st: ( x x ( (2 x=: x x B( B( ( x by th dfinition of B (+, s ppndix 4. By th induction hypothsis, all th trms a ( x ( and a (2 x (2 ar in B (, whn (ax (ax (ax is in B ( B (, implying that ax is in B (+.. It follows from 2. that h is a subalgbra with rspct to µ. Lt x, y, z h. Thn sinc x is primitiv, it follows from (x=x + x and th slf-distributivity idntity (8 that: x( yz=( x y ( z + ( y ( xz=( x y z+ y( xz. proving th lft Libniz idntity. Th morphism statmnt is clar, sinc ach morphism of rac bialgbras is a morphism of C 3 -coalgbras and prsrvs primitivs. (6 Libniz algbras hav bn invntd by. M. Bloh in 965, and thn rdiscovrd by J.-L. Loday in 992 in th sarch of an xplanation for th absnc of priodicity in algbraic K-Thory [0,]. s an immdiat consqunc, w gt that th functor Prim inducs a functor from th catgory of all rac bialgbras ovr K to th catgory of all Libniz algbras ovr K. Rmar 2.2: Dfin st-li lmnts to b lmnts a in a rac bialgbra B such that (a=a a. Thans to th fact that is a morphism of coalgbras, th st of st-li lmnts Sli(Bis closd undr. In fact, Sli(B is a rac, and on obtains in this way a functor Sli: RacBialg Racs. Proposition 2.3: Th functor of st-lis Sli: RacBialg Racs has th functor K[ ]: Racs RacBialg (s Exampl 2.3 as its lftadjoint. Proof: This follows from th adjointnss of th sam functors, sn as functors btwn th catgoris of pointd sts and of C 4 -coalgbras, obsrving that th C 4 -coalgbra morphism inducd by a morphism of racs rspcts th rac product. Obsrv that th rstriction of Sli: RacBialg Racs to th subcatgory of connctd, cocommutativ Hopf algbras Hopf (whr th Hopf algbra is givn th rac product dfind in qn (2 givs th usual functor of group-li lmnts. (ugmntd rac bialgbras for any Libniz algbra Lt (h,[,] b a Libniz algbra ovr K, i.. h is a K-modul quippd with a K-linar map [,]:h h satisfying th (lft Libniz idntity (. Rcall first that ach Li algbra ovr K is a Libniz algbra giving ris to a functor from th catgory of all Li algbras to th catgory of all Libniz algbras. Furthrmor, rcall that ach Libniz algbra has two canonical K-submoduls: Q(h {x h N \{0}, λ,,λn K, x,, x N N such that x= λ [ x, x ]}, (8 r r r r= z(h {x h y h:[x,y]=0}. (9 It is wll-nown and not hard to dduc from th Libniz idntity that both Q(h and z(h ar two-sidd ablian idals of (h,[,], that Q(h z(h, and that th quotint Libniz algbras: h:= h/ Q( h and gh (:= h/ zh ( (20 ar Li algbras. Sinc th idal Q(h is clarly mappd into th idal Q(h by any morphism of Libniz algbras h h (which is a priori not th cas for z(h!, thr is an obvious functor h h from th catgory of all Libniz algbras to th catgory of all Li algbras. In ordr to prform th following constructions of rac bialgbras for any givn Libniz algbra (h,[,], choos first a two-sidd idal z h such that: Q(h z z(h, (2 Lt g dnot th quotint Li algbra h/z, and lt p:h g b th natural projction. Th data of z h, i.. of a Libniz algbra h togthr with an idal z such that Q(h z z(h, could b calld an augmntd Libniz algbra. Thus w ar actually ociating an augmntd rac bialgbra to vry augmntd Libniz algbra. In fact, w will s that J Gnralizd Li Thory ppl, an opn accss journal ISSN: Volum 0 Issu

5 Citation: lxandr C, Bordmann M, Rivièr S, Wagmann F (206 Structur Thory of Rac-Bialgbras. J Gnralizd Li Thory ppl 0: 244. doi:0.472/ Pag 5 of 20 this augmntd rac bialgbra dos not dpnd on th choic of th idal z and thrfor rfrain from introducing augmntd Libniz algbras in a mor formal way. Th Li algbra g naturally acts as drivations on h by mans of (for all x, y h P(x.y [x, y]=:ad x (y (22 bcaus z z(h. Not that: h/ z(h {ad x Hom K (h,h x h}. (23 as Li algbras. Considr now th C 5 -coalgbra (B=S(h,,, which is actually a commutativ cocommutativ Hopf algbra ovr K with rspct to th symmtric multiplication. Th linar map p:h g inducs a uniqu morphism of Hopf algbras: Φ = S( p : S( h S( g (24 satisfying Φ ( x x= px ( px ( (25 for any nonngativ intgr and x,,x h. In othr words, th ociation S:V S(V is a functor from th catgory of all K-moduls to th catgory of all commutativ unital C 5 -coalgbras. Considr now th univrsal nvloping algbra U(g of th Li algbra g. Sinc K by umption, th Poincaré-Birhoff-Witt Thorm (in short: PBW holds [2]. Mor prcisly, th symmtrisation map ω: S(g U(g, dfind by: ω( S( = U(, and ω ( = σ( σ(, (26 g g! σ S is an isomorphism of C 5 -coalgbras (in gnral not of ociativ algbras [3]. W now nd an action of th Hopf algbra H=U(g on B, and an intrtwining map Φ: B U(g. In ordr to gt this, w first loo at g-moduls: Th K-modul h is a g-modul by mans of qn (22, th Li algbra g is a g-modul via its adjoint rprsntation, and th linar map p:h g is a morphism of g-moduls sinc p is a morphism of Libniz algbras. Now S(h and S(g ar g-moduls in th usual way, i.. for all \{0},,,, g, and x,,x h.( x x := x (. x x, (27 r r =.( := [. ], (28 r r = and of cours. S(h =0 and. S(g =0. Rcall that U(g is a g-modul via th adjoint rprsntation ad (u=.u=u u (for all g and all u U(g. It is asy to s that th map Φ (25 is a morphism of g-moduls, and it is wll-nown that th symmtrization map ω (26 is also a morphism of g-moduls. Dfin th K-linar map Φ: S(g U(g by th composition: Φ:= Φ. (29 Thn Φ is a map of C 5 -coalgbras and a map of g-moduls. Thans to th univrsal proprty of th univrsal nvloping algbra, it follows that S(h and U(g ar lft U(g-moduls, via (for all,, g, and for all a S(h (.a=.( 2.(.a (30 and th usual adjoint rprsntation (2 (for all u U(g ad ( u = (ad ad ( u, (3 and that Φ intrtwins th U(g-action on C=S(h with th adjoint action of U(g on itslf. Finally it is a routin chc using th abov idntitis (27 and (2 that S(h bcoms a modul coalgbra. W can rsum th prcding considrations in th following; Thorm 2.: Lt (h,[,] b a Libniz algbra ovr K, lt z b a twosidd idal of h such that Q(h z z(h, lt g dnot th quotint Li algbra h/z by g, and lt p:h g b th canonical projction.. Thn thr is a canonical U(g-action l on th C 5 -coalgbra B:=S(h (maing it into a modul coalgbra laving invariant and a canonical lift of p to a map of C 5 -coalgbras, Φ: S(h U(g such that qn (4 holds. Hnc th quadrupl (S(h, Φ, U(g, l is an augmntd rac bialgbra whos ociatd Libniz algbra is qual to (h,[,] (indpndntly of th choic of z. Th rsulting rac multiplication µ of S(h (writtn µ(a b=a b is also indpndnt on th choic of z and is xplicitly givn as follows for all positiv intgrs,l and x,,x, y,,y h: ( x s s x ( y y l= (adx ad ( ( x ( l! y y (32 σ σ σ S whr ad s x dnots th action of th Li algbra h/z(h (s qn (23 on S(h according to qn ( In cas z=q(h, th construction mntiond in. is a functor h UR (h from th catgory of all Libniz algbras to th catgory of all augmntd rac bialgbras ociating to h th rac bialgbra: UR (h (S(h, Φ, U(g, l and to ach morphism f of Libniz algbras th pair (S( f, U( f whr f is th inducd Li algbra morphism. 3. For ach nonngativ intgr, th abov construction r rstricts to ach subcoalgbra of ordr, S( h ( = r=0s( h, to dfin an augmntd rac bialgbra (S( h (, Φ(, U( g, U( g S( h ( which in cas z=q(h dfins a functor h UR ( (h (UR (h ( from th catgory of all Libniz algbras to th catgory of all augmntd rac bialgbras. Rmar 2.3: This thorm should b compard to Proposition 3.5. [3]. In [3], th authors wor with th vctor spac N K h, whil w wor with th whol symmtric algbra on th Libniz algbra. In som sns, w xtnd thir Proposition 3.5 to all ordrs. Howvr, as w shall s blow, N is alrady nough to obtain a lft-adjoint to th functor of primitivs. Th abov rac bialgbra ociatd to a Libniz algbra h can b sn as on vrsion of an nvloping algbra of h. Dfinition 2.3: Lt h b a Libniz algbra. W will call th augmntd rac bialgbra (S(h, Φ, U(g, l th nvloping algbra of infinit ordr of h. s such, it will b dnotd by UR (h. This trminology is justifid, for xampl, by th fact that h is idntifid to th primitivs in S(h (cf Proposition 2.2. This is also justifid by th following thorm th goal of which is to show that th nvloping algbra UR (h fits into th following diagram of functors: Li[r] U [d] i Hopf[d] j Lib[r] UR RacBialg J Gnralizd Li Thory ppl, an opn accss journal ISSN: Volum 0 Issu

6 Citation: lxandr C, Bordmann M, Rivièr S, Wagmann F (206 Structur Thory of Rac-Bialgbras. J Gnralizd Li Thory ppl 0: 244. doi:0.472/ Pag 6 of 20 Hr, i is th mbdding functor of Li algbras into Libniz algbras, and j is th mbdding functor of th catgory of connctd, cocommutativ Hopf algbras into th catgory of rac bialgbras, using th adjoint action (s qn (2 as a rac product. Thorm 2.2: Lt g b a Li algbra. Th PBW isomorphism U(g S(g inducs an isomorphism of functors: j U UR i. Proof: Th nvloping algbra UR (h is by dfinition th functorial vrsion of th rac bialgbra S(h, i.. ociatd to th idal Q(h. But in cas h is a Li algbra, Q(h={0}. Thn th map p is simply th idntity, and UR (h=j(u(h. s a rlativly asy corollary w obtain from th prcding construction th computation of univrsal rac bialgbras. Mor prcisly, w loo for a lft adjoint functor for th functo Prim, sn as a functor from th catgory of all rac bialgbras to th catgory of all Libniz algbras. For a givn Libniz algbra (h,[,] dfin th subcoalgbra of ordr of th first componnt of UR ( (h (s th third statmnt of Thorm 2., i.. UR(h UR (h ( :K h (33 With = K which is rac subbialgbra according to Proposition 2.2. Its structur rads for all λ,λ K and for all x, y h (λ+x=λ +x + x, (34 ε(λ+x=λ, (35 µ((λ+x (λ +x =λλ +λx +[x, x ]. (36 For th particular cas of a Li algbra (h,[,], th abov construction can b found in [3]. Morovr, for any othr Libniz algbra (h,[,] and any morphism of Libniz algbras f: h h dfin th K-linar map UR(f:UR(h UR(h as th first componnt of UR ( (f (cf. th third statmnt of Thorm 2. by UR( f( λâ + x = λâ + f( x, (37 which is clarly is a morphism of rac bialgbras. Hnc UR is a functor from th catgory of all Libniz algbras to th catgory of all rac bialgbras. Now lt (C, C,ε C, C,µ C b a rac bialgbra, and lt f: hprim(c b a morphism of Libniz algbras. Dfin th K-linar map f ˆ : UR( h C by fˆ( λ+ x = λ C + f ( x, (38 and it is again a routin chc that it dfins a morphism of rac bialgbras. Morovr, du to th almost trivial coalgbra structur of UR(h, it is clar that any morphism of rac bialgbras UR(h C is of th abov form and is uniquly dtrmind by Prim( fˆ = f. Hnc w hav shown th following: Thorm 2.3: Thr is a lft adjoint functor, UR, for th functor Prim (ociating to ach Rac bialgbra its Libniz algbra of all primitiv lmnts. For a givn Libniz algbra (h,[,], th objct UR(h which w shall call th Univrsal Rac Bialgbra of th Libniz algbra (h,[,] has th usual univrsal proprtis. Nxt w can rfin th abov univrsal construction by taing into account th augmntd rac bialgbra structur of UR (h to dfin anothr univrsal objct. Considr th mor dtaild catgory of all augmntd rac bialgbras. gain, th functor Prim applid to th coalgbra B (and not to th Hopf algbra H givs a functor from th first catgory to th catgory of all Libniz algbras, and w s again a lft adjoint of this functor, calld UR. Hnc, a natural candidat for a univrsal augmntd rac bialgbra ociatd to a givn Libniz algbra h is: UR( := UR ( = (,, U(,ad s h ( h K h Φ h. (39 ( U( h ( K h Th third statmnt of Thorm 2. tlls us that this is a wlldfind augmntd rac bialgbra, and that UR is a functor from th catgory of all Libniz algbras to th catgory of all augmntd rac bialgbras. Now lt (B,Φ,H,l b an augmntd rac bialgbra, and lt f: h Prim(B b a morphism of Libniz algbras. Clarly, as has bn shown in Thorm 2.3, th map f ˆ : UR( h B givn by qn (37 is a morphism of rac bialgbras. Obsrv that th morphism of C 3 -coalgbras Φ snds th Libniz subalgbra Prim(B of B into K-submodul of all primitiv lmnts of th Hopf algbra H, Prim(H, which is nown to b a Li subalgbra of H quippd with th commutator Li bract [,] H. Morovr this rstriction is a morphism of Libniz algbras. Indd, for any x, y Prim(B w hav Φ ([x,y ] =Φ (x y =Φ ((Φ (x.y =ad Φ (x (Φ (y =Φ (x Φ (y Φ (y Φ (x =[Φ (x,φ (y ] H. It follows that th two-sidd idal Q(Prim(B of th Libniz algbra Prim(B is in th rnl of th rstriction of Φ to Prim(B, whnc th map inducs a wll-dfind K-linar morphism of Li algbras Φ : Prim( B Prim( H. It follows that th composition Φ f : h Prim( H H is a morphism of Li algbras, and by th univrsal proprty of univrsal nvlopping algbras thr is a uniqu morphism of ociativ unital algbras ψ := U( Φ f : U( h H. But w hav for all, h, H (ψ( = H (ψ( ψ( = H ( (ψ( =(ψ(( H + H ψ( (ψ( H + H ψ( = ( ψ ψ( + ( ψ ψ( + U( h U( h U( h U( h = ( ψ ψ( ( ( ψ ψ( ( U( h U( h = ( ( ( ψ ψ U( h sinc ψ maps primitivs to primitivs whnc ψ is a morphism of coalgbras. It is asy to chc that ψ prsrvs counits, whnc ψ is a morphism of C 5 -Hopf-algbras. For all λ K and x h w gt: ( ψ Φ ( λ+ x = ψ( λ + px ( = λ + ( Φ f( px ( ( U( h = λ ( ( = ( ˆ H +Φ f x Φ f( λ+ x, showing th first quation ψ Φ ˆ ( = Φ f of th morphism quation (5. Morovr for all λ K, x h, and u U( h w gt fˆ( u.( λ+ x= fˆ( λε ( u + ux. = λε ( u + f( ux. U( h U( h Lt x,,x h such that u=p(x p(x. Thn f(u.x=f([x,[x 2, [x,x] ]=f(x (f(x 2 (f(x f(x =(Φ (f(x Φ (f(x.(f(x =(ψ(p(x (p(x.(f(x=(ψ(u.(f(x, and thrfor fˆ( u.( λ+ x = ( ψ( u.( fˆ( λ+ x showing th scond quation ψ Φ ˆ ( = Φ f of th morphism quation (5. It follows that th pair ( fˆ, ψ is a morphism of augmntd rac bialgbras. W thrfor hav th following H C J Gnralizd Li Thory ppl, an opn accss journal ISSN: Volum 0 Issu

7 Citation: lxandr C, Bordmann M, Rivièr S, Wagmann F (206 Structur Thory of Rac-Bialgbras. J Gnralizd Li Thory ppl 0: 244. doi:0.472/ Pag 7 of 20 Thorm 2.4: Thr is a lft adjoint functor, UR, for th functor Prim (ociating to ach augmntd rac bialgbra its Libniz algbra of all primitiv lmnts. For a givn Libniz algbra (h,[,], th objct UR(h which w shall call th Univrsal ugmntd Rac Bialgbra of th Libniz algbra (h,[,] has th usual univrsal proprtis. Th rlationship btwn th diffrnt notions (taing into account also Rmar (2.2 is rsumd in th following diagram: U ( i ( j Prim L i@< x >[ >[ d]h opf@ >[ d] [ x > S li K[ ] ( UR [ r] G rp[ >[ d]l ib@< x >[ r] R acbialg[] x >[ r] R acs[] l Prim S li K[ ] whr UR is not lft-adjoint to Prim, whil UR is, but dos not rndr th squar commutativ. Thr is a similar diagram for augmntd notions. Rlation with bar-unital di(coalgbras In th bginning of th nintis th nvloping structur ociatd to Libniz algbras has bn th structur of dialgbras. W shall show in this sction that rac bialgbras and crtain cocommutativ Hopf dialgbras ar strongly rlatd. Lft-unital bialgbras and right Hopf algbras: Lt (B,,ε,,µ b a nonociativ lft-unital C 3 -bialgbra. It will b calld lft-unital C 3 -bialgbra iff µ is ociativ. In gnral, (B,,ε,, nd not b unital, i.. w do not hav in gnral a=a. Howvr, it is asy to s that th sub-modul B of B is a C 3 -subcoalgbra of (B,,ε,, and a subalgbra of (B, such that (B,,ε,,µ is a unital (i.. lft-unital and rightunital bialgbra. Hr, ε, and µ dnot th obvious rstrictions and corstrictions. In a compltly analogous way right-unital C 3 -bialgbras ar dfind. lft-unital (rsp. right unital cocommutativ C 3 -bialgbra (B,,ε,,µ will b calld a cocommutativ right Hopf algbra (rsp. a cocommutativ lft Hopf algbra, (B,,ε,,µ, S, iff thr is a right antipod S (rsp. lft antipod S, i.. thr is a K-linar map S: B B which is a morphism of C 3 -coalgbras (B,,ε, to itslf such that id*s=ε (rsp. S*id= (40 whr * dnots th convolution product (s ppndix 4 for dfinitions. It will bcom clar a postriori that right or lft antipods ar always uniqu, s Lmma 2.2. first cl of xampls is of cours th wll-nown cl of all cocommutativ Hopf algbras (H,,ε,,µ, S for which is a unit lmnt, and S is a right and lft antipod. Scondly it is asy to chc that vry C 4 -coalgbra (C,,ε, quippd with th lft-trivial multiplication (rsp. right trivial multiplication µ 0 (s qn (5 and trivial right antipod (rsp. trivial lft antipod S 0 dfind by S 0 (x=ε(xfor all x C (in both c is a cocommutativ right Hopf algbra (rsp. cocommutativ lft Hopf algbra calld th cocommutativ lft-trivial right Hopf algbra (rsp. right-trivial lft Hopf algbra dfind by th C 4 -coalgbra (C,,ε,. W hav th following lmntary proprtis showing in particular that ach right (rsp. lft antipod is uniqu: Lmma 2.2: Lt (,,ε,,µ, S b a cocommutativ right Hopf algbra.. S*(S S=ε, S (S S=id*ε, S*ε=S and S S S=S, which for ach a implis (a a ( S(a (2 =ε(a= (a S(a ( a (2. It follows that right antipods ar uniqu. 2. For all a,b :S(ab=S(bS(a. 3. For any lmnt c, c is a gnralizd idmpotnt if and only if c= (c S(c ( c (2 iff thr is x with c= (x S(x ( x (2, and all ths thr statmnts imply that c is a gnralizd lft unit lmnt. Proof:. Sinc S is a coalgbra morphism, it prsrvs convolutions whn composing from th right. This givs th first quation from statmnt 2.2. Hnc th lmnts id, S, and S S satisfy th hypothss of th lmnts a,b,c of Lmma 5. in th lft-unital convolution smigroup (Hom K (,,*,ε, whnc th scond and third quations of statmnt 2.2. ar immdiat, and th fourth follows from composing th scond from th right with S and using th third. Clarly S is uniqu according to Lmma gain th lmnts µ, S µ and (id*ε µ satisfy th hypothss on th lmnts a,b,c of Lmma 5. in th lft-unital convolution smigroup (Hom K (,,*,(ε ε (using th fact that µ is a morphism of coalgbras whnc S µ is th uniqu right invrs of µ. computation shows that also µ τ (S S is a right invrs of µ, whnc w gt statmnt 2.2. by uniqunss of right invrss (Lmma Th scond statmnt obviously implis th third, and it is asy to s by straight-forward computations that th third statmnt implis th first and th scond. Convrsly, if c is a gnralizd idmpotnt, i.. c=(µ (c, w gt sinc is a morphism of C 3 - coalgbras that ( (2 ( (2 ( (2 Sc ( c = Sc ( (( µ ( c = Sc ( ( µ ( c ( c ( c ( c Lmma 2.2,. ( (2 (3 ( (2 (3 = Sc ( c c = Sc ( c c = c, ( c ( c and all th thr statmnts ar quivalnt. In ordr to s that vry such lmnt c is a gnralizd lft unit lmnt pic y and = ( ( (2 = ( ( (2 Lmma 2.2,. = ε( = ε( ( x ( x cy Sx x y Sx x y xy cy sinc obviously ε(c=ε(x, so c is a gnralizd lft unit lmnt. Thr is th following right Hopf algbra analogu of th Suschwitsch dcomposition thorm for right groups (s ppndix 5: Thorm 2.5: Lt (,,ε,,µ, S a cocommutativ right Hopf algbra. Thn th following holds:. Th K-submodul (,,ε, µ, S is a unital Hopf subalgbra of (,,ε,,µ, S. 2. Th K submodul E {x x is agnralizd idmpotnt} is a right Hopf subalgbra of qual to th lft-trivial right Hopf algbra dfind by th C 4 -coalgbra ( E,, ε, ε 3. Th map ( x. E E E Ψ: E : x x Sx ( x ( (2 (3 is an isomorphism of right Hopf algbras whos invrs Ψ is th rstriction of th multiplication map. Proof:. It is asy to s that quippd with all th rstrictions is a unital bialgbra. Not that for all a ( S *id ( a = (( S*id ( a = ( S*id * ( ε ( a J Gnralizd Li Thory ppl, an opn accss journal ISSN: Volum 0 Issu

8 Citation: lxandr C, Bordmann M, Rivièr S, Wagmann F (206 Structur Thory of Rac-Bialgbras. J Gnralizd Li Thory ppl 0: 244. doi:0.472/ Pag 8 of 20 Lmma 2.2 = ( ε( a=( ε( a=( ε ( a Whnc S is also a lft antipod. It follows that is a Hopf algbra. 2. Sinc th proprty of bing an gnralizd idmpotnt is a K-linar condition, it follows that E is a K-submodul. Morovr sinc ach c E is of th gnral form c= (x S(x ( x (2, x ι, and sinc th map ι: dfind by ι (x= (x S(x ( x (2 is an idmpotnt morphism of C 3 -coalgbras, w gt (c= (ι(c=(ι ι( (x showing that E is a C 3 -subcoalgbra of. Furthrmor, sinc vry lmnt of E is a gnralizd lft unit lmnt (Lmma 2.2, 3., th rstriction of th multiplication of µ of to E E is lft trivial. Finally, Lmma 2.2, 2. ( (2 ( (2 Sc ( = S( ( c= SSc ( ( c = Sc ( SSc ( ( ι ( c ( c Lmma ,. Lmma ,. ( (2 = Sc ( c = ε ( c= S0(, c ( c showing that th th rstriction of S to E is th trivial right antipod. 3. It is clar from th two prcding statmnts that Ψ is a wlldfind linar map into th tnsor product of two cocommutativ right Hopf algbras. W hav for all x Lmma 2.2,. ( (2 (3 ( (2 ( µ ( x= x S( x x = ε( x x = x Ψ ( x ( x and for all a, c E th trm (Ψ µ(a c is qual to ( ( (2 (2 (3 (3 ( a c ( S( c S( a a c ( a( c ( (2 (3 ( (2 = ( ( ( = ( ( ( =( ( c ( c ac Sc c a Sc c a c bcaus all th trms S(a (2 a (3 and th componnts c (, of itratd comultiplications of gnralizd idmpotnts can b chosn in E (sinc th lattr has bn shown to b a subcoalgbra, and ar thus gnralizd lft unit lmnts (Lmma 2.2, 3.. Hnc Ψ is a K-linar isomorphism. Morovr, it is asy to s from its dfinition that Ψ is a morphism of C 3 -coalgbras. Nxt w comput for all a, a and c, c E : Ψ (((a c((a c =Ψ ((aa ε(cc =(caa c, and -sinc c is a gnralizd lft unit lmnt Ψ (((a c Ψ ((a c =acac =ε(caa c, showing that Ψ and hnc Ψ is a morphism of lft-unital algbras. Finally w obtain ( (2 (3 ( S S0( Ψ( x= ( Sx ( ( ε ( Sx ( x = Sx (, ( x ( (2 (3 Ψ( Sx ( = ( Sx ( ( SSx ( ( Sx ( = Sx (, ( x thans to Lmma 2.2, and intrtwins right antipods. Not that th K-submodul of all gnralizd lft unit lmnts of a right Hopf algbra is givn by K ( Φ ( E + and thus in gnral much biggr than th submodul E of all gnralizd idmpotnts. s it is asy to s that vry tnsor product H C of a unital cocommutativ Hopf algbra H and a C 4 -coalgbra C (quippd with th lft-trivial multiplication and th trivial right antipod is a right Hopf algbra, it is a fairly routin chc using th prcding Thorm 2.5 that th catgory of all cocommutativ right Hopf algbras is quivalnt to th product catgory of all cocommutativ Hopf algbras and of all C 4 -coalgbras. In th squl, w shall nd th dual lft Hopf algbra vrsion whr all th formulas hav to b put in rvrs ordr: Hr vry lft Hopf algbra is isomorphic to C H. Dialgbras and Rac Bialgbras: Rcall that a dialgbra ovr K is a K modul D quippd with two ociativ multiplications,: (writtn ab ab and ab ab satisfying for all a,b,c : (a b c=(a b c, (4 a (b c=a (b c, (42 (a b c=a (b c. (43 n lmnt of is calld a bar-unit lmnt of th dialgbra (,, and (,,, is calld a bar-unital dialgbra iff in addition th following holds a=a, (44 a =a, (45 for all a. Morovr, w shall call a bar-unital dialgbra (,,, balancd iff in addition for all a a = a. (46 Clarly ach ociativ algbra is a dialgbra upon stting = qual to th givn multiplication. Th cl of all (bar-unital and balancd dialgbras forms a catgory whr morphisms prsrv both multiplications and map th initial bar-unit to th targt bar-unit. Ths algbras had bn introducd to hav a sort of ociativ analogu for Libniz algbras. Mor prcisly, thr is th following important fact, which can asily b chcd: Proposition 2.4: Lt (,, b a dialgbra. Thn th K-modul quippd with th bract [,]:, writtn [a,b], [a,b] a b b a (47 is a Libniz algbra, dnotd by. In fact, this construction is wll-nown to giv ris to a functor from th catgory of all dialgbras to th catgory of all Libniz algbras in complt analogy to th obvious functor from th catgory of all ociativ algbras to th catgory of all Li algbras. n important construction of (bar-unital dialgbras is th following: Exampl 2.5: Lt (B, B b a unital ociativ algbra ovr K, and lt b a K-modul which is a B-bimodul, i.. thr ar K-linar maps B and B (writtn (b x bx and (x b xb quipping with th structur of a lft B-modul and a right B-modul such that (bxb =b(xb for all b,b B and for all x B. Suppos in addition that thr is a bimodul map Φ: B, i.. Φ(bxb =bφ(x b for all b,b B and for all x. Thn it is not hard to chc that th two multiplications,: dfind by x y Φ(xy and xy xφ(y (48 quip with th structur of a dialgbra. If in addition thr is an lmnt such that Φ(= B, thn (,,, will b a bar-unital dialgbra. W shall call this structur (, Φ, B an augmntd dialgbra. J Gnralizd Li Thory ppl, an opn accss journal ISSN: Volum 0 Issu

9 Citation: lxandr C, Bordmann M, Rivièr S, Wagmann F (206 Structur Thory of Rac-Bialgbras. J Gnralizd Li Thory ppl 0: 244. doi:0.472/ Pag 9 of 20 In fact, vry dialgbra (,, ariss in that fashion: Considr th K submodul I whos lmnts ar linar combinations of arbitrary product xprssions p(a,.,a r,(a r b r a r b r,a r+,,a n (whr all rasonabl parnthss and symbols and ar allowd for any two strictly positiv intgr r n, and a,, an, br. It follows that th quotint modul /I is quippd with an ociativ multiplication inducd by both and. Lt b qual to /I if is bar-unital: In that cas, th bar-unit of projcts on th unit lmnt of /I; and lt b qual to /I K (adjoining a unit lmnt in cas dos not hav a bar-unit. Thans to th dfining quations (4, (42, (43, it can b shown by induction that for any strictly positiv intgr n, any a,,a n,a, and any product xprssion mad of th prcding lmnts upon using or p(a,,a n a=a a n a=(a a n a, ap(a, a n =a a a n =a(a a n, proving in particular that I acts trivially from th lft (via and from th right (via on such that thr is a wll-dfind -bimodul structur on such that th natural map Φ : is a bimodul morphism. Hnc (, Φ, is always an augmntd dialgbra, and th ignmnt (, Φ, is nown to b a faithful functor. Not also that this construction allows to adjoin a bar-unit to a dialgbra (,,: Considr th K-modul := with th obvious -bimodul structur α.(b+β=α.b+αβ and (b+β.α=b.α+βα for all αβ, and b. Obsrv that th obvious map Φ : dfind by Φ ( b+ β = Φ ( b + β is an -bimodul map, and that = is a bar-unit. Th bar-unital augmntd dialgbra (, Φ, is asily sn to b balancd. Thr ar nonbalancd bar-unital dialgbras as can b sn from th augmntd bar-unital dialgbra xampl (B B, B B,µ B,B whr (B,,µ B is any unital ociativ algbra and th bimodul action is dfind by b.(b b 2.b (bb (b 2 b for all b, b, b, b 2 B. gain, in cas th dialgbra (,,, is bar-unital and balancd, not that = is an ociativ unital subalgbra of whos multiplication is inducd by both and, i.. a b =a b for all a, b. Sinc th K-linar map π : :aaa dscnds to a surjctiv morphism of ociativ algbras by th abov, it is clar that th idal I contains th rnl of π. On th othr hand, if a Kr(π thn 0=π(a=a, and obviously a=a a I, thus inducing a usful isomorphism, and thus a subalgbra injction i: : Φ( a a which is a right invrs to th projction, i.. Φ i = id. In this wor, w also hav to ta into account coalgbra structurs and thus dfin th following: Dfinition 2.4: Lt (,,ε, b cocommutativ C 3 -coalgbra (a C 4 - coalgbra and two K-linar maps,:. Thn (,,ε,,, will b calld a cocommutativ bar-unital di-coalgbra if and only if. (,,, is a bar-unital balancd dialgbra. 2. Both and ar morphisms of C 3 -coalgbras. If in addition thr is a morphism of C 3 -coalgbras S: such that (,,ε,,, S is a cocommutativ right Hopf algbra and (,,ε,,, S is a cocommutativ lft Hopf algbra, thn (,,ε,,,, S is calld a cocommutativ Hopf dialgbra. W hav usd a rlativly simpl notion of on singl compatibl coalgbra structur motivatd from diffrntial gomtry, s Sction 3. In contrast to that, F. Goichot uss two a priori diffrnt coalgbra structurs. Morovr, a slightly mor gnral contxt would hav bn to dmand th xistnc of two diffrnt antipods, a right antipod S for, and a lft antipod S for. Th thory including th clification in trms of ordinary Hopf algbras could hav bn don as wll, but w hav rfraind from doing so sinc it is not hard to s that such a mor gnral Hopf dialgbra is balancd iff S=S. This fact is crucial in th following rfinmnt of Proposition 2.4: Proposition 2.5: Lt (,,ε,,,, S b cocommutativ Hopf dialgbra. Thn th submodul of all primitiv lmnts of, Prim(, is a Libniz subalgbra of quippd with th bract (47. Proof: Lt x,y b primitiv. Thn, using that and ar morphisms of coalgbras, w gt (x y y x= ( x y y x+(x y y x +( x y + y ( x y ( x ( x y = (x y y x+( x y y x bcaus is balancd, and thrfor xy yx is primitiv. Th first rlationship with rac bialgbras is th following simpl gnralization of a cocommutativ Hopf algbra quippd with th adjoint rprsntation: Proposition 2.6: Lt (,,ε,,,, S b cocommutativ Hopf dialgbra. Dfin th following multiplication µ: by ( (2 µ ( a b := a b:= ( a b ( Sa (. (49 ( a Thn w hav th following:. Th map dfins on th K-modul two lft modul structurs, on with rspct to th algbra (,,, and on with rspct to th algbra (,,, maing th Hopf-dialgbra (,,ε,,,, S a modul- Hopf dialgbra, i.. a(bc=(abc=(abc (50 ( ( (2 (2 ( a b= ( a b ( a b ( a,( b (5 a b c a b a c ( (2 ( = ( ( ( a ( (2 ( a (52 a( bc= ( a b ( a c (53 2. (,,ε,,µ is a cocommutativ rac bialgbra. Proof:. First of all w hav µ=µ (µ S (id τ, ( id whr µ and µ stand for th multiplication maps and, and this is clarly a composition of morphisms of C 3 -coalgbras whnc µ is a morphism of C 3 -coalgbras proving qn (5. Nxt, thr is clarly b=b for all b, and, sinc th dialgbra is balancd, w gt for all a ( (2 ( (2 a = ( a ( Sa ( = a ( Sa ( ( a ( a ( (2 = ( a ( Sa ( = ε( a = ε( a. ( a Nxt, lt a,b,c. Thn J Gnralizd Li Thory ppl, an opn accss journal ISSN: Volum 0 Issu

10 Citation: lxandr C, Bordmann M, Rivièr S, Wagmann F (206 Structur Thory of Rac-Bialgbras. J Gnralizd Li Thory ppl 0: 244. doi:0.472/ Pag 0 of 20 a bc a b c Sb Sa ( ( (2 (2 ( = ( (( ( ( ( a,( b ( ( (2 (2 = ((( a b c ( Sb ( Sa ( ( a,( b ( ( (2 (2 = ((( a b c ( Sb ( Sa ( ( a,( b ( ( (2 (2 = ((( a b c ( S( a b ( a,( b =( a b c, Proving qs (50. Nxt, for all a, a, a, w gt (54 ( (2 ( (2 (3 (4 ( a a ( a a = (( a a Sa ( (( a a Sa ( ( a ( a ( (2 (3 (4 = ( a a (( Sa ( a ( a Sa ( ( a ( (2 (3 = ( a a (( ε ( a ( a Sa ( ( a ( (2 = ( a ( a a S( a = a ( a a, ( a whr in th scond to last quation w hav usd th lft antipod idntity for th cas and th fact that (a S(a ( a (2 is a gnralizd lft unit lmnt for th cas. It follows that (, is an -modulalgbra proving qs (52 and ( It rmains to prov slf-distributivity: For all a,b,c, w gt (50 ( (2 ( (2 ( a b ( a c = (( a b a c, ( a ( a and in th nd ( a b a = (( a b Sa ( a ( (2 ( (2 (3 ( a ( a ( (2 (3 = ( a b ( Sa ( a ( a a b a ab ( (2 = ( ( ε ( = ( a proving th slf-distributivity idntity. Th nxt thorm rlats augmntd cocommutativ rac bialgbras with cocommutativ Hopf dialgbras: Thorm 2.6: Lt (B,Φ B,H,l b a cocommutativ augmntd rac bialgbra. Thn th K-modul (B H, B H,ε B ε H, B H,Φ,H will b an augmntd cocommutativ Hopf dialgbra by mans of th following dfinitions. Hr w us Exampl 2.5 and ta h,h H and bb:. Φ:B H H:(b hφ(b h ΦB(bh. 2. h.(b h (h ((h (.b ((h (2h and (b h.h b (hh. 3. S(b h B S H (ΦB(bh. Morovr, th Libniz bract on th K-modul of all primitiv lmnts of B H, a Prim(B Prim(H, is computd as follows for all x,y Prim(B and all,η in th Li algbra Prim(H (writing x and for th mor prcis x H and B [x+,y+η]=([x,y]+.y+([φb(x,η]+[,η] (55 whr ach bract is of th form (47. Not that this Libniz algbra is split ovr th Li subalgbra Prim(H, th complmntary two-sidd idal {x ΦB(xx Prim(B} bing in th lft cntr of a. For an xplicit formula, s th nd of th proof of th thorm. Proof: It is clar from th dfinitions that condition 2.6 dfins a H-bimodul structur on C H maing it into a modul C 3 -coalgbra. Morovr, w comput for all h, h, h H and b B Φ( h.( b h. h = Φ((( h. b (( h hh = Φ (( h. b( h hh ( (2 ( (2 B ( h ( h = ad ( Φ ( b ( h hh = ( h Φ ( b S (( h ( h hh (2 ( (2 (3 ( h ( B B H ( h ( h ( (2 = ( h ΦB( b ε H(( h hh = h ΦB( b hh = h Φ( b h h, ( h whnc Φ is a morphism of H-bimoduls. Nxt, w gt for all b B and h H: ( ( (2 (2 (id B H * S( b h = ( b h S( b h ( b( h ( ( (2 (2 = Φ( b h.( B SH( ΦB( b h ( b( h ( ( (2 (2 = ( ΦB( b h.( B SH( ΦB( b h ( b( h ( ( (2 (2 (3 (3 = ε H( ΦB( b h B Φ ( B( b h SH( ΦB( b h ( b( h = B (ε B (bε H (h H =(ε B ε H (b h( B H, proving th right antipod idntity, and ( ( (2 (2 ( S* id B H( b h= S( b h ( b h ( b( h ( ( (2 (2 = ( B SH( ΦB( b h. Φ( b h ( b( h ( ( (2 (2 = ( B SH( ΦB( b h.( ΦB( b h ( b( h ( ( (2 (2 = B ( SH( ΦB( b h ΦB( b h ( b( h B (ε B (bε H (h H =(ε B ε H (b h( B H, proving th lft antipod idntity. Finally for all h H w gt h h h h h ( (2.( B H= ( ε H( B = B =( B H. ( h implying that th bar-unital dialgbra is balancd. Formula (55 is straight-forward: [x H + B, y H + B η] =(x H (y H (y H (x H +(x H ( B η ( B η(x H +( B (y H (y H ( B +( B ( B η ( B η( B =(Φ B (x.y H +y Φ B (x y Φ B (x +ε B (x B η+ B (Φ B (xη B (ηφ B (x +(.y H +y y ε H ( B η+ B (η B (η =[x,y] H + B [Φ B (x,η]+(.y H + B [,η], bcaus primitivs ar illd by counits, and th formula is provd. lngthy, but straight-forward rasoning shows that th abov construction igning (B,Φ B,H,l (B H,Φ,H dfins a covariant functor from th catgory of all cocommutativ rac bialgbras to th catgory of all cocommutativ Hopf dialgbras. J Gnralizd Li Thory ppl, an opn accss journal ISSN: Volum 0 Issu