DOUBLE-BOSONIZATION AND MAJID S CONJECTURE, (I): RANK-INDUCTIONS OF ABCD HONGMEI HU,1 AND NAIHONG HU,1

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1 DOUBLE-BOSONIZATION AND MAJID S CONJECTURE, (I): RANK-INDUCTIONS OF ABCD HONGMEI HU, AND NAIHONG HU, arxv:505.06v [math.qa] 0 Dec 05 Abstract. Majd developed [M3] the double-bosozato theory to costruct U q (g) ad expected to geerate ductvely ot just a le but a tree of quatum groups startg from a ode. I ths paper, the authors cofrm the Majd s frst expectato (see p. 78 [M3]) through gvg ad verfyg the full detals of the ductve costructos of U q (g) for the classcal types,.e., the ABCD seres. Some examples low raks are gve to elucdate that ay quatum group of classcal type ca be costructed from the ode correspodg to U q (sl ).. Itroducto The veto of quatum groups s oe of the outstadg achevemets of mathematcal physcs ad mathematcs the late tweteth cetury, whch arose the work of L. D. Faddeev wth hs school for solvg tegrable models + dmeso by the quatum verse scatterg method. I the hstory of developmet, a major evet was the dscovery of quatzed uversal evelopg algebras U q (g) over the complex feld by V. G. Drfeld [D] ad M. Jmbo [J] depedetly aroud 985. A strkg feature of quatum group theory s the close coectos wth may braches of mathematcs ad physcs, such as Le groups, Le algebras ad ther represetatos, represetato theory of Hecke algebras, lk varat theory, coformal feld theory, ad so o. Ths attracts may mathematcas to fd some better way a sutable frame to uderstad the structure of quatum groups defed tally by geerators ad relatos. For stace, the frst we must meto s the famous FRT-costructo [FRT] of U q (g) for the classcal types based o the R-matrces of the vector represetatos assocated to the classcal smple Le algebras g, whch s a atural aalogue of the matrx realzato of the classcal Le algebras. It s a belef that the commutato relatos quatum groups expressed by meas of the R-matrces are of fudametal mportace for the theory. Afterwards, Majd redscovered the so-called Radford-Majd bosozato arsg from a framework of a braded category over a Hopf N. H., supported by the NNSFC (Grat No. 73). correspodg author.

2 H. HU AND N. HU algebra (see [Ra], [RT], [M], [M], [M9]), ad the, based o the FRT-costructo ad extedg the Drfeld double to the geeralzed quatum double assocated to a dual par of Hopf algebras equpped wth a (ot ecessarly o-degeerate) parg, Majd [M3] 996 developed the double-bosozato theory to gve a drect costructo of U q (g) through vewg the Lusztg s algebra f ([L]) as a braded group a specal braded categorym H (or H M), where (H= kq=k[k ± ], A=kQ ) s a weakly quastragular dual par (see p. 69 of [M3]), where Q (resp. Q ) s a (dual) root lattce ofg. A aalogous costructo sprt the Yetter-Drfeld category of a Hopf algebra was gve depedetly by Sommerhäuser [So]. Besdes the costructos above, the early 90s, Rgel [R] realzed the postve part of quatum groups by quver represetatos ad Hall algebras, whch spred Lusztg s caocal bases theory [L, L]. Rosso [Ro] also realzed the postve part of U q (g) by troducg the quatum shuffle algebra a braded category, ad gave a recpe of axomatc ductve costructo based o the quatum shuffle operato defed o the (braded) tesor algebra of a braded vector space whch s sometmes a bt coveet for makg practcal calculatos, cotrast wth the Majd s double-bosozato [M3]. Brdgelad [Br] recetly realzed the etre quatum groups of type ADE usg the Rgel-Hall algebras. From aother way, Fag-Rosso [FR] realzed the whole quatum groups by meas of a ew theory o the quatum quas-symmetrc algebras due to Ja-Rosso [JR]. More geeral, the whole axomatc descrpto for the mult-parameter quatum groups as well as the Hopf -cocycle deformato uder the machery of mult-parameter quas-symmetrc algebras have bee acheved Hu-L-Rosso [HLR]. I ths paper, let us focus o the double-bosozato theory [M3]. Assocated to ay mutually dual braded groups B, B covarat uder a backgroud quastragular Hopf algebra H, there s a ew quatum group o the tesor space B H B by double-bosozato [M3], cosstg of H exteded by B as addtoal postve roots ad ts dual B as addtoal egatve roots. The costructo s more powerful tha the quatum double sce oe ca reach drectly to the U q (g) (the quatum double s a bt bg ad oe has to make quotet). Specally, Majd vewed U q ( ± ) as the mutually dual braded groups braded category of rght Carta subalgebra H-modules, the recovered U q (g) by hs double-bosozato theory. Also, based o two examples low raks gve [M3, M8], Majd clamed that may ew quatum groups, as well as the ductve (by rak) costructo of U q (g) ca be obtaed prcple by ths theory [M3] (we call t the Majd s expectato), sce he maged hs double-bosozato framework allows to geerate a tree of quatum groups ad at each ode of the tree, there are may choces to adjo a par of braded groups covarat uder the correspodg quatum group at that

3 DOUBLE-BOSONIZATION AND MAJID S CONJECTURE, (I): RANK-INDUCTIONS OF ABCD 3 ode. Actually, t s a combatoral ad represetato-theoretcal challege to elaborate the full tree structure, whch brgs us the frst motvato. We wat to descrbe detals the ductve costructo of quatum groups U q (g) for all complex semsmple Le algebrasg, whch just elaborates some ma braches of the tree. We wll lmt ourself to cosder the classcal ABCD seres ths paper, whch s orgazed as follows. I secto, we recall some basc facts about the FRT-costructo ad the Majd s double-bosozato costructo. I secto 3, we aalyse explctly the procedure of the ductve costructo of U q (sl ), ad fd some tps low rak cases. That s, although the etre FRT-matrx m ± s hard to kow from the vector represetato, we oly eed to kow those dagoal ad mor dagoal elemets. Ths spres us how to proceed the geeral rak-ductve costructo from U q (sl ) to U q (sl + ). At the ode dagram of U q (sl ), we ca choose a par of braded groups correspodg to the stadard R-matrx arsg from the vector represetato. Smlarly, we ca do the same thg for the BCD seres. These demostrate that the Majd s double-bosozato framework does allow to geerate four brachg-les of odes dagram of quatum groups. Furthermore, the last secto, we wll gve some examples to show how to grow the tree (wth ABCD-braches) of odes dagram of quatum groups out of the same root ode at type A va type-crossg costructo startg from type A r for r=,, 3.. FRT-costructo ad Majd s double-bosozato costructo I ths paper, let k be the complex feld,gafte-dmesoal complex smple Le algebra wth smple rootsα. Letλ be the fudametal weght correspodg toα. Carta matrx ofgs (a j ), where a j = (α,α j ) (α,α ), ad d = (α,α ). Let (H, R) be a quastragular Hopf algebra, where R s the uversal R-matrx, R= R () R (), R = R () R (). Deote by, η, ǫ, S ts coproduct, cout, ut, ts atpode, respectvely. We shall use Sweedler s otato: for h H, (h)=h h. Wrte H op (H cop ) the opposte (co)algebra structure of H, respectvely. LetM H ( H M) be the braded category cosstg of rght (left) H-modules, respectvely. If there exsts a coquastragular Hopf algebra A such that (H, A) s a weakly quastragular dual par, them H ( H M) s equvalet to the braded category A M (M A ) cosstg of left (rght) A-comodules, respectvely. For the detaled descrpto of these theores, we left to the readers to refer to Drfeld s ad Majd s papers [D], [M4], [M5], ad so o. By a braded group, followg [M4], we mea a braded balgebra or Hopf algebra some braded category. I order to dstgush from the ordary Hopf algebras, deote by, S ts coproduct ad atpode, respectvely. A vertble matrx soluto of the quatum Yag-Baxter equato (QYBE) R R 3 R 3 = R 3 R 3 R s called a R-matrx. For later use, oe eeds a certa exteso Uq ext (g) of U q (g). Ths s costructed by adjog

4 4 H. HU AND N. HU formally certa products of the elemets K ±, =,, to U q (sl ), the elemets K ± to U q (sp ), ad K ± K± ad K ± K to U q (so ). The precse deftos of them ca be foud [KS]. Deote by T V a rreducble represetato of U q (g), where V s the correspodg module wth a bass{ x }... FRT-costructo. Oe ca obta a vertble (basc) R-matrx from the quastragular Hopf algebra U q (g) ad ts (vector) represetato. Coversely, startg from a vertble (basc) R-matrx, f does there exst a quastragular Hopf algebra to recover such a R-matrx through a sutable represetato? Frst of all, by Faddeev-Reshetkh-Takhtaja [FRT], the followg fact s basc ad well-kow. Defto.. Gve a vertble matrx soluto R of the QYBE, there s a balgebra A(R), amed the FRT-balgebra, whch s geerated by ad t j, for, j, wth the relatos RT T = T T R, (T)=T T adǫ(t)=iusg stadard otato [FRT], where the matrx T= (t j ), T = T I, T = I T. Observe that A(R) s a coquastragular balgebra wth R : A(R) A(R) k such that R(t j tk l )=Rk jl. Here Rk jl deotes the etry at row (k) ad colum ( jl) matrx R. Secodly, the dual space A(R) = Hom(A(R), k), of A(R) duces the multplcato of A(R). I [FRT], U R s defed to be the subalgebra of A(R) geerated by L ± = (l ± j ), wth relatos (PRP)L ± L± = L± L± (PRP), (PRP)L+ L = L L+ (PRP), where l ± j s defed by (l+ j, tk l )=Rk l j, (l j, tk l )=(R ) k jl. The algebra U R s a balgebra wth coproduct (L ± )=L ± L ± ad coutε(l ± j )=δ j. Specally, whe R s the classcal R-matrx, balgebra A(R) has a quotet coquastragular Hopf algebra, deoted by Fu(G q ) oro q (G), ad U R also has a correspodg quotet quastragular Hopf algebra, whch s somorphc to the exteded quatzed evelopg algebra Uq ext (g). Moreover, there exsts a (o-degeerate) dual parg, betweeo q (G) ad Uq ext (g). The way of gettg the resultg quastragular algebras Uq ext (g) s the socalled FRT-costructo of the quatzed evelopg algebras (for the classcal types). Motvated by the work of [FRT], Majd bult the theory of the weakly quastragular dual pargs assocated wth R-matrces [M3] a more geeral cotext. Remark.. The R-matrces used Majd s papers [M, M3] are a bt dfferet from the stadard oes as [FRT], whch are the cojugatos P Pof the ordary R- matrces by the permutato matrx P : P(u v)=v u. It ca be checked drectly that A(P R P)=A(R) op, where (P R P) j kl = R j lk. Note that we wll use Majd s otato for R-matrces as [M3] the remag sectos of ths paper.

5 DOUBLE-BOSONIZATION AND MAJID S CONJECTURE, (I): RANK-INDUCTIONS OF ABCD 5.. Majd s double-bosozato. Majd [M3] proposed the cocept of a weakly quastragular dual par va hs sght o more examples o matched pars of balgebras or Hopf algebras [M7]. Ths allowed hm to establsh a theory of double-bosozato a broad framework that geeralzed the FRT s costructo whch was lmted to the classcal types. Defto.. Let (H, A) be a par of Hopf algebras equpped wth a dual parg, ad covoluto-vertble algebra/ at-coalgebra maps R, R : A Hobeyg R(a), b = R (b), a, R h=r ( L h) R, R h= R ( L h) R for a, b A, h H. Here s the covoluto product hom(a, H) ad ( L h)(a) = h (), a h (), ( R h)(a)=h () h (), a are left, rght dfferetato operators regarded as maps A Hfor fxed h. Let C, B be a par of braded groups M H, whch are called dually pared f there s a tertwer ev : C B k such that ev(cd, b)=ev(d, b () )ev(c, b () ), ev(c, ab)= ev(c (), a)ev(c (), b), a, b B, c, d C. The C op/cop (wth opposte product ad coproduct) s a Hopf algebra H M, whch s dual to B the sese of a ordary dual parg, wth H-bcovarat: h c, b = c, b h for all h H. Let C= (C op/cop ) cop, the C s a braded group H M, where H s (H, R ). Wth these, Majd gave the followg double-bosozato theorem. Theorem.. (Majd) O the tesor space C H B, there s a uque Hopf algebra structure U = U( C, H, B) such that H B(bosozato) ad C H (bosozato) are sub-hopf algebras by the caocal clusos, wth cross relato bh=h () (b h () ), ch=h () (c h () ), (C) here b B, c C, h H. If there exsts a coquastragular Hopf algebra A such that (H, A) s a weakly quastragular dual par, ad b, c are prmtve elemets, the some relatos smplfy to [b, c]=r(b () ) c, b () c (), b R(c () ); (C) b=b () R(b () )+ b, c=c + R(c () ) c (). Let Ũ(R) be the double cross product balgebra of A(R) op [M7] geerated by m ± wth the balgebra structure gve by (C3) Rm ± m± = m± m± R, Rm+ m = m m+ R, ((m± ) j )=(m± ) a j (m± ) a, ǫ((m± ) j )=δ j, (C4) where (m ± ) j are the FRT-geerators, m± are called FRT-matrces.

6 6 H. HU AND N. HU Proposto.. () (Ũ(R), A(R)) s a weakly quastragular dual par wth (m + ) j, tk l =Rk jl, (m ) j, tk l =(R ) k l j, R(T)=m+, R(T)=m. Specally, whe R s the classcal R-matrx, the the weakly quastragular dual par ca be desceded to a mutually dual par of quotet Hopf algebras (Uq ext (g),o q (G)) (where G s oe of coected ad smply coected algebrac groups of classcal types) wth the correct modfcato [M3]: (m + ) j, tk l =λrk jl, (m ) j, tk l =λ (R ) k l j. Such λ s called a quatum group ormalzato costat. () Suppose that R s aother matrx such that () R R 3 R 3 = R 3 R 3R, R 3 R 3 R = R R 3R 3, () (PR+)(PR )=0, () R R = R R, where P s the permutato matrx wth etres P j kl = δ l δ jk. The a braded-vector algebra V(R, R) geerated by R j ab ea e b forms a braded group wth geerators,{e =,, }, ad relatos e e j = a,b (e )=e + e,ǫ(e )=0, S (e )= e,ψ(e e j )= R j ab ea e b braded category a,b A(R) M. Uder dualty f j, e =δ j, a braded-covector algebra V (R, R ) geerated by ad{ f j j=,, }, ad relatos f f j = f b f a R j ab forms aother braded group wth a,b ( f )= f + f,ǫ( f )=0, S ( f )= f,ψ( f f j )= f b f a R a b j braded category a,b M A(R). Remark.. I fact, by Remark., Ũ(R) determed uquely by relatos (C4) wth the R-matrx [M3] s the opposte of U R costructed by FRT-approach. So the quatzed evelopg algebra U q (g) recovered by Majd (Proposto 4.3 [M3]) satsfes E K j = q a j K j E, F K j = q a j K K j F, [E, F j ]=δ K j, where q q q = q d, whch s the opposte of the ordary U q (g). By abuse of otato, we stll use U q (g) stead of U q (g) op the remag sectos of ths paper. Wth the ewλr-matrx Proposto., order that V(R, R), V (R, R ) are stll braded groups, the (U ext(g),o q(g)) must be cetrally exteded to the par q ( (g)=u q (g) k[c, c ], Oq (G)=O q (G) k[g, g ] ) Uext q wth acto e c=λe, f c=λ f, c, g =λ. By Theorem., we have the followg Corollary.. U = U(V (R, R ), Uext q (g), V(R, R)) s a ew quatum group wth the cross relatos: e (m + ) j j k =λrab (m+ ) a k eb, (m ) j ek =λr k ab ea (m ) b j, (m+ ) j f k=λ f b (m + ) a Rab jk, f (m ) j k =λ(m ) j b f ar ab k, c f =λ f c, e c=λce, [c, m ± ]=0, [e (m, f j ]=δ + ) j c c(m ) j j ; q q

7 DOUBLE-BOSONIZATION AND MAJID S CONJECTURE, (I): RANK-INDUCTIONS OF ABCD 7 ad the coproduct: c=c c, e = e a (m + ) ac + e, f = f +c(m ) a f a, ǫe =ǫ f = 0, where oe ca ormalze e such that the factor q q satsfes the stuato you eed. So the Majd s double-bosozato costructo ca lead to ew quatum groups. After gvg a example of obtag U q (sl 3 ) from Uq ext(sl ) [M3], Majd expected that the ovel resultg quatum group s the quatum group of hgher-oe rak the classcal ABCD s seres. I the curret paper, we wll solve such Majd s expectato the classcal types that has bee a ope questo amed by Majd ([M3]) sce the md of 90 s. I the ext secto, we wll gve full detals of the rak-ductve costructo the ABCD seres. 3. Rak-ductve costructo of quatum groups for classcal types I order to explore the structure of the resultg quatum group Corollary., we eed to kow how to get the explct form of the FRT-matrx m ±, whch ca be obtaed by the followg lemma. Lemma 3.. Correspodg to the vertble matrx R obeyg the QYBE, we have S (l ± j )= (m ± ) j the quotet Hopf algebra, S s the correspodg atpode. Proof. If there exsts a quotet Hopf algebra of Ũ(R), accordg to Remark., we obta the followg relatos ths quotet Hopf algebra: RL ± L± = L± L± R, RL L+ = L+ L R, ad (L ± )=L ± L ±. We ca descrbe these relatos vew of the etres the matrx R ad L ±. For example, for ay fxed, j, k, l, we have (RL ± L± ) j kl = (L± L± R) j kl, the we obta the followg equalty o the left had sde (RL ± L± ) kl= j R j ab (I L± ) ab m (L± I) m kl = R j ab δ am(l ± ) b (l ± ) mk δ l = R j mb (l± ) bl (l ± ) mk = R m(l j ± ) l (l ± ) mk, ad the equalty by the rght had sde (L ± L± R) j kl = (L± I) j ab (I L± ) ab m Rm kl = (l ± ) a δ jb δ am (l ± ) b R m kl = (l ± ) m (l ± ) j R m kl. So R m(l j ± ) l (l ± ) mk = (l ± ) m (l ± ) j R m. Takg the atpode S o both sdes, we obta Uder the otato (m ± ) j = S (l± j ), we have kl R j ms ((l ± ) mk )S ((l ± ) l )=S ((l ± ) j )S ((l ± ) m )R m kl. R j m(m ± ) m k (m± ) l )=(m± ) j )(m ± ) m )Rm kl,.e., Rm± m± = m± m± R.

8 8 H. HU AND N. HU We also obta Rm + m = m m+ R by a smlar argumet. O the other had, ((m ± ) j )= (S (l± j ))=P (S S ) (l± j )=P (S S )(l± a l± a j ) = P((m ± ) a (m± ) a j )=(m± ) a j (m± ) a. So, the geerators (m ± ) j satsfy relato (C4). Ths completes the proof. 3.. Iductve costructo of U q (sl ). Sce m ± ad R-matrx wll become larger ad larger wth the growg of rak, t s a challege to descrbe explctly the procedure of geeral ductve costructo. We wat to fd some tps o some cocrete examples. Majd already descrbed explctly the case of U(V (R, R ), Uext q (sl ), V(R, R)) U q (sl 3 ) [M3]. Frst of all, order to capture much more hts comg from the theory of Majd s double-bosozato costructo, what follows, we wll descrbe detal a example of rak case: U(V (R, R ), Uext q (sl 3 ), V(R, R)) U q (sl 4 ). Example 3.. Let us start wth R-matrx datum q q 0 q q q q R= q q 0 q q q q Take R = q R, the R, R gve braded groups V (R, R ) = k f =,, 3 ad V(R, R)=k e =,, 3. Idetfy e 3, f 3, (m + ) 3 3 c wth the addtoal smple root vectors E 3, F 3 ad group-lke elemet K 3, the the resultg quatum group (V (R, R ), Uext q (sl 3 ), V(R 3, R)) s exactly the quatum group U q (sl 4 ) wth K, =, adjoed. Proof. The quatum group ormalzato costat for R eeded for the weakly quastragular structure o Uq ext (sl 3 ) sλ=q 4 3 obtaed by the facts [FRT]. Moreover, the m ± -matrx correspodg to the vector represetato ca be obtaed by Lemma 3.. m + = 3 K K 3 (q q )E K 3 K 3 q (q q )[E, E ] q K 3 K 3 0 K 3 K 3 (q q )E K 3 K K 3 K 3,

9 DOUBLE-BOSONIZATION AND MAJID S CONJECTURE, (I): RANK-INDUCTIONS OF ABCD 9 K 3 K m = (q q )K 3 K 3 F K 3 K 3 0, q(q q 3 )K K 3 [F, F ] q (q q )K 3 K 3 3 F K K 3 where [E, E ] q = E E q E E, [F, F ] q = F F qf F. By Corollary., we get [e 3, f 3 ]= (m+ ) 3 3 c c(m ) 3 3 q q,= [E 3, F 3 ]= K 3 K3 q q. E 3 K 3 = e 3 (m + ) 3 3 c = λr 33 ab (m+ ) a 3 eb c = R 33 ab (m+ ) a 3 c e b = R (m+ ) 3 3 c e 3 = q K 3 E 3. From the expresso of (m + ), we have (m+ ) K =(m + ), (m+ ) 3 3 K =(m + ). Assocatg wth the cross relato e 3 (m + ) =λr3 ab (m+ ) a eb, we obta e 3 (m + ) = q 3 (m + ) e3, e 3 (m + ) = q 3 (m + ) e 3 K = K e 3, E 3 K = K E 3, e3, = e 3 (m + ) 3 3 = q 3 (m + ) 3 e 3 e3, 3 K = q K e 3. E 3 K = q K E 3. I order to explore the relatos betwee F 3 ad K, =,, 3, we have K 3 F 3 = (m + ) 3 3 c f 3 = λ (m+ ) 3 3 f 3c = λ λ f 3(m + ) 3 3 R33 33 c = f 3 (m + ) 3 3 c R = q F 3 K 3 through the cross relato (m + ) f 3=λ f b (m + ) a Rab 3. Smlarly, (m + ) + + K = (m + ), =,, (m + ) f 3= q 3 f 3 (m + ) f 3 K = K f 3, F 3 K = K F 3,, =,, = (m + ) 3 3 f 3= q 3 f 3 (m + ) 3 3. f 3 K = qk f 3. F 3 K = qk F 3. The the relatos betwee K 3 ad E, F, =, are gve as follows E K 3 = E K 3 K 3 c = q 3 q 3 K 3 K 3 E c = K 3 K 3 c E = K 3 E, E K 3 = E K 3 K 3 c = q 3 q 4 3 K 3 K 3 E c = q K 3 K 3 c E = q K 3 E, F K 3 = F K 3 K 3 c = q 3 q 3 K 3 K 3 F c = K 3 K 3 c F = K 3 F, F K 3 = F K 3 K 3 c = q 3 q 4 3 K 3 K 3 F c = qk 3 K 3 c F = qk 3 F. Moreover, (E 3 )=E 3 K 3 + E 3, (F 3 )=F 3 +K3 F 3 ca be obtaed by Corollary.. The most mportat relatos are the q-serre relatos. Sce the geerators E s belog to (m + ) +, =,, we wll cosder the followg equaltes (m + ) = (q q )E (m + ), e 3 (m + ) =λr3 ab (m+ ) a eb =λq(m + ) e3, e 3 (m + ) =λr3 ab (m+ ) a eb =λq(m + ) e3. = e 3 E = E e 3,= E 3 E = E E 3.

10 0 H. HU AND N. HU (m + ) 3 = (q q )E (m + ) 3 3, e 3 (m + ) 3 =λr3 ab (m+ ) a 3 eb =λq(m + ) 3 e3 +λ(q )(m + ) 3 3 e, = e = e 3 E q E e 3. e 3 (m + ) 3 3 =λr33 ab (m+ ) a eb =λq (m + ) 3 3 e3. So, we eed to explore the relato betwee e ad e 3. Note that e e 3 = R 3 ab ea e b = q R 3 ab ea e b = q R 3 3 e3 e = q e 3 e, the combg wth e = e 3 E q E e 3, we obta (E 3 ) E (q+q )E 3 E E 3 + E (E 3 ) = 0. O the other had, we eed to kow aother relato betwee e ad E, whch ca be explored by the followg cross relatos (m + ) 3 = (q q )E (m + ) 3 3, e (m + ) 3 =λq (m + ) 3 e, e (m + ) 3 3 =λq(m+ ) 3 = (E ) E 3 (q+q )E E 3 E + E 3 (E ) = 0. 3 e3, e = e 3 E q E e 3. The q-serre relato of F, 3 ca be obtaed smlarly. The the resultg quatum group s just the quatzed evelopg algebra U q (sl 4 ), where e, f, =, ca be expressed by the q-commutators wth geerators E, F, =,, 3. Remark 3.. I the above cocrete example, we fd that t s ot ecessary to kow the etre m ± -matrx for determg the structure of resultg quatum group, except for the dagoal ad mor dagoal etres. At least, correspodg to the vector represetato of U q (g), all the smple root vectors E, F ad group-lke elemets K are cluded the dagoal ad mor dagoal etres. Other etres m ± are flled by o-smple root vectors geerated by q-commutators wth geerators E, F ad K. Moreover, the explct expressos of the dagoal ad mor dagoal etres m ± ca be easly obtaed by Lemma 3.. For the A seres, the data R, R, m ± as above are deduced from vector represetato, whch spres us to start wth the vector represetato whe cosderg the geeral rakductve costructo. The R VV -matrx of vector represetato T V satsfes the quadratc equato (PR VV q I)(PR VV + q + I)=0. So settg R=q + R VV, R = q R, the we have (PR+I)(PR I)=0, ad R j kl = qqδ j δ k δ jl + (q )δ l δ jk θ( j ), where k>0, θ(k)= 0 k 0. O the other had, accordg to Lemma 3., we obta the followg

11 DOUBLE-BOSONIZATION AND MAJID S CONJECTURE, (I): RANK-INDUCTIONS OF ABCD Lemma 3.. Correspodg to the vector represetato, the dagoal ad mor dagoal etres FRT-matrx m ± of Uq ext(sl ) are gve by (m + ) + = (q q )E K K (m ) + (m + ) = K K K K = (q q )K K K K (m ) = K K K K K K (+) K K (+) + K ( ),, K ( ),. + K ( ) F,, K ( ),. Obvously, we observe that (m + ) K = (m + ) + + for by the above lemma. Wth these, we have the followg Theorem 3.. For type A, letλ=q +, ad detfy e, f, (m + ) c wth the addtoal smple root vectors E, F ad group-lke elemet K. The the resultg quatum group U(V (R, R ), Uext q (sl ), V(R, R )) s exactly the U q (sl + ) wth K ± adjoed. Proof. (m + ) = K K K ca be obtaed by Lemma 3.. [E, F ] = K K, q q (E )=E K + E, ad (F )=F +K F ca be deduced easly from Corollary.. O the other had, we have the followg cross relatos by Corollary. (m ) + = (q q )(m ) + (m ) + e =λr,+,+ e (m ) +, (m ) + + e =λr,+,+ e (m ) + +, + F, = F e = e F,= [E, F ]=0,. (m + ) + = (q q )E (m + ) + +, (m + ) + f =λ f (m + ) + R+, +,, = E f = f E,= [E, F ]=0,. (m + ) + + f =λ f (m + ) + + R+, +,, Wth these, we get K K [E, F ]=δ q q, [E, F ]=δ K K. (A) q q The relatos betwee E, F ad K, also ca be obtaed by the cross relatos Corollary., (m + ) K = (m + ) + +,, e (m + ) j j =λr j j (m + ) j E j e, K j = K j E, j, = R j j = q, j, R= q E. K = q (A) K E.

12 H. HU AND N. HU (m + ) K = (m + ) + +,, (m + ) f =λ f (m + ) R, F K j = K j F, j, = (A3) R j j = q, j, R = F q. K = qk F. I the U q (sl ), E K = q K E, E K ± = q K ± E, E K j = K j E, j ±; F K = q K F, F K ± = qk ± F, F K j = K j F, j ±. The we get the relatos betwee K ad E, F, where. E K = q q q + K K c E = K E, F K = q q q + K K c F = K F, E K = q q ( ) K K (A4) K c E = q K E, F K = q q ( ) K K K c F = qk F. We wat to explore the relatos betwee e ad E,, ad observe that E just belogs to the etry (m + ) +, so (m + ) + = (q q )E (m + ) + +, e (m + ) + =λr (m+ ) + e,, e (m + ) =λr,, (m+ ) e +λr,, (m+ ) E e, E = E E,, e (m + ) + + =λr+, +, (m+ ) + e = e E q E e. + e, R j j = q, j, R= q, R,, = q. (m ) + = (q q )(m ) + + F, f (m ) + =λr (m ) + f,, f (m ) =λr,, (m ) f +λr, f (m ) + + =λr,+,+ (m ) + + f,, (m ) f, F F = F F,, f = q f F F f. Note that e e = R, e a e b = q R, ab, e e = q e e ad f f = f b f a R ab, = q f f R,, = q f f. The we obta (E ) E (q+q )E E E + E (E ) = 0, (F ) F (q+q )F F F + F (F ) (A5) = 0. O the other had, (m + ) = (q q )E (m + ), e (m + ) =λq (m + ) e, = e E = qe e. e (m + ) =λq(m+ ) e. (m ) = (q q )(m ) F, f (m ) =λq (m ) f, = f F = qf f. f (m ) =λq(m ) f.

13 DOUBLE-BOSONIZATION AND MAJID S CONJECTURE, (I): RANK-INDUCTIONS OF ABCD 3 Combg wth e = e E q E e ad f = q f F F f, we get (E ) E (q+q )E E E + E (E ) = 0, (F ) F (q+q )F F F + F (F ) = 0. The other elemets e, f j ca be detfed wth o-smple root vectors geerated by q- commutators wth E, F, K,. Wth the above equaltes (A) (A6), we prove that the resultg quatum groups s U q (sl + ). 3.. Iductve costructo of U q (g) for the BCD seres. The rak-ductve costructo of U q (sl ) gves us the cofdece to cosder the BCD seres. Ther Dyk dagrams are gve respectvely by the followg dagrams, ad the arrow s pot to the shorter of the two roots the dagrams. < : B ( ) > 3 : C ( 3) : D ( 4) Correspodg to ther vector represetatos, the matrx PR VV satsfes the followg cubc equato [FRT, KS]: (PR VV + q I)(PR VV qi)(pr VV ǫq ǫ N I)=0. The set R=qR VV, R = RPR (ǫq ǫ N+ +q )R+(ǫq ǫ N+3 +)P, we get (PR+I)(PR I)=0. Moreover, there s a uque formula for the matrx etres of R, R j kl = qqδ j δ j δ k δ jl + (q )θ( j l)(δ l δ jk K j lk ). Here K j lk =ǫc j Cl k, ad Cm t =ǫ m δ mt q ρ m, where = N+, let N= f N s eve, ad N= + f N s odd.ρ = N f < ;ρ = ρ f,ǫ=ǫ = =ǫ N = for g=so N, adρ = N +,ρ = ρ f <,ǫ = =ǫ =,ǫ=ǫ + = =ǫ N = for g=sp N. For ther vector represetatos, the dagoal ad mor dagoal etres we eed m ± ca be obtaed by Lemma 3.. Lemma 3.3. () For U ext q (so + ) : (m + ) = K K K K +, (m + ) + + =,, (m + ) + = (q q )E + K K K,, (m + ) + = c 0E, (m ) + = (q q )K K K F +, m (+) = c 0 F, (m + ) (m ) =, (A6)

14 4 H. HU AND N. HU where c 0 = (q + q ) (q q ). () For U ext q (sp ) : (m + ) = K K K +, (m + ) (m+ ) =,, (m + ) + = (q q )E + K K K,, (m + ) + = (q q )E K, (m ) = K K K+, (m ) (m ) =,, (m ) + = (q q )K K K F +,, (m ) + = (q q )K F. (3) For U ext q (so ) : (m + ) = (K K )K 3 K +, (m + ) (m+ ) =,, (m + ) = K K, (m + ) =K K, (m+ ) + = (q q )E (K K ), (m + ) + = (q q )E + (K K )K 3 K,, (m ) = (K K )K 3 K+, (m ) (m ) =,, (m ) = K K, (m ) = K K, (m ) + = (q q )(K K )F, (m ) + = (q q )(K K )K 3 K F +,. Wth these, we have the followg Theorem 3.. Wth quatum group ormalzato costatλ=q. () Type B Idetfy e +, f +, (m + ) + +c wth the addtoal smple root vectors E +, F + ad the group-lke elemet K +. The the resultg quatum group U(V (R, R ), Uext q (so + ), V(R, R)) s the quatum group U q (so +3 ). () Type C Idetfy e, f, (m + ) c wth the addtoal smple root vectors E +, F + ad the grouplke elemet K +. The the resultg quatum group U(V (R, R ), Uext q (sp ), V(R, R)) s the quatum group U q (sp + ) wth K ± adjoed. (3) Type D Idetfy e, f, (m + ) c wth the addtoal smple root vectors E +, F + ad the grouplke elemet K +. The the resultg quatum group U(V (R, R ), Uext q (so ), V(R, R)) s the quatum group U q (so + ) wth K ± K±, K± K adjoed.

15 DOUBLE-BOSONIZATION AND MAJID S CONJECTURE, (I): RANK-INDUCTIONS OF ABCD 5 Proof. The proof of Theorem 3. meas that the relatos of egatve part ca be obtaed a smlar way, so we oly focus o the relatos of the postve part. () (m + ) + +c = K K K follows from Lemma 3.3. From the detfcato the above theorem, t s easly deduced from Corollary. that [E +, F + ]= K + K+, q q (E + )=E + K + + E +, ad (F + )=F + + K+ F +. O the other had, we have E + K + = e + (m + ) + + c =λra + + b (m + ) a + eb c = R+ + +(m+ ) + +c e + = q K + E +. The relatos betwee the addtoal smple root vector e + ad other K, ca be deduced from e + (m + ) p p=λr p,+ ab (m + ) a pe b =λr p,+ p,+ (m+ ) p pe +, p +. Combg wth (m + ) + + K + = (m + ),, ad R + + =, Rp p + + = q, p +, we obta e+ K = q K e +, e + K = K e +, for, amely, E + K = q K E +, E + K = K E +,. We observe that F belogs to (m ) +, so the relatos betwee E + ad F ca be obtaed by the equalty (m ) j ek =λr k a b ea (m ) b j Corollary.. (m ) + = (q q )(m ) + + F,, (m ) + e + =λra + + b ea (m ) b = e + (m ) +,, = [E +, F ]=0,. (m ) p pe + =λra + p b ea (m ) b p= e + (m ) p p, p, We wll explore the q-serre relatos of the postve part. We also observe that E + belogs to (m + ) +,, so e + (m + ) =λr a + b (m + ) a eb =λr + + (m+ ) e+ +λr + (m + ) e, e + (m + ) + =λr a + b (m + ) a + eb =λr + + (m+ ) + e+ =λq(m + ) + e+,. Puttg the expresso of (m + ) +, to the above equaltes, we get e + E j = E j e +, j, e = e + E q E e +. So, we eed to kow the relatos betwee e ad e +, E. We have e + e = R a b + e a e b = (q+ǫq ǫ N+ )e e + + (ǫq ǫ N+ + )e + e, so e + e = qe e +. Combg wth e = e + E q E e +, we get (e + ) E (q+q )e + E e + + E (e + ) = 0, amely, (E + ) E (q+q )E + E E + + E (E + ) = 0.

16 6 H. HU AND N. HU O the other had, accordg to the equalty e (m + ) =λr a b (m+ ) a eb =λr (m+ ) e = (m + ) e ad e (m + ) =λr (m+ ) e = q (m + ) e, we get e E = qe e. Combg wth e = e + E q E e + aga, we obta (E ) E + (q+q )E E + E + E + (E ) = 0. Wth these relatos, we prove that the resultg quatum group s U q (so +3 ). () ad (3) ca be proved a smlar way. 4. Type-crossg costructos of types BCD startg from type A Up to ow, we have got the geeral ductve costructos of the classcal quatum groups wth the same type as the above secto. However, Majd clamed that hs double-bosozato allows to create ot oly a le of odes dagram but also a tree of odes dagram of quatum groups. At each ode of the tree, we have more choces to adjo sutable braded groups to obta possble dfferet ew quatum groups of hgher rak oe. I ths secto, we wll gve some examples to demostrate ths fact. As we kow, at the source ode correspodg to U q (sl ), Majd [M3] chose a par of braded groups geerated by the vector represetato of U q (sl ) to gve U q (sl 3 ) as above. I what follows, we wll gve 3 kds of type-crossg costructos: from type A to type B, from type A to type C 3, ad from type A 3 to type D 4. R VV Correspodg to a rreducble represetato of U q (g), the matrx R VV s duced by ( q β ) r j r = B VV (T V T V )(R). HereR= j [r j ] qβ! q j(r j+) β j E r j β j F r j β j s the ma j r,,r =0 j= part of the uversal R-matrx of U h (g). B VV deotes the lear operator o V V gve by B VV (v w) := q (µ,µ ) v w for v V µ, w V µ. 4.. Type-crossg costructo from type A to type B. Now, we stll start from the ode dagram of U q (sl ) ad choose other braded groups to gve U q (so 5 ) the followg. Example 4.. Startg from a 3-dmesoal represetato T V of U q (sl ), whch s gve by E (x )=[] q x +, E (x 3 )=0, F (x + )= x, F (x )=0, =,, where x, x, x 3 s the

17 DOUBLE-BOSONIZATION AND MAJID S CONJECTURE, (I): RANK-INDUCTIONS OF ABCD 7 base of V wth correspodg weghts α, 0,α. The we get a 9 9 R VV -matrx datum q q q q 0 q q 0 ( q )(q q ) R VV = q q q q q Clearly, R VV s vertble, ad accordg to the submodules decomposto of module V, t s easy to see that PR VV obeys the mmal polyomal (PR VV + q I)(PR VV q I)(PR VV q 4 I)=0. Settg R=q R VV, R = RPR q R q 4 R+(q + )P, the (PR+I)(PR I)=0. Wth these R ad R, choose a par of braded groups V (R, R ), V(R, R), adλ=q. Idetfy e 3, f 3, (m + ) 3 3 c wth the addtoal smple root vectors E, F ad the group-lke elemet K. The the resultg quatum group U(V (R, R ), Uext q (sl ), V(R, R)) s the quatum group U q (so 5 ). Proof. Correspodg to ths represetato, we get the m ± -matrx as follows K (q q q( q )E ) m + [] q E K = 0 (q q )E K, 0 0 K K m 0 0 = (q 4 )F 0. (q 4 q )(q 4 )K F (q q )K F K Wth the detfcato as above, [E, F ]= K K, (E q q )=E K + E, ad (F )=F +K F ca be deduced from Corollary.. E K = e 3 (m + ) 3 3 c =λr (m+ ) 3 3 e3 c = R (m+ ) 3 3 c e 3 = q 4 K E, E K = q K E ca be deduced from e 3 (m + ) =λr 3 3 (m+ ) e3 = q (m + ) e3. O the other had, E K = E (m + ) 3 3 c = E K c = q K c E = q K E. Accordg to the equalty (m ) e3 =

18 8 H. HU AND N. HU λr 3 3 e3 (m ) = e3 (m ), we obta e3 F = F e 3 combg wth (m ) = (q4 )F, amely, [E, F ]=0. We wll explore the q-serre relato betwee e 3 ad E. The equalty (q+q )e = q E e 3 e 3 E ca be gve by e 3 (m + ) =λr3 ab (m+ ) a eb = q (m + ) e3 + (q q )(m + ) e. Combg wth e 3 e = q e e 3, whch s deduced from e e 3 = R 3 ab ea e b = (q 4 +q +)e e 3 (q + )e 3 e, we obta (E ) E (q + q )E E E + E (E ) = 0. O the other had, we eed to kow the relato betwee e ad E. e (m + ) =λr ab (m+ ) a eb = (m + ) e + (q q )(m + ) e, e (m + ) =λr ab (m+ ) a eb =λr (m+ ) e = q (m + ) e. = e = (E q+q e e E ), e E = q E e. Combg wth (q+q )e = q E e 3 e 3 E, we obta (E ) 3 E 3 (E ) E E + 3 E E (E ) E (E ) 3 = 0. q q Wth these relatos, the Carta matrx of the resultg quatum group s So the resultg quatum group s U q (so 5 ), ad the proof s complete Type-crossg costructo from type A to type C 3. I Example 3., we get U q (sl 4 ) startg from U q (sl 3 ) by choosg the braded groups geerated by a 3-dmesoal vector represetato va the Majd s double-bosozato costructo. I the example below, we wll choose aother par of braded groups geerated by a 6-dmesoal rreducble module to gve U q (sp 6 ) startg from the ode dagram of U q (sl 3 ). Example 4.. The par of braded groups we wat to have s obtaed from the 6-dmesoal rreducble represetato T V of U q (sl 3 ), whch s defed by E F x x x 3 x x 4 x 5 = = x (q+q )x 4 x 5 (q+q )x x x 3, E, F x x 4 x 5 x 3 x 5 x 6 x 3 = x 5, (q+q )x 6 = x (q+q )x 4 x 5 where{ x 6} s a bass of V wth the correspodg weghts λ, λ + α, λ +α +α, λ + α, λ + α +α, λ + α + α, respectvely. We ca,

19 DOUBLE-BOSONIZATION AND MAJID S CONJECTURE, (I): RANK-INDUCTIONS OF ABCD 9 obta a matrx R VV correspodg to the 6-dmesoal represetato, ad the PR VV obeys the mmal polyomal (PR VV q 8 3 I)(PRVV + q 4 3 I)(PRVV q 0 3 I)=0. Settg R=q 4 3 R VV, R = RPR (q + q 4 )R+(q + )P, we have (PR+I)(PR I)=0, adλ=q 4 3. Idetfy e 6, f 6, (m + ) 6 6 c wth the addtoal smple root vectors E 3, F 3 ad the group-lke elemet K 3, the the resultg quatum group U(V (R, R ), Uext q (sl 3 ), V(R, R)) 3 s the quatum group U q (sp 6 ) wth K (=, ) adjoed. Proof. The elemets matrx m ± we eed ca be obtaed by Lemma 3., whch are gve by the followg equaltes (m + ) = (q q 3 )E K K 3, (m + ) = K 3 3 K, (m + ) 5 6 = (q q )E K 3 K 3 4, (m+ ) 6 6 = K 3 4 K 3, (m+ ) 4 4 = K 3 K 3, (m ) 5 3 = q(q q )K 3 3 K F, (m ) 5 5 = K 3 K 3, (m ) 6 5 = (q q 3 )K K 4 3 F, (m ) 6 6 = K 3 3 K 4. We oly focus o the relatos of the postve part. Note that E 3 K 3 = e 6 (m + ) 6 6 c =λr (m+ ) 6 6 e6 c = R (m+ ) 6 6 c e 6 = q 4 (m + ) 6 6 c e 6 = q 4 K 3 E 3. Combg wth e 6 (m + ) =λr 6 6 (m+ ) e6, we have (m + ) 6 6 K = (m+ ) 4 4, e 6 (m + ) 6 6 = q 8 3 (m + ) 6 e 6 K = K e 6, E 3 K = K E 3, 6 e6, = e 6 (m + ) 4 4 = q 4 3 (m + ) 4 e 4 e6. 6 K = q K e 6 =. E 3 K = q K E 3. O the other had, E K 3 = E K 3 4 K 3 c = q 3 q 8 3 K 3 4 K 3 c E = q K 3 E, E K 3 = E K 3 4 K 3 c = q 4 3 q 4 3 K 3 4 K 3 c E = K 3 E. We wll explore the q-serre relatos. Sce E, E belog to (m + ), (m+ ) 5 6, respectvely, the (m + ) = (q q )E (m + ), e 6 (m + ) = q 4 3 (m + ) e6, e 6 (m + ) = q 4 3 (m + ) e6. (m + ) 5 6 = (q q )E (m + ) 6 6, e 6 (m + ) 5 6 = q 3 (m + ) 5 6 e6 + (q q )q 3 (m + ) 6 6 e5, e 6 (m + ) 6 6 = q 8 3 (m + ) 6 6 e6. = e 6 E = E e 6,= E 3 E = E E 3. = e 5 = q 4 3 (q E e 6 e 6 E ).

20 0 H. HU AND N. HU So we eed to kow the relato betwee e 5 ad e 6. e 6 e 5 = q e 5 e 6 s obtaed by e e j = R j ab ea e b ad R = q, R = +q, the we have (E 3 ) E (q + q )E 3 E E 3 + E (E 3 ). O the other had, we eed to obta the relato betwee e 5 ad E. (m + ) 5 6 = (q q )E (m + ) 6 6, e 5 (m + ) 5 6 = q 3 (m + ) 5 6 e5 + (q q )q 4 3 (m + ) 6 6 e4, = e 4 = (q+q )(E e 5 e 5 E ). e 4 (m + ) 6 6 = q 4 3 (m + ) 6 6 e4. The the relato betwee e 5 ad E must be deduced from the relato betwee e 4 ad E, whch s gve by the followg cross relatos (m + ) 5 6 = (q q )E (m + ) 6 6, e 4 (m + ) 5 6 = q 3 (m + ) 5 6 e4, = e 4 E = q E e 4 = (E e 5 e 5 E )E = q E (E e 5 e 5 E ). e 4 (m + ) 6 6 = q 4 3 (m + ) 6 6 e4. Combg wth e 5 = q 4 3 (q E e 6 e 6 E ), we obta (E ) 3 e 6 (q ++q )(E ) e 6 E + (q + +q )E e 6 (E ) e 6 (E ) 3 = 0, amely, (E ) 3 E 3 3 (E ) E 3 E + 3 E E 3 (E ) E 3 (E ) 3 = 0. q q From these relatos, t follows that the resultg quatum group s U q (sp 6 ) Type-crossg costructo from type A 3 to type D 4. I the followg example, we ca choose the dfferet braded groups geerated by a 6-dmesoal U q (sl 4 )-module to gve U q (so 8 ) based o the ode dagram of U q (sl 4 ). Example 4.3. There s a 6-dmesoal rreducble represetato T V of U q (sl 4 ), gve by E x x = x 4 3 x, E x 5 x = x 5 x, E 3 x 6 x = x 3 4 x ; 5 F x 4 = x, F x = x, F 3 x 3 = x. x 5 x 3 x 6 Here{x 6} s a bass of V wth correspodg weghts λ +α, λ +α + α, λ +α +α +α 3, λ + α +α, λ + α +α +α 3, λ + α + α +α 3, respectvely. The we get aother matrx R VV, ad the PR VV matrx obeys the mmal polyomal (PR VV + q I)(PR VV q I)(PR VV qi)=0. x 5 x 5 x 4

21 DOUBLE-BOSONIZATION AND MAJID S CONJECTURE, (I): RANK-INDUCTIONS OF ABCD Settg R=qR VV, R = RPR (q + )R+(q + )P, the we have (PR+I)(PR I)=0, adλ=q. Idetfy e 6, f 6, (m + ) 6 6 c wth the addtoal smple root vectors E 4, F 4 ad the group-lke elemet K 4. The the resultg quatum group U(V (R, R ), Uext q (sl 4 ), V(R, R)) s the quatum group U q (so 8 ) wth K, 3 adjoed. Proof. The etres matrx m ± we eed are lsted by the followg equaltes (m + ) = (q q )E K K 3, (m + ) = K K 3, (m + ) 3 = (q q )E 3 K K 3, (m+ ) 3 3 = K K 3, (m + ) 4 = (q q )E K K 3, (m + ) 4 4 = K K 3, (m + ) 6 6 = K K K 3, (m ) = q(q q )K K 3 F, (m ) = K K 3, (m ) 3 = q(q q )K K 3 F 3, (m ) 3 3 = K K 3, (m ) 5 3 = q(q q )K K 3 F, (m ) 5 5 = K K 3. The cross relatos ca be easly obtaed as above, so we oly descrbe the q-serre relatos of the postve part. E, E 3 belog to (m + ) 4, (m+ ) 3, respectvely, so we have (m + ) 4 = (q q )E (m + ) 4 4, e 6 (m + ) 4 =λr 6 6 (m+ ) 4 e6 = (m + ) 4 e6, = e 6 E = E e 6,= E 4 E = E E 4. e 6 (m + ) 4 4 = (m+ ) 4 4 e6. (m + ) 3 = (q q )E 3 (m + ) 3 3, e 6 (m + ) 3 =λr 6 6 (m+ ) 3 e6 = (m + ) 3 e6, = e 6 E 3 = E 3 e 6,= E 4 E 3 = E 3 E 4. e 6 (m + ) 3 3 = (m+ ) 3 3 e6. E belogs to (m + ), the we obta (m + ) = (q q )E (m + ), e 6 (m + ) = q (m + ) e6 + q (q q )(m + ) e5, = e 5 = q E e 6 e 6 E. e 6 (m + ) = (m+ ) e6, e 5 (m + ) = q (m + ) e5. Combg wth e 6 e 5 = qe 5 e 6, whch s obtaed by e e j = R j ab ea e b ad R 5 5 q, R = 3, we have E (E 4 ) (q+q )E 4 E E 4 + (E 4 ) E = = O the other had, combg wth e 5 E = qe e 5 deduced from e 5 (m + ) =λr 5 5 (m+ ) e5, we obta E 4 (E ) (q+q )E E 4 E + (E ) E 4 = 0.

22 H. HU AND N. HU Remark 4.. Observg the costructos of C 3 ad D 4, we clam that C + ad D + ca be costructed drectly from the ode dagram A. The pars of braded groups or the R- matrces data R, R we should choose wll be obtaed by the symmetrc square ad the secod exteror power of the vector represetato of A. The verfcato of the clam wll rely o some sklls, full detals of t wll be developed a sequel. Refereces [Br] T. Brdgelad, Quatum groups va Hall algebras of complexes, A. of Math. () 77 ()(03), [D] V. G. Drfeld, Hopf algebras ad the quatum Yag-Baxter equato, Dokl. Akad. Nauk. SSSR 83 (5)(985), [FRT] L. D. Faddeev, N. Yu. Reshetkh ad L. A. Takhtaja, Quatzato of Le groups ad Le algebras, Legrad Math. J. (990), [FR] X. Fag ad M. Rosso, Mult-brace cotesor Hopf algebras ad quatum groups, arxv:0.3096v. [G] J. E. Grabowsk, Braded evelopg algebras assocated to quatum parabolc subalgebras, Comm. Algebra 39 (0) (0), [HLR] N. H. Hu, Y. N. L ad M. Rosso, Mult-parameter quatum groups va quatum quas-symmetrc algebras, arxv: [JR] R. Q. Ja ad M. Rosso, Braded cofree Hopf algebras ad quatum mult-brace algebras, J. ree agew. Math. 667 (0), [J] M. Jmbo, A q-dfferece aalog of U(g) ad the Yag-Baxter equato, Lett. Math. Phys. 0 () (985), [KS] A. Klmyk ad K. Schmüdge, Quatum Groups ad Ther Represetatos, Sprger-Verlag, Berl Hedelberg (997). [L] G. Lusztg, Caocal bases arsg from quatzed evelopg algrbras, J. Amer. Math. Soc. 3 (990), [L] G. Lusztg, Itroducto to Quatum Groups, Progress Math. 0, Brkhauser, Bosto (993). [M] S. Majd, Braded matrx structure of the Sklya algebra ad of the quatum Loretz group, Comm. Math. Phys. 56 (993), [M] S. Majd, Cross products by braded groups ad bosozato, J. Algebra 63 (994), [M3] S. Majd, Double-bosozato of braded groups ad the costructo of U q (g), Math. Proc. Cambrdge. Phlos. Soc. 5 (999), 5 9. [M4] S. Majd, Braded groups, J. Pure ad Appled Algebra 86 (993), 87. [M5] S. Majd, Algebras ad Hopf algebras braded categores, Lecture Notes Pure ad Appl. Math 58 (994), [M6] S. Majd, Braded mometum the q-pocaré group, J. Math. Phys. 34 (993), [M7] S. Majd, More examples of bcrossproduct ad double cross product Hopf algebras, Isr. J. Math 7 (990), [M8] S. Majd, New quatum groups by double-bosozato, Czech. J. Phys. 47 () (997), [M9] S. Majd, Some commets o bosozato ad bproducts, Czech. J. Phys. 47 () (997), 5 7.

23 DOUBLE-BOSONIZATION AND MAJID S CONJECTURE, (I): RANK-INDUCTIONS OF ABCD 3 [Ra] D. Radford, Hopf algebras wth projecto, J. Algebra 9 (985), [RT] D. Radford ad J. Towber, Yetter-Drfeld categores assocated to a arbtrary balgebra, J. Pure Appl. Algebra 87 (3) (993), [R] C. Rgel, Hall algebras ad quatum groups, Ivet. Math. 0 (990), [Ro] M. Rosso, Quatum groups ad quatum shuffles, Ivet. Math. 33 () (998), [So] Y. Sommerhäuser, Deformed evelopg algebras, New York J. Math. (996), [Y] D. N. Yetter, Quatum groups ad represetatos of moodal categores, Math. Proc. Camb. Phl. Soc. 08, (990), Schoolof Mathematcal Sceces, Uverstyof Scecead Techologyof Cha, J Zha Road 96, Hefe 3006, PR Cha E-mal address:hmhu04@6.com Departmetof Mathematcs, Shagha Key Laboratoryof Pure Mathematcsad Mathematcal Practce, East Cha Normal Uversty, Mhag Campus, Dog Chua Road 500, Shagha 004, PR Cha E-mal address:hhu@math.ecu.edu.c