CONSTRUCTING POINTED WEAK HOPF ALGEBRAS BY ORE-EXTENSION

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1 Acta Mathematca Scenta 2014,34B(2): CONSTRUCTING POINTED WEAK HOPF ALGEBRAS BY ORE-EXTENSION Hajun CAO ( ) School of Scence, Shandong Jaotong Unversty, Jnan , Chna E-mal :hjcao99@163.com Fang LI (Óà) Department of Mathematcs, Zhejang Unversty, Hangzhou , Chna E-mal : fangl@zju.edu.cn Manman ZHANG (éé) School of Scence, Zhejang Normal Unversty, Hangzhou , Chna E-mal : zhmm1216@163.com Abstract The man work of ths artcle s to gve a nontrval method to construct ponted semlattce graded weak Hopf algebra A t = n A α t from a Clfford monod S = [Y ; G α, ϕ α,β ] =0 by Ore-extensons, and to obtan a co-frobenus semlattce graded weak Hopf algebra H(S, n, c, χ, a,b) through factorng A t by a semlattce graded weak Hopf deal. Key words Semlattce graded weak Hopf algebra; Clfford monod; Ore-extenson 2010 MR Subject Classfcaton 16W30; 16W50 1 Introducton Because of the mportant role of Hopf algebra n the theory of quantum group and related mathematcal physcs along wth the deepenng of researches, the meanng of some weaker concepts of Hopf algebras s understood and pad close attenton more and more. A well known example s weak Hopf algebra, whch s ntroduced n [3] for studyng the non-nvertble soluton of the quantum Yang-Baxter equaton based on ths class of balgebras(see [3, 6]). Due to the mportance of the quantum Yang-Baxter equaton n the theoretcal physcs, ts soluton s the keystone n research. The theory of sngular soluton extends largely the scope of research feld. On the other aspect, there s a tght relaton between weak Hopf algebra and regular monod, for example, a semgroup algebra s a weak Hopf algebra f and only f the semgroup s a regular monod. The so-called semlattce graded weak Hopf algebra whch group-lke elements form a Clfford monod was ntroduced n [4] and a sngular soluton of the quantum Yang-Baxter Receved December 7, The project s supported by the Natonal Natural Scence Foundaton of Chna ( , , and J ), the Specalzed Research Fund for the Doctoral Program of Hgher Educaton of Chna ( ), and the Zhejang Provncal Natural Scence Foundaton of Chna (LZ13A010001).

2 No.2 H.J. Cao et al: CONSTRUCTING POINTED WEAK HOPF ALGEBRAS 253 equaton was obtaned by quantum G-double. Obvously, t s necessary to fnd more nontrval weak Hopf algebras n order to study these two aspects deeply. In [1], there s a general constructon producng ponted Hopf algebras through Ore-extenson from a group algebra. Ths gves us an dea to construct semlattce graded weak Hopf algebras through Ore-extenson from a semgroup algebra. As preparatons, we ntroduce some concepts about semgroup S: for an element x S, we call x s regular f there exsts a x S satsfyng xx x = x, x xx = x, x s called the regular nverse of x, and denote V (x) = {x S xx x = x, x xx = x } the set of all regular nverse elements of x, E(S) = {x S x 2 = x} the set of all dempotents of S. If all the elements of S are regular, we call S a regular semgroup. If V (S) has only one element for any x S, S s called an nverse semgroup, at ths case, we usually denote x 1 = x. A semlattce s a commutatve semgroup S of dempotents wth a partal order x y x y = x and x y s the greatest lower bound of {x, y}, but xy wll replace x y usually. A semgroup S s called a Clfford semgroup f t s a regular semgroup and all of ts dempotents le n ts center C(S). An equvalent defnton s that a Clfford semgroup S s a semlattce of groups, whch means the set of maxmal subgroups {G α : α Y } of S can be ndexed by elements of a semlattce Y such that S = G α and G α G β G αβ for each α, β Y. For each α, β Y wth αβ = β, there exsts a group homomorphsm ϕ α,β : G α G β. The homomorphsms are such that ϕ α,α s the dentty map on G α, and f αβ = β, βγ = γ, then, ϕ β,γ ϕ α,β = ϕ α,γ. For any α, β Y and a G α, b G β, the multplcaton n S can be gven by a b = ϕ α,αβ (a)ϕ β,αβ (b). A weak Hopf algebra H wth weak antpode T s called a semlattce graded weak Hopf algebra f H = H α s a semlattce gradng sum where H α are Hopf sub-algebras of H wth antpodes T Hα for all α Y and there are Hopf algebra homomorphsms ϕ α,β from H α to H β f αβ = β, such that for a H α and b H β, the multplcaton a b n H can be gven by a b = ϕ α,αβ (a)ϕ β,αβ (b). Thus, the set of group-lke elements of H s the Clfford monod G(H) = [Y ; G(H α ), ϕ α,β ]. The Clfford monod algebra s a trval example of ths knd of weak Hopf algebras. Moreover, throughout, k wll be an algebracally closed feld of characterstc 0; n fact, we only need that k contans enough roots of unty. The set of non-zero elements of k s denoted by k. All maps,, etc., are k-lnear. In order to compute comultplcaton on products and powers of (g, h)-prmtves, we wll requre q-bnomal coeffcents, (C l n ) q, q k. Note that ths s a formal notaton, (C l n) q s a polynomal n q. For n, l ntegers wth 0 l n, the q-bnomal coeffcents are defned by (C l n ) q = (n) q!/(l) q!(n l) q!. If l s a postve nteger, (l) q = 1 + q + + q l 1, (l) q! = (l) q (l 1) q (1) q. By defnton, (0) q! = 1. For more detal, we refer the reader to [1]. Suppose that a and b are elements of a k-algebra and ba = qab. Then, the expanson of (a + b) n s descrbed by the followng. Lemma 1.1 For q 0, ba = qab, (1) (a + b) n = n (Cn) l q a n l b l ; l=0 (2) (a + b) n = a n + b n f q s a prmtve nth root of unty. Remark 1.2 Note that n Lemma 1.1 (2), t s essental that q be a prmtve nth root.

3 254 ACTA MATHEMATICA SCIENTIA Vol.34 Ser.B For a weak Hopf algebra H, G(H) = {g H (g) = g g, ε(g) = 1} s the monod of grouplke elements of H and H 0, H 1,, H n, s denoted as the coradcal fltraton of H. H s called ponted f H 0 = kg(h). If g, h are group-lke elements of a weak Hopf algebra H, then x H s called (g, h)-prmtve of H f (x) = x g +h x and P g,h = {x H (x) = x g +h x} s the k-space of (g, h)-prmtves of H. For a ponted semlattce graded weak Hopf algebra H = H α, we know that for any α Y, H α s a ponted Hopf algebra, then by Taft-Wlson theorem[10], (H α ) 1 = (H α ) 0 P α, where (H α ) 0 = kg α s the coradcal of H α, (H α ) 1 s the next term of the coradcal, and P α = P g,h, where P g,h k(g h) = Pg,h. g,h G α Proposton 1.3 For a ponted semlattce graded weak Hopf apgebra H = H α, f {H n } s the coradcal fltraton of H and (H α ) n s the coradcal fltraton of H α for any α Y, then, H n = (H α ) n. Proof As H s a ponted semlattce graded weak Hopf algebra, H 0 = kg(h) = kg α = (H α ) 0. For H 1 = 1 (H H 0 + H 0 H), (H1 ) H H 0 + H 0 H. Obvously, for any x (H α ) 1, (x) H α kgα + kg α Hα H kg + kg H. Hence, x 1 (H kg + kg H) = H 1, that s H α H 1. On the other hand, f x H 1, then (x) H H 0 + H 0 H. As H1 s a sub-coalgebra of H and x H = H α, there s some α Y such that x H α and (x) H α Hα, thus (x) (H α H) (H α kg) + (H α kg) (H α H) H α kg α + kg α H α. Hence, x 1 (H α kgα + kg α Hα ) = (H α ) 1, that s H 1 (H α ) 1. Smlarly, by nducton on the fltraton of H, we can prove H n = (H α ) n. Hence, H 1 = ( kg α ) ( P α ) wth P α = P g,h, P g,h k(g h) = Pg,h. g,h G α Defnton 1.4 A semlattce graded weak Hopf algebra H = H α s called co- Frobenus, f for any α Y, H α s co-frobenous, that s, Hα has a left (and a rght) ntegral. Clearly, Clfford monod s a co-frobenus weak Hopf algebra. 2 Constructon of Ore-Extenson Recall that for a k-algebra A, an algebra endomorphsm ϕ of A, and a ϕ-dervaton δ of A(that s a lnear map δ : A A such that δ(ab) = δ(a)b + ϕ(a)δ(b) for all a, b A), the Ore extenson A[X, ϕ, δ] s A[X] as an Abelan group, wth multplcaton nduced by Xa = δ(a) + ϕ(a)x for all a A. The followng s an obvous extenson of the unversal property for polynomal rngs. Lemma 2.1 ([1]) Let A[X, ϕ, δ] be an Ore extenson of A and : A A[X, ϕ, δ] the ncluson morphsm. Then, for any algebra B, any algebra morphsm f : A B and every element b B such that bf(a) = f(δ(a))+f(ϕ(a))b for all a A, there exsts a unque algebra morphsm f : A[X, ϕ, δ] B such that f(x) = b and f = f. Let S be a fnte semgroup, and we wrte aj b f there exst x, y, u, v n S 1 = S {1}

4 No.2 H.J. Cao et al: CONSTRUCTING POINTED WEAK HOPF ALGEBRAS 255 such that xay = b and ubv = a. Obvously, J s an equvalent relaton on S. A J -class of some element x S s the set {y S xj y}, that s the equvalent class ncludng x under the relaton J. A J -class s called regular J -class f all ts elements are regular (see [12]). If Γ : S Hom(A, A) s a representaton of S, by lnear transformatons of a fntedmensonal vector space A over the complex feld C, then the character χ Γ of Γ s the functon S C defned by χ Γ (x) = TrΓ(x) for any x S, where TrΓ(x) denotes the sum of the dagonal entres n any matrx expresson of Γ(x). Let a S and ā = ae, where e s the unque dempotent n the cyclc subsemgroup a whch generated by a, we shall say that two elements a, b S are conjugate f and only f b = x āx, ā = x bx for some regular element x wth nverse x. Then, a functon S C s called a class functon f t s constant on each conjugacy class of S, and all the class functons form an algebra over C under pontwse addton and mutplcaton, we denote ths algebra by cfs. The character rng of S, whch denoted by chs, s a subrng of the class functon algebra whch s spanned by the rreducble characters of S [9]. Lemma 2.2 ([9]) Let S be a fnte semgroup, let J 1,, J r be the regular J -classes of S, and let H 1,, H r be the maxmal subgroups of J 1,, J r, respectvely. Then, chs = chh 1 chh r. For a Clfford semgroup S, we can see that J ( = 1, 2,, r) are the maxmal subgroups G α, α Y of S. Hence, chs = chg α. Next, we wll construct a ponted weak Hopf algebra by startng wth the coradcal, formng Ore extensons, and then factorng out a weak Hopf deal. Let S = [Y ; G α, ϕ α,β ] be a fnte commutatve Clfford monod (the method for constructng some concrete Clfford monod can be seen n [7]). As S s fnte, Y s fnte, let Y = {α 1, α 2,, α n }, moreover, becaus S s a monod, we can suppose that 1 G α1, where α 1 s the dentty of Y. Then, for the Clfford monod algebra ks, we have A = ks = n kg α s a semlattce graded weak Hopf algebra and chs chg α1 chg α2 chg αn. As for any Abelan group G, chg = G, then for any c G, there s a c chg respondng to c. Let 1 c 1 G α1, then there exsts a c 1 chg α1 whch can be mapped nto chs by χ 1, where χ 1 = (c 1, 1,, 1) chs. Let ϕ 1 be an algebra automorphsm of A defned by ϕ 1 (g) = χ 1, g g for all g S, that s: f g G α1, ϕ 1 (g) = c 1, g g; or f g G α1, ϕ 1 (g) = g. Consder the Ore extenson A 1 = A[X 1, ϕ 1, δ 1 ] = A[X 1, ϕ 1, 0] wth δ 1 = 0. Apply Lemma 2.1 on B = A 1 A 1, f = ( ) A, b = c 1 X 1 + X 1 1, we can defne algebra homomorphsms : A 1 A 1 and ε : A 1 k by (X 1 ) = c 1 X 1 + X 1 1, ε(x 1 ) = 0 wth B = k, f = ε A, b = 0. It s easly to check that and ε defne a balgebra structure on A 1. The weak antpode T of A extends to a weak antpode on A 1 by T(X 1 ) = c 1 1 X 1, T(X 1 g α ) = (cg α ) 1 X 1 because of (d T d)(x 1 ) = c 1 T(c 1 )X 1 + c 1 T(X 1 )1 + X 1 T(1)1 = c 1 c 1 1 X 1 + c 1 ( c 1 1 X 1) + X 1 = X 1, (T d T)(X 1 ) = T(c 1 )c 1 T(X 1 ) + T(c 1 )X 1 + T(X 1 ) Hence, A 1 s a weak Hopf algebra. = c 1 1 c 1( c 1 1 X 1) + c 1 1 X 1 + T(X 1 ) = T(X 1 ).

5 256 ACTA MATHEMATICA SCIENTIA Vol.34 Ser.B If we let A α 1 = kg α [X 1 e α, ϕ α 1, δα ], where δ α = 0 and ϕ α 1 (g α ) = χ 1 Gα, g α g α, then, A α 1 s a subweak Hopf algebra of A 1 whch s a Hopf algebra for each α Y under the defned, ε, and antpode T. We have X 1 e α g α = X 1 g α = χ 1, g α g α X 1, X 1 e α = χ 1, e α e α X 1 = e α X 1, (X 1 e α ) = c 1 e α X 1 e α + X 1 e α e α, ε(x 1 e α ) = ε(x 1 )ε(e α ) = 0, T(X 1 e α g α ) = T(X 1 g α ) = (c 1 g α ) 1 X 1 = T(e α g α )T(X 1 ), m(d T) (X 1 e α ) = c 1 e α T(X 1 e α ) + X 1 e α T(e α ) = c 1 e α (e α c 1 1 X) + X 1e α = 0 = ε(x 1 e α ). Ths s to say that A α 1 s a Hopf algebra wth antpode T α A 1 for any α Y. Secondly, we need to prove A α 1 Aβ 1 Aαβ 1 for any α, β Y. Obvously, B α 1 = {g α(x 1 e α ) n g α G α, n N} forms a bass of A α 1. If we defne an algebra homomorphsm φ 1 α,β : Aα 1 A β 1 from A α 1 to Aβ 1 n case of αβ = β, whch satsfes φ1 α,β (g α) = ϕ α,β (g α ) for any g α G α, and φ α,β (X 1 e α ) = X 1 e β. Then, for any g α (X 1 e α ) n1 B α 1, g β (X 1 e β ) n2 B β 1, g α (X 1 e α ) n1 g β (X 1 e α ) n2 = g α X n1 1 e αβϕ β,αβ (g β )X n2 1 e αβ = g α χ 1, ϕ β,αβ (g β ) n1 ϕ β,αβ (g β )X n1 1 Xn2 1 e αβ = χ 1, ϕ β,αβ (g β ) n1 (g α g β )(X 1 e αβ ) n1+n2 = ϕ α,αβ (g α ) χ 1, ϕ β,αβ (g β ) ϕ β,αβ (g β )X n1 1 Xn2 1 e αβ = ϕ α,αβ (g α )φ 1 α,αβ ((X 1e α ) n1 )ϕ β,αβ (g β )φ 1 β,αβ ((X 1e β ) n2 ) = φ 1 α,αβ(g α (X 1 e α ) n1 )φ 1 β,αβ(g β (X 1 e β ) n2 ). Ths just means that for any a α A α 1, b β A β 1, we have a α b β = φ 1 α,αβ (a α)φ 1 β,αβ (b β). Thus, A 1 = A α 1 s a semlattce graded weak Hopf algebra under the above decomposton. Next, let 1 c 2 G α1, c 2 chg α1, then c 2 can be naturally mapped nto chs by χ 2, such that χ 2 = c 2, 1,, 1 chs. At the same tme, let γ 12 k, and defne an automorphsm ϕ 2 of A 1 = A[X 1, ϕ 1, 0] by ϕ 2 (X 1 ) = γ 12 X 1 for any g S, ϕ 2 (g) = χ 2, g g. We seek a ϕ 2 - dervaton δ 2 of A 1 such that δ 2 s zero on ks and δ 2 (X 1 ) ks. We want the Ore extenson A 2 = A 1 [X 2, ϕ 2, δ 2 ] to have a semlattce graded weak Hopf algebra structure wth X 2 beng a (1, c 2 )-prmtve for some c 2 S. Suppose that A 2 s a weak Hopf algebra, then and ε are all algebra homomorphsms. As X 2 X 1 = δ(x 1 ) + ϕ 2 (X 1 )X 2 = δ 2 (X 1 ) + γ 12 X 1 X 2, applyng to both sdes of the above equaton, we see that (X 2 X 1 ) = c 2 c 1 X 2 X 1 + c 2 X 1 X 2 + X 2 c 1 X 1 + X 2 X 1 1 = (δ 2 (X 1 )) + γ 12 c 1 c 2 X 1 X 2 + γ 12 c 1 X 2 X 1 + γ 12 X 1 c 2 X 2 + γ 12 X 1 X 2 1. Hence, c 2 X 1 = γ 12 X 1 c 2 = γ 12 χ 1, c 2 c 2 X 1 = γ 12 c 1, c 2 c 2 X 1, γ 12 c 1 X 2 = X 2 c 1 = χ 2, c 1 c 1 X 2 = c 2, c 1 c 1 X 2, we wll obtan γ 12 = c 2, c 1 = c 1, c 2 1, and (δ 2 (X 1 )) = c 2 c 1 X 2 X 1 + X 2 X 1 1 γ 12 c 1 c 2 X 1 X 2 γ 12 X 1 X 2 1 = c 1 c 2 δ 2 (X 1 ) + δ 2 (X 1 ) 1. Thus, δ 2 (X 1 ) s a (1, c 1 c 2 ) prmtve of ks, then, δ 2 (X 1 ) kg α1. So, we must have δ 2 (X 1 ) k(c 1 c 2 1) because there s no trval (g, h) prmtve n ks, that s δ 2 (X 1 ) = b 12 (c 1 c 2 1) for some scalar b 12. If c 1 c 2 1 = 0, we can defne b 12 to be 0, that s δ 2 (X 1 ) = 0, and δ 2 s clearly a ϕ 2 -dervaton. Suppose that δ 2 0. It remans to check that δ 2 s a ϕ 2 -dervaton of A 1. In order that

6 No.2 H.J. Cao et al: CONSTRUCTING POINTED WEAK HOPF ALGEBRAS 257 δ 2 s well defned, we must have, for any g S, δ 2 (gx 1 ) = ϕ 2 (g)δ 2 (X 1 ) = χ 2, g gδ 2 (X 1 ) = χ 1, g 1 δ 2 (X 1 )g. If g G α1, then δ 2 (gx 1 ) = c 2, g gδ 2(X 1 ) = c 1, g 1 δ 2 (X 1 )g. As δ 2 (X 1 ) ks, δ 2 (X 1 )g = gδ 2 (X 1 ), c 1, g 1 = c 2, g. If g G α and α α 1, then δ 2 (gx 1 ) = gδ 2 (X 1 ) = δ 2 (X 1 )g s always satsfed. Thus, f we want δ 2 to be well-defned, we need to suppose c 2, g = c 1, g 1 for any g G α1, that s ϕ 2 (g) = c 2, g g = c 1, g 1 g. Hence, c 1 c 2 = 1 and γ 12 = c 1, c 2 1 = c 2, c 2 = c 2, c 1 = c 1, c 1 1. Now, we compute δ 2 (X1 2) = δ 2(X 1 )X 1 + ϕ 2 (X 1 )δ 2 (X 1 ) = b 12 (1 + c 1, c 1 )c 1 c 2 X 1 b 12 (1 + c 1, c 1 1 )X 1. By nducton, we see that for every postve nteger n N, ( n 1 ) ( n 1 ) δ 2 (X1 n ) = b 12 c 1, c 1 k c 1 c 2 X1 n 1 b 12 c 1, c 1 k X1 n 1. A straghtforward (tedous) composton now ensures that for any g, g S, n 1, n 2 N, δ 2 (gx n1 1 g X n2 1 ) = δ 2(gX n1 1 )g X n2 1 + ϕ 2 (gx n1 1 )δ 2(g X n2 1 ). As B 1 = {gx n 1 g S, n N} forms a bass of A 1 and by the above equatons, for any a, b A 1, δ 2 (ab) = δ 2 (a)b+ϕ 2 (a)δ 2 (b). Then, we have proved that δ 2 s a ϕ 2 -dervaton of A 1 f t satsfes c 1c 2 = 1 when b To ths condton, we gve a condton for A 2 = A 1 [X 2, ϕ 2, δ 2 ] beng a weak Hopf algebra, that s, c 1 c 2 = 1 when b Summarzng, A 2 s a weak Hopf algebra wth generators g S = [Y ; G α, ϕ α,β ], X 1, X 2, such that the elements of S are commutng group-lkes, X j s a (1, c j )-prmtve, where c j G α1, j = 1, 2, and the followng relatons hold gx j = χ j, g 1 X j g, X 2 X 1 γ 12 X 1 X 2 = b 12 (c 1 c 2 1) (where γ 12 = c 1, c 2 1 = c 2, c 1 ), and f δ(x 1 ) 0, c 1 c 2 = 1, γ 12 = c 1, c 2 1 = c 2, c 2 = c 2, c 1 = c 1, c 1 1. The weak antpode of A 2 can be defned by T(X j ) = c 1 j X j, T(g) = g 1, T(X j g) = (c j g) 1 X j. Smlar to the decomposton of A 1, let A α 2 = Aα 1 [X 2e α, ϕ α 2, δα 2 ], where ϕα 2 = ϕ 2 A α 2, δ2 α = δ 2 A α 2, T2 α = T A α. Then, smlar to the above proof, we obtan the result that for any α Y, 2 A α 2 s a Hopf algebra, and A 2 = n A α 2. If α β, we defne a Hopf algebra homomorphsm φ 2 α,β : Aα 2 Aβ 2 satsfyng φ2 α,β (g) = ϕ α,β (g) for any g S and φ 2 α,β (X je α ) = X j e β, j = 1, 2. Moreover, we also prove that A 2 = n A α 2 s a semlattce graded weak Hopf algebra under ths defnton. We contnue formng Ore extensons. Defne an algebra automorphsm ϕ j of A j 1 by ϕ j (g) = χ j, g g, where χ j = c j, 1,, 1 chs, c j G α1, ϕ j (X ) = c j, c X where c G α1, and X are (1, c )-prmtves. The dervaton δ j of A j 1 s 0 on ks and δ j (X ) = b j (c c j 1). If c c j = 1, we defne b j = 0. Denote X p beng X p1 1 Xp2 2 Xpt t and (Xe α ) p beng (X 1 e α ) p1 (X 2 e α ) p2 (X t e α ) pt, where p N t. After t steps, we have a semlattce graded weak Hopf algebra A t. Defnton 2.3 A t = n A α t s the semlattce graded weak Hopf algebra generated by g S = [Y ; G α, ϕ α,β ](whch s a Clfford monod) and X j, j = 1,, t, where (1) the elements of S are commutatng group-lkes; (2) the X j are (1, c j )-prmtves;

7 258 ACTA MATHEMATICA SCIENTIA Vol.34 Ser.B (3) X j g = χ j, g g, where χ j = c j, 1,, 1 chs; (4) X j X k = c j, c k X k X j + b kj (c k c j 1) for 1 k j t; (5) c k, c j c j, c k = 1 for j k; (6) f b j 0, then c c j = 1; (7) for any α Y, A α t s a Hopf algebra generated by g G α and X j e α, j = 1,, t. The weak antpode T of A t s defned by T(g) = g 1, T(X j ) = c 1 j X j, and T(X j g) = (c j g) 1 X j. The relatons show that B t = {gx n g S, n N t } forms a bass of A t and Bt α = {g α (Xe α ) n g α G α, n N t } forms a bass of A α t for any α Y, where n = (n 1, n 2,, n t ) N t, and X n = X n1 1 Xn2 2 Xnt t, (Xe α ) n = X n e α. Because of q j = c j, c j, (X j 1)(c j X j ) = χ j, c j c j X j X j = q j (c j X j )(X j 1), for any k Z +, (Xj n) = (X j) n = (c j X j + X j 1) n = n (Cn k) q j (c j X j ) n k (X j 1) k and expanson of ths power follows the rules n Lemma 1.1. Because (c j X j ) n k = c n k j (X j 1) k = Xj k 1, we wll get (Xn j ) = n (Cn) k qj c n k = n j X k j Xn k j X n k j, (Cn) k qj c k j Xn k j Xj k. Smlarly, by nducton, we prove that for any n = (n 1, n 2,, n t ) N t, X n = X n1 1 Xn2 2 Xnt t, we have (gx n ) =(g g) d ( n1 k 1=0 (C k1 n 1 ) q1 c k1 1 Xn1 k1 1 X k1 1 ) ( nt k t=0 (C kt n t ) qt c kt t Xnt kt t X kt t α d gc d1 1 cd2 2 cdt t X p d gx d, (2.1) where d = (d 1, d 2,, d t ) Z +, the jth entry d j n the t-tuple d ranges from 0 to n j, and α d are scalars resultng from the q-bnomal expanson descrbed n Lemma 1.1 and the commutaton relatons. Therefore, we obtan the followng result. Proposton 2.4 The semlattce graded weak Hopf algebra A t = n has the followng propertes: A α t ) (1) The (n + 1)th term, (A t ) n n the coradcal fltraton of A t s generated by gx p, g S, p N t, p 1 + p p t n. At the same tme, for any α Y, the (n + 1)th term, (A α t ) n n the coradcal fltraton of A α t s generated by g(xe α) p, g G α, p N t, p 1 + p p t n, and n (A α t ) n = (A t ) n. In partcular, A t s a ponted weak Hopf algebra wth coradcal ks, A α t s a ponted Hopf algebra wth coradcal kg α for any α Y. (2) For g S, the njectve envelop of kg n the category of rght A t -comodules s the k-space C g, whch s spanned by all gc p1 1 c p2 2 c pt t X p1 1 Xp2 2 Xpt t = gc p1 1 c p2 2 c pt t X p, p = (p 1, p 2,, p t ) N t. (3) A t s not a co-frobenus weak Hopf algebra. Proof (1) An nducton argument usng equaton (2.1) shows that for all n, {gx p g S, p N t, p p t n} n+1 ks. Thus, ks = A t, and (A t ) 0 ks. As ks s a cosemsmple coalgebra, t s drectly the coradcal (A t ) 0 = ks.

8 No.2 H.J. Cao et al: CONSTRUCTING POINTED WEAK HOPF ALGEBRAS 259 (2) Agan by equaton (2.1), C g s a rght A t -subcomodule of A t, and kg s essental n C g. On the other hand, A t = C g, g S. Thus, the C g s are njectve. g (3) For any α Y, g G α, the njectve envelop of kg n the category of rght A α t - comodules s the k-space Cg α, whch s spanned by all g(c 1 e α ) p1 (c 2 e α ) p2 (c t e α ) pt (Xe α ) p, p = (p 1, p 2,, p t ) N t. Then, Cg α are nfnte dmensonal Aα t -comodules, followng from [10], A α t s not co-frobenus, hence A t s not co-frobenus weak Hopf algebra. In order to obtan a co-frobenus weak Hopf algebra, we factor A t by a weak Hopf deal. At frst, we need the followng defnton. Defnton 2.5 An deal J = J α of a semlattce graded weak Hopf algebra H = H α s called semlattce graded weak Hopf deal, f J s a b-deal of H(that s deal and codeal of H) and satsfes T(J) J. Moreover, for any α Y, J α s a Hopf deal of H α and satsfes for any α, β Y, J α J β J α, H α J β J αβ, and J α H β J αβ. Theorem 2.6 Let H = H α be a semlattce graded weak Hopf algebra, and J = J α s a semlattce graded weak Hopf deal of H, then, H/J = H α /J α s a semlattce graded weak Hopf algebra. Proof Obvously, H/J s a weak Hopf algebra because H/J s a balgebra and satsfes T(J) J. We just need to prove that t has semlattce graded weak Hopf algebra structure, that s to prove that for any α, β Y, (H α /J α )(H β /J β ) H αβ /J αβ. For any h+j α H α /J α, k + J β H β /J β, (h + J α )(k + J β ) = hk + hj β + J α k + J α J β = hk + J αβ H αβ /J αβ. Then, H/J = H α /J α s a semlattce graded weak Hopf algebra. Lemma 2.7 Let n 1, n 2,, n t 2, a = (a 1, a 2,, a t ) {0, 1} t. The deal J(a) α of A α t generated by (X 1 e α ) n1 a 1 ((c 1 e α ) n1 e α ),, (X t e α ) nt a t ((c t e α ) nt e α ) s a Hopf deal f and only f q j = c j, c j s a prmtve n j th root of unty for 1 j t. Under these condtons, J(a) = n J(a) α becomes a semlattce graded weak Hopf deal of A t. Proof We need to prove that for any α Y, J(a) α s an deal, codeal of A α t and satsfes T(J(a) α ) J(a) α. Moreover J(a) s an deal and codeal of A t whch satsfes T(J(a)) J(a). As (c j e α ) nj 1 s an (e α, (c j e α ) nj )-prmtve, t follows that (X j e α ) nj a j ((c j e α ) nj e α ) s an (e α, (c j e α ) nj )-prmtve f and only f (X j e α ) nj s (e α, (c j e α ) nj )-prmtve. As ((X j e α ) nj ) = n j k j=0 (C kj n j ) qj (c j e α ) kj (X j e α ) nj kj (X j e α ) kj accompany wth Remark 1.2, ths occurs f and only f for every 0 < k j < n j, (Cn kj j ) qj = 0 f and only f q j s a prmtve n j th root of unty. Moreover, because T(X j e α ) = (c j e α ) 1 X j = (c j e α ) 1 (X j e α ), nducton on n shows that: T((X j e α ) nj ) = ( 1) nj q nj(n j 1) 2 j (c j e α ) nj (X j e α ) nj. Now, as q nj j = 1, checkng the cases n j even and n j odd, we see that ( 1) nj q nj(nj 1) 2 j = 1, and hence T((X j e α ) nj ) = (c j e α ) nj (X j e α ) nj. So, T((X j e α ) nj a j ((c j e α ) nj e α )) = (c j e α ) nj ((X j e α ) nj a j ((c j e α ) nj e α )) J(a) α.

9 260 ACTA MATHEMATICA SCIENTIA Vol.34 Ser.B Moreover, for any X j e α A α t, (X e β ) n a ((c e β ) n e β ) J(a) β, (X j e α )((X e β ) n a ((c e β ) n e β )) = X j (X n = (X j e αβ )((X e αβ ) n a ((c e αβ ) n e αβ )) J(a) αβ. a (c n 1))e α e β Ths just means that for any α Y, we have A α t J(a)α J(a) α, and A α t J(a)β J(a) αβ, then A t J(a) J(a). Smlarly, we can get J(a) α A α t J(a) α and J(a)A t J(a). Hence, for any α Y, J(a) α s an deal of A α t and J(a) s an deal of A t. From the above proof, t s easly to check that J(a) s a codeal of A t and J(a) α s a codeal of A α t for any α Y, whch satsfy T(J(a)) J(a) and T(J(a) α ) J(a) α. At ths case, we prove that J(a) s a weak Hopf deal of A t and J(a) α s a Hopf deal of A α t for any α Y. Smlarly, by the above proof, for any α, β Y, J(a) α J(a) β J(a) αβ, that s, J(a) s a semlattce graded weak Hopf deal. However, we want H to be a ponted semlattce graded weak Hopf algebra wth coradcal ks, where some addtonal restrctons are requred. We denote by x the mage of X n H/J, then x e α s the mage of X e α n H/J, and x p = x p1 1 xpt t, p = (p 1,, p t ) N t. Proposton 2.8 Assume that J(a) = n J α a as n Lemma 2.7 s a semlattce graded weak Hopf deal. Then, J(a) ks = 0 f and only f for each, ether a = 0 or (c )n = 1. If ths s the case, then {g(xe α ) p g G α, p N t, 0 p j n j 1} s a bass of A α t /Jα a. Proof By Lemma 2.7, we know that J(a) s a semlattce graded weak Hopf deal f and only f for any 0 t, q = c, c s a prmtve n th root of unty. Now, suppose that J(a) ks = 0, then for any α Y, J(a) α kg α = 0. Hence, for each g G α1, we have (X n It follows that X n a (c n 1))g = c, g n (X n c, g n a (c n c, g n a (c n 1)) J(a) α1. 1) J(a) α1. Then, f a 0, whch by our conventon mples that for any g G α1, c, g n = 1, and thus c n = 1. Conversely, assume that c n = 1 whenever a 0, J(a) ks = 0, that s, for any α Y, J(a) α kg α = 0. By defnton, for any g G α, (X e α ) n g = χ, g g(x e α ) n. In partcular, (X e α ) n g = g(x e α ) n f a 0. Also, f < j, then (X j e α )(X e α ) nα = ϕ j ((X e α ) n )X j e α + δ j ((X e α ) n ) ( n 1 ) = c j, c n (X e α ) n (X j e α ) + b j c, c k c c j (X e α ) n 1 ( n 1 b j c, c )(X k e α ) n 1. So, f b j = 0, then (X j e α )(X e α ) n = c j, c n (X e α ) n (X j e α ), where c j, c n = c, c n = 1 when a 0. If b j 0, then c c j = 1, hence c, c s a prmtve n th root of unty, so that (X j e α )(X e α ) n = (X e α ) n (X j e α ). A smlar argument works for > j such that (X j e α )(X e α ) n = (X e α ) n (X j e α ). Thus, (X e α ) n s a central element of A α t f a 0. It follows that: (X j e α )((X e α ) n a ((c e α ) n e α )) = c j, c n ((X e α ) n a ((c e α ) n e α ))(X j e α ), so that J(a) α s equal to the left deal generated by {(X j e α ) nj a j ((c j e α ) nj e α ) 1 j t},

10 No.2 H.J. Cao et al: CONSTRUCTING POINTED WEAK HOPF ALGEBRAS 261 and A α t s a free left module wth bass {X p e α 0 p j n j 1} over the subalgebra B α generated by G α and (X 1 e α ) n1,, (X t e α ) nt. We now show that no non-zero lnear combnaton of elements of the form gx p, p N t,g G α, 0 p j n j 1 les n J(a) α. Otherwse, there exst f j A α t, not all zero, such that ((X j e α ) nj a j ((c j e α ) nj e α ))f j = α g,p gx p, 1 j t where g G α, p N t. As A α t s a free left B α-module wth bass {X p e α 0 p j n j 1}, each f j can be expressed n terms of ths bass, and we fnd that ((X j e α ) nj a j ((c j e α ) nj 1 j t e α ))F j kg α {0} for some F j B α. Now, B α s somorphc to the algebra R α obtaned from kg α by a sequence of Ore-extensons wth zero dervatons n the ndetermnate Y = (X e α ) n, so that for any g G α, Y g = (X e α ) n g = χ n, g gy, Y j Y = (X j e α ) nj (X e α ) n = c nj j, c n X n X nj j e α = c nj j, c n Y Y j. Thus, there exst some G j R α such that 1 j t (Y j a ((c j e α ) nj e α ))G j kg α {0}. By nducton on the number of ndetermnate Y j, there exsts a kg α -algebra homomorphsm θ : R α kg α such that θ(y j ) = (c j e α ) nj e α f a j 0 and θ(y j ) = 0 otherwse. Then, θ( (Y j a j ((c j e α ) nj e α ))G j )=0, a contradcton. 1 j t From now on, we assume that n j 2, q j = c j, c j s a prmtve n j th root of unty, and c nj j = 1 whenever a j 0, and we study the new semlattce graded weak Hopf algebra H = A t /J(a) = n A α t /J(a) α. Defnton 2.9 Let t 1, S = [Y ; G α, ϕ α,β ] s a fnte commutatve Clfford monod, n = (n 1, n 2,, n t ) N t, c = (c 1, c 2,, c t ) G t α 1, χ = (χ 1, χ 2,, χ t ) chs t as above, where χ j = (c j, 1,, 1) chs, a = (a 1, a 2,, a t ) {0, 1} t, and b = (b j ), 1 j t. Defne H = H(S, n, c, χ, a, b) to be the semlattce graded weak Hopf algebra generated by the commutng group-lkes g S, and the (1, c j )-prmtves x j, 1 j t, where, as well, (1) for any g S, x j g = χ j, g gx j ; (2) x nj j = a j (c nj j 1); (3) x k x j = c k, c j x j x k + b jk (c j c k 1) for any 1 j < k t; (4) c j, c k c k, c j = 1 for j k, c j, c j s a n j th prmtve root of unty; (5) a j = 0 whenever c nj j = 1; f a j 0, c nj j = 1, and then χ nj j = 1; (6) b j = 0 f c c j = 1; f b j 0, c c j = 1, and then χ χ j = 1; (7) for any α Y, defne H α = A α t /J(a)α = H(G α, n, ce α, χ Gα, a, b) to be the Hopf algebra generated by the commutng group-lkes g G α and the (e α, c j e α )-prmtves x j e α, 1 j t, whch s a subweak Hopf algebra of H, where ce α = (c 1 e α, c 2 e α,, c t e α ) G t α. Remark 2.10 To construct a new semlattce graded weak Hopf algebra, t suffce to have c G t α 1 and χ = (χ 1, χ 2,, χ t ) chs t, where χ = (c, 1,, 1) chs, such that for any 1 t, c, c s a root of unty not equal to 1, and for any j, c, c j c j, c = 1. Then, n s the order of c, c, and we choose a and b such that a = 0 whenever c n = 1, a = 0 whenever c n 1, b j = 0 whenever c c j = 1, and b j = 0 whenever c c j 1. The remanng a s and b js are arbtrary. By the above Proposton 2.8, {gx p g S, p N t, 0 p j n j 1} s a bass of H, then the comultplcaton of H on a general bass s gven by: (gx p ) = α d gc d1 1 cd2 2 cdt t x p d d

11 262 ACTA MATHEMATICA SCIENTIA Vol.34 Ser.B gx d, where d = (d 1, d 2,, d t ) Z t wth 0 d j p j. Here, the scalars α d are non-zero products of q j -bnomal coeffcents and powers of c j, c. In partcular, for k Z +, (x k j ) = (Ck d) q j c d j xk d j x d j. 0 d k Proposton 2.11 H = H(S, n, c, χ, a, b) has the followng propertes: (1) H s ponted semlattce graded weak Hopf algebra, and the (r + 1)th term n the coradcal fltraton of H s H r = gx p g S, p N t, p 1 + p p t r. When n = n 1 +n 2 + +n t t such that the coradcal fltraton has n 1 +n 2 + +n t t+1 terms, then H = H n. (2) H 1 = ( kg α ) ( P α ) = H 0 P, where P α = P g,h, P g,h k(g h) = g,h G α P g,h, and P = P α. We obtan the fact that {gx j g S, j = 1, 2,, t} s a bass of P, and that {gx j g G α, j = 1, 2,, t} s a bass of P α for any α Y. Thus, for any g S, P 1,g = k(g 1) unless g = c j for some 1 j t, and for any g G α, P eα,g = k(g e α ) unless g = c j e α for some 1 j t. Moreover, the k-dmenson of P s S t, and the k-dmenson of P α s G α t. Remark 2.12 Unlke A t, H = H(S, n, c, χ, a, b) = n H(G α, n, ce α, χ Gα, a, b) s a co- Frobenus weak Hopf algebra. Example 2.13 If H = H(S, n, c, χ, a, b), then H op and H cop are also of ths type. Indeed, H op = H(S, n, c, χ 1, a, b ), H cop = H(S, n, c 1, χ, a, b ), where for any < j, b j = c j, c j b j. Also, the second somorphsm can be gven by the followng map f: f : H cop H(S, n, c 1, χ, a, b ) wth g g, x j z j = c 1 j x j. Where for any < j, b j = c j, c b j, and z j s a (1, c 1 j )- prmtve. References [1] Beatte M, Dǎscǎlescu S, Grünenfelder L. Constructng ponted Hopf algebra by Ore exten-sons. J Algebra, 2000, 225: [2] Kassel C. Quantum Groups. New York: Sprnger-Verlag, 1995 [3] L F. Weak Hopf alg ebras and some new solutons of quantum Yang-Baxter equaton. J Algebra, 1998, 208: [4] L F, Cao H J. Semlattce graded weak Hopf algebra and ts related quantum G-double. J Math Physcs, 2005, 46(8): 1 17 [5] L F, Cao H J. Semlattce graded weak Hopf algebra and ts quantum double//advance n rng theory proceedngs. 2005: [6] L F, Duplj S. Weak Hopf algebras and sngular solutons of quantum Yang-Baxter equaton. Comm Math Phys, 2002, 225: [7] L F, Zhang Y Z. Quantum Doubles From A Class of Noncocommutatve Weak Hopf Algebras. J Math Physcs, 2004, 45(8): [8] Ln B I P. Semperfect coalgebra. J Algebra, 1977, 49: [9] McAlster D B. Characters of fnte semgroups. J Algebra, 1972, 22: [10] Montgomery S. Hopf algebras and ther actons on rngs//cbms Regonal Conference Seres n Mathematcs. Provdence. RI: Amercan Mathematcal Socety, 1993: 82 [11] Montgomery S. Indecomposable coalgebras, smple comodules, and ponted Hopf algebras. Proceedng of Amercan Mathematcal Socety, 1995, 123(8): [12] Petrch M. Inverse Semgroups. New York: John Wley & Sons, 1984