ORE EXTENSIONS OF HOPF GROUP COALGEBRAS

J. Korean Math. Soc. 51 (2014), No. 2, pp. 325 344 http://dx.doi.org/10.4134/jkms.2014.51.2.325 ORE EXTENSIONS OF HOPF GROUP COALGEBRAS Dingguo Wang and Daowei Lu Abstract. The aim of this paper is to generalize the theory of Hopf-Ore extension on Hopf algebras to Hopf group coalgebras. First the concept of Hopf-Ore extension of Hopf group coalgebra is introduced. Then we will give the necessary and sufficient condition for the Ore extensions to become a Hopf group coalgebra, and certain isomorphism between Ore extensions of Hopf group coalgebras are discussed. 1. Introduction Ore extensions are main kinds of ring extensions to construct a class of noncommutative rings and algebras. From the point of view of quantum group and Hopf algebra theory, Ore extensions are important for constructing examples of Hopf algebras which are neither commutative nor cocommutative. In recent years, many new examples (often finite dimensional) with special properties were constructed by means of Ore extensions, such as pointed Hopf algebras, co-frobenius Hopf algebras, and quasitriangular Hopf algebras (see, e.g., [1, 2, 5]). Panov [6] introduced the concept and equivalent description of a Hopf-Ore extension and given the classifications of Hopf-Ore extensions for some typical Hopf algebras. Multiplier Hopf algebras were introduced by Van Daele [10] as a generalisation of Hopf algebras to the case where the underlying algebra is not necessarily unital. Lihui Zhao and Diming Lu [13] generalized the notion of Ore extension of Hopf Algebras to regular multiplier Hopf algebras and obtained the corresponding result. Hopf group coalgebras were introduced by V. G. Turaev [8, 9]. Hopf group coalgebras generalize usual coalgebras and Hopf algebras, in the sense that we recover these notions in the situation where the group is trivial. Virelizier [11] started an algebraic study of this topic, this was continued by Zunino [14, 15] and Wang [12]. It is natural to investigate the Ore extensions of Hopf group coalgebras. This was the motivation of our paper. The first problem we face is how to define the Ore extensions of Hopf group coalgebras. Here we could resort to the method used in [3] which constructed Received June 5, 2013. 2010 Mathematics Subject Classification. 16W30, 16S36. Key words and phrases. Hopf algebras, Hopf π-coalgebras, Ore extension. 325 c 2014 Korean Mathematical Society

326 DINGGUO WANG AND DAOWEI LU the concept of differential calculus on Hopf group coalgebras, that is, let A = {A α } α π be a Hopf group coalgebra, and R = {R α = A α [y α ;σ α,δ α ]} α π, where for any α π, A α [y α ;σ α,δ α ] is the Ore extension of A α. Then we make R = {R α } α π a Hopf group coalgebra. The comultiplication and counit could be extended from A to R naturally. In Section 2, we recall the definition of Hopf group coalgebras and some basic facts about Hopf group coalgebras. In Section 3, we first introduce the notion of an Ore extension for a Hopf group coalgebra, and extend and ε from A to R such that R = {R α } α π becomes a Hopf group coalgebra. In the main theorem of this article, we give a necessary and sufficient condition for Ore extensions of a Hopf group coalgebra to be a Hopf group coalgebra. In Section 4, we will consider the isomorphisms between Ore extensions for different Hopf group coalgebras and give the sufficient conditions. 2. Hopf group coalgebra For convenience of the reader we recall the standard definitions of Hopf algebras, see for instance [4, 7]. Throughout this paper, we let π be a discrete group(with neutral element 1) and k be a field. All algebras are supposed to be over k. If U and V are k-space, τ U,V : U V V U will denote the flip defined by τ U,V (u v) = v u. Definition 2.1. A π-coalgebra(over k) is a family C = {C α } α π of k-spaces endowed with a family k-linear maps (the comultiplication) and a k-linear map ε : C 1 k (the counit) such that is a coassociative in the sense that, for any α,β,γ π, (b) for all α π, ( α,β id Cγ ) αβ,γ = (id Cα β,γ ) α,βγ ; (id Cα ε) α,1 = id Cα = (ε id Cα ) 1,α. Note that (C 1, 1,1,ε) is a coalgebra in the usual sense. Sweedler s notation. We extend the Sweedler notation in the following way: for any α,β π and c C αβ, we write α,β (c) = (c) c (1,α) c (2,β) C α C β. Or shortly, if we leave the summation implicitly, α,β (c) = c (1,α) c (2,β). The coassociativity axiom gives that, for any α,β,γ π and c C αβγ, c (1,αβ)(1,α) c (1,αβ)(2,β) c (2,γ) = c (1,α) c (2,βγ)(1,β) c (2,βγ)(2,γ). This element of C α C β C γ is written as c (1,α) c (2,β c (3,γ). For any c C α1 α n, by iterating the procedure we define inductively c (1,α1) c (n,αn).

ORE EXTENSIONS OF HOPF GROUP COALGEBRAS 327 Definition 2.2. Let C = ({C α,,ε}) be a π-coalgebra and A be an algebra with multiplication m and unit element 1 A. For any f Hom k (C α,a) and g Hom k (C β,a), we define their convolution product by f g = m(f g) α,β Hom k (C αβ,a). Using (2.1), one verifies that the k-space Conv(C,A) = α π Hom k (C α,a), endowedwiththeconvolutionproduct andtheunitelementε1 A,isaπ-graded algebra, called convolution algebra. Definition 2.3. A Hopf π-coalgebra is a π-coalgebra H = ({H α,,ε}) endowed with a family S = {S α : H α H α 1} α π of k-linear maps (the antipode) such that each H α is an algebra with multiplication m α and 1 α H α ; (b) ε : H 1 k and α,β : H αβ H α H β (for all α,β π) are algebra homomorphisms; (c) for any α π, m α (S α 1 id Hα ) α 1,α = ε1 α = m α (id Hα S α 1) α,α 1 : H 1 H α. Note that (H 1,m 1,1 1, 1,1,ε,S 1 ) is a Hopf algebra in the usual sense of the word. We call it the neutral component of H. And axiom (c) says that S α is the inverse of id Hα in the convolution algebra Conv(H 1,H α ). Lemma 2.4. Let H = ({H α } α π,,ε,s) be a Hopf π-coalgebra. Then β 1,α 1S αβ = τ Hα 1,H β 1(S α S β ) α,β for any α,β π, (b) εs 1 = ε, (c) S α (ab) = S α (b)s β for any α π and a,b H α, (d) S α (1 α ) = 1 α 1 for any α π. 3. Main theorem In this section, we will define the Ore extension of a Hopf group coalgebra and prove the criterion for an Ore extension of a Hopf group coalgebra to be a Hopf group coalgebra. Definition 3.1. Let A be a k-algebra. Consider an endomorphism σ of the algebra A over k and a σ-derivation δ of A. This means that (3.1) δ(ab) = σδ(b) + δb. The Ore extension R = A[y;σ,δ] of the k-algebra R generated by the variable y and the algebra A with the relation (3.2) ya = σy +δ for any a A. Now with the definition of Hopf-Ore extension [6], we can define the Hopf group coalgebra Ore extension.

328 DINGGUO WANG AND DAOWEI LU Definition 3.2. Let A = {A α } α π be a Hopf group coalgebra, the family R = {R α = A α [y α ;σ α,δ α ]} α π of k-spaces is called the Hopf Group coalgebra Ore extension if R = {R α } α π is also a Hopf group coalgebra, where for any α π, A α [y α ;σ α,δ α ] is the Ore extension of A α, and there exist r 1 α,r 2 α A α such that α,β (y αβ ) = y α r 2 β +r1 α y β, and β,α (y βα ) = y β r 2 α +r1 β y α. Note that R 1 = A 1 [y 1 ;σ 1,δ 1 ] is the Hopf-Ore extension in the sense of [6]. Now that R = {R α } α π is a Hopf group coalgebra, if we apply the axioms of (2.1) to Definition 3.2, we have on one hand ( α,β id Rγ ) αβ,γ (y αβγ ) = ( α,β id Rγ )(y αβ r 2 γ +r1 αβ y γ) = α,β (y αβ ) r 2 γ + α,β(r 1 αβ ) y γ = (y α r 2 β +r1 α y β) r 2 γ + α,β(r 1 αβ ) y γ = y α r 2 β r2 γ +r1 α y β r 2 γ + α,β(r 1 αβ ) y γ. On the other hand (id Rα β,γ ) α,βγ (y αβγ ) = (id Rα β,γ )(y α r 2 βγ +r1 α y βγ) We obtain = y α β,γ (r 2 βγ )+r1 α (y β r 2 γ +r1 β y γ) = y α β,γ (r 2 βγ )+r1 α y β r 2 γ +r1 α r1 β y γ. (3.3) α,β (r 1 αβ ) = r1 α r1 β, α,β(r 2 αβ ) = r2 α r2 β for any α,β,γ π. Lemma 3.3. If R = {R α } α π is the Hopf group coalgebra Ore extension of the Hopf group coalgebra A = {A α } α π as above, then for any α,β π r 1 α and r 2 α are invertible in A α, (b) the equalities: (3.4) α,β ((r 1 αβ ) 1 ) = (r 1 α ) 1 (r 1 β ) 1, and (3.5) α,β ((r 2 αβ) 1 ) = (r 2 α) 1 (r 2 β) 1. Proof. (1) First of all, from (3.3) we can see that r 1 1 is a group-like element in A 1, so ε(r 1 1 ) = 1. m α (S α 1 id) α 1,α (r1 1 ) = m α(s α 1 id)(r 1 α 1 r1 α ) = S α 1(r 1 α 1)r1 α = ε(r 1 1 )1 α = 1 α.

ORE EXTENSIONS OF HOPF GROUP COALGEBRAS 329 So we have S α 1(r 1 α 1 ) = (r 1 α) 1, and similarly we get S α 1(r 2 α 1 ) = (r 2 α) 1 for any α π. (2) By Lemma 2.4, We have α,β ((r 1 αβ) 1 ) = α,β (S (αβ) 1(r 1 (αβ) 1)) = α,β (S β 1 α 1(r1 β 1 α 1)) = τ(s β 1 S α 1)( β 1 α 1(r1 β 1 α 1)) = τ(s β 1(r 1 β 1) S α 1(r1 α 1)) = (r 1 α ) 1 (r 1 β ) 1. Similarly we can prove the other equality. Replacing the generating elements y α by y α = y α(r 2 α ) 1 and r 1 α (r2 α ) 1 by r α, we see that And (3.6) α,β (y αβ) = α,β (y αβ (rαβ) 2 1 ) = α,β (y αβ ) α,β ((rαβ) 2 1 ) = (y α rβ 2 +rα 1 y β ) ((rα) 2 1 (rβ) 2 1 ) = y α (rα) 2 1 1+rα(r 1 α) 2 1 y β (rβ) 2 1 = y α 1+r α y β. α,β (r αβ ) = α,β (rαβ 1 (r2 αβ ) 1 ) = α,β (rαβ 1 ) α,β((rαβ 2 ) 1 ) = (rα 1 r1 β ) ((r2 α ) 1 (rβ 2 ) 1 ) = r 1 α (r2 α ) 1 r 1 β (r2 β ) 1 = r α r β. Preserving the above notations, we assume in what follows that the elements {y α } α π in the Hopf group coalgebra-ore extension satisfying the relations (3.7) α,β (y αβ ) = y α 1+r α y β for some elements r α A α satisfying α,β (r αβ ) = r α r β. As usual, Ad rα = r α as α 1(r 1 α ) = r α a(r α ) 1. Lemma 3.4. If R = {A α [y α ;σ α,δ α ]} α π is a Hopf group coalgebra-ore extension of the Hopf group coalgebra A = {A α } α π, then (3.8) S α 1(y α 1) = (r α ) 1 y α, where (r α ) 1 = S α 1(r α 1). Proof. (r α ) 1 = S α 1(r α 1) is easy to check. Now we have m α (S α 1 id Aα ) α 1,α(y 1 ) = ε(y 1 )1 α = 0,

330 DINGGUO WANG AND DAOWEI LU by (3.7), so then S α 1(y α 1)+S α 1(r α 1)y α = 0, S α 1(y α 1) = S α 1(r α 1)y α = (r α ) 1 y α. Now we will give the main result. Theorem 3.5. The Ore extension R = {A α [y α ;σ α,δ α ]} α π of the Hopf group coalgebra A = {A α } α π is a Hopf group coalgebra-ore extension if and only if there exists a group-like element r = {r α } α π such that the following conditions hold: there is a character χ : A 1 k such that for any α π (3.9) σ α = χ(a (1,1) )a (2,α), where a A α ; (b) the following relation holds: (3.10) χ(a(1,1) )a (2,α) = Ad rα (a (1,α) )χ(a (2,1) ), (c) the σ α -derivation δ satisfies the relation (3.11) α,β δ αβ = δ α (a (1,α) ) a (2,β) +r α a (1,α) δ β (a (2,β) ). Proof. The proofis presented under three headings. At step 1 weshow that the comultiplication = { α } α π can be extended to R = {A α [y α ;σ α,δ α ]} α π by (3.7) if and only if relations (3.9)-(3.11) hold. At step 2 we prove that R 1 admits an extension of the counit from A 1 (in fact this has been proved in [6]). At step 3 we show that R has antipode S extending the antipode S A by (3.8). Step 1. Comultiplication. Assume that the comultiplication A can be extended to R = {A α [y α ;σ α,δ α ]} α π by (3.7). Then the homomorphism preserve the relation (3.12) y α a = σ α y α +δ α for any α π and a A α, i.e., (3.13) α,β (y αβ ) α,β = α,β σ αβ α,β (y αβ )+ α,β δ αβ for any a A αβ. We have α,β (y αβ ) α,β (y α 1+r α y β )(a (1,α) a (2,β) ) y α a (1,α) a (2,β) +r α a (1,α) y β a (2,β) σ α (a (1,α) )y α a (2,β) +δ α (a (1,α) ) a (2,β) +r α a (1,α) σ β (a (2,β) )y β +r α a (1,α) δ β (a (2,β) )

ORE EXTENSIONS OF HOPF GROUP COALGEBRAS 331 (σ α (a (1,α) ) a (2,β) )(y α 1)+r α a (1,α) σ β (a (2,β) )y β +δ α (a (1,α) ) a (2,β) +r α a (1,α) δ β (a (2,β) ) (σ α (a (1,α) ) a (2,β) )(y α 1)+(r α a (1,α) r 1 α σ β (a (2,β) )(r α y β ) +δ α (a (1,α) ) a (2,β) +r α a (1,α) δ β (a (2,β) ). And α,β σ αβ α,β (y αβ )+ α,β δ αβ = α,β σ αβ (y α 1+r α y β )+ α,β δ αβ = α,β σ αβ (y α 1)+ α,β σ αβ (r α y β )+ α,β δ αβ. It is clear that preserves (3.12) if and only if the following relations hold: (3.14) α,β σ αβ (σ α (a (1,α) ) a (2,β) ), (3.15) α,β σ αβ Ad rα (a (1,α) ) σ β (a (2,β) ), α,β δ αβ δ α (a (1,α) ) a (2,β) +r α a (1,α) δ β (a (2,β) ) for any a A αβ. The last equation coincides with (3.11). Let us show that (3.14) and (3.15) imply (3.10) and (3.11). Define a family of maps {χ α : A 1 A α } α π by χ α σ α (a (1,α) )S α 1(a (2,α 1 )) for any a A 1. Obvious compulation gives α,1 (χ α ) α,1 (σ α (a (1,α) )S α 1(a (2,α 1 ))) α,1 (σ α (a (1,α) )) α,1 S α 1(a (2,α 1 )) [σ α (a (1,α)(1,α) ) a (1,α)(2,1) ][S α 1(a (2,α 1 )(2,α 1 )) S 1 (a (2,α 1 )(1,1))] σ α (a (1,α)(1,α) )S α 1(a (2,α 1 )(2,α 1 )) a (1,α)(2,1) S 1 (a (2,α 1 )(1,1)) σ α (a (1,α) )S α 1(a (4,α 1 )) a (2,1) S 1 (a (3,1) )

332 DINGGUO WANG AND DAOWEI LU = χ α 1, and 1,α (χ α ) = 1,α (σ α (a (1,α) ) 1,α S α 1(a (2,α 1 )) [σ 1 (a (1,α)(1,1) ) a (1,α)(2,α) ][τ(s α 1 S 1 ) α 1,1(a (2,α 1 ))] [σ 1 (a (1,α)(1,1) ) a (1,α)(2,α) ][S 1 (a (2,α 1 )(2,1)) S α 1(a (2,α 1 )(1,α 1 ))] σ 1 (a (1,α)(1,1) )S 1 (a (2,α 1 )(2,1)) a (1,α)(2,α) S α 1(a (2,α 1 )(1,α 1 )) σ 1 (a (1,1) )S 1 (a (4,1) ) a (2,α) S α 1(a (3,α 1 )) = σ 1 (a (1,1) )S 1 (a (2,1) ) 1 α = χ 1 1 α, so we have χ α = (ε id) 1,α (χ α ) = ε(σ 1 (a (1,1) )S 1 (a (2,1) ))1 α ε(σ 1 (a (1,1) )ε(s 1 (a (2,1) ))1 α = ε(σ 1 )1 α. ε(σ 1 (a (1,1) )ε(a (2,1) )1 α Also we have for any α π, χ α = ε(χ 1 )1 α. Indeed from Theorem 1.3 of [6], we know that for any a A 1, χ 1 belongs to k, thus χ α could be identified with an element in k as well. One can regard χ α as a mapping χ : A 1 k. Since for any α π, σ α is an endomorphism, it follows that χ α (ab) = (b) = (b) σ α (a (1,α) b (1,α) )S α 1(a (2,α 1 )b (2,α 1 )) σ α (a (1,α) )σ α (b (1,α) )S α 1(b (2,α 1 ))S α 1(a (2,α 1 )) σ α (a (1,α) )χ α (b)s α 1(a (2,α 1 )) = χ α χ α (b),

χ α (a+b) = ORE EXTENSIONS OF HOPF GROUP COALGEBRAS 333 (b) = (b) σ α ((a+b) (1,α) )S α 1((a+b) (2,α 1 )) σ α (a (1,α) )S α 1(a (2,α 1 ))+σ α (b (1,α) )S α 1(b (2,α 1 )) = χ α +χ α (b). One can recoverσ α from χ α. In fact for anya A α, 1,α a (1,1) a (2,α) and χ α (a (1,1) )a (2,α) = σ α (a (1,1)(1,α) )S α 1(a (1,1)(2,α 1 ))a (2,α) = σ α. σ α (a (1,α) )S α 1(a (2,α 1 ))a (3,α) This proves (3.9). Substituting σ α into (3.15), we obtain Ad r1 (a (1,1) ) σ α (a (2,α) ) 1,α (χ α (a (1,1) )a (2,α) ) Ad r1 (a (1,1) ) χ α (a (2,α)(1,1) )a (2,α)(2,α) Ad r1 (a (1,1) ) χ α (a (2,1) )a (3,α). And because for any a A 1, χ α k1, we have 1,α (χ α (a (1,1) )a (2,α) ) = χ α (a (1,1) ) 1,α (a (2,α) ) = χ α (a (1,1) )a (2,α)(1,1) a (2,α)(2,α) χ α (a (1,1) )a (2,1) a (3,α), α,1 (χ α (a (1,1) )a (2,α) ) = Ad rα (a (1,α) ) σ 1 (a (2,1) ) = Ad rα (a (1,α) ) χ 1 (a (2,1)(1,1) )a (2,1)(2,1) Ad rα (a (1,α) ) χ 1 (a (2,1) )a (3,1),

334 DINGGUO WANG AND DAOWEI LU and α,1 (χ α (a (1,1) )a (2,α) ) χ α (a (1,1) α,1 (a (2,α) ) χ α (a (1,1) a (2,α)(1,α) a (2,α)(2,1) χ α (a (1,1) )a (2,α) a (3,1), that is So we have Ad rα (a (1,α) ) χ 1 (a (2,1) )a (3,1) χ α (a (1,1) )a (2,α) a (3,1). χ α (a (1,1) )a (2,α) a (3,1) S 1 (a (4,1) ), Ad rα (a (1,α) )χ 1 (a (2,1) )a (3,1) S 1 (a (4,1) ) = χ α (a (1,1) )a (2,α). Ad rα (a (1,α) )χ 1 (a (2,1) ) This proves (3.10). We have proved that conditions (3.9)-(3.11) are necessary conditions of the comultiplication. On the other hand, if conditions (3.9)-(3.11) hold, then for any a A αβ α,β (σ αβ ) χ(a (1,1) ) α,β (a (2,αβ) ) χ(a (1,1) )a (2,αβ)(1,α) a (2,αβ)(2,β) χ(a (1,1) )a (2,α) a (3,β) σ α (a (1,α) ) a (2,β) and α,β (σ αβ ) α,β (Ad rαβ (a (1,αβ) )χ(a (2,1) )) Ad rα (a (1,αβ)(1,α) ) Ad rβ (a (1,αβ)(2,β) )χ(a (2,1) ) Ad rα (a (1,α) ) Ad rβ (a (2,β) )χ(a (3,1) ) Ad rα (a (1,α) ) σ β (a (2,β) ).

ORE EXTENSIONS OF HOPF GROUP COALGEBRAS 335 This proves the relations (3.14) and (3.15) hold and the comultiplication A can be extended to a homomorphism : R R R. Since ( α,β id Aγ ) αβ,γ = (id Aα β,γ ) α,βγ and since ( α,β id) αβ,γ (y αβγ ) = (id β,γ ) α,βγ (y αβγ ) for any α,β,γ π and a A αβγ. The mapping : R R R is a comultiplication. Step 2. Counit. For this part, from [6] we have known that, as R 1 admits a comultiplication, there exists a counit extending ε A1 and satisfying ε(y 1 )=0. It follows that ε admits an extension to R if and only if ε(δ 1 ) = 0, for any a A 1. Step 3. Antipode. Let R be as in Step 1. Recall that S = {S α : A α A α 1} α π with S α being an antiautomorphism and β 1,α 1S αβ = τ Hα 1,H β 1(S α S β ) α,β for any α,β π. If R admits an antipode S which can be extended(as an antiautomorphism) from A to R by means of (3.8), then S preserves (3.12). This means that for any a A α (3.16) S α S α (y α ) = S α (y α )S α σ α +S α δ α. On the other hand, if relation (3.16) holds, then S can be extended as an antiautomorphism from A to R by means of (3.8). Using the expression b = ci y i α of the arbitrary element b R α, one can readily see that the mappings S = {S α: Rα R α 1} defines above is an antipode of R. Hence the existence of an antipode of R satisfying (3.8) is equivalent to (3.16). It follows from (3.8) that for any a A α S α (r α 1) 1 y α 1 = (r α 1) 1 y α 1S α (σ α )+S α δ α, S α (r α 1) 1 y α 1 = (r α 1) 1 σ α 1(S α (σ α ))y α 1 (r α 1) 1 δ α 1(S α (σ α ))+S α δ α. Condition (3.16) holds if and only if the following two conditions hold: (3.17) S α (r α 1) 1 = (r α 1) 1 σ α 1(S α (σ α )), (3.18) (r α 1)S α δ α = δ α 1(S α (σ α )). Let us prove (3.17). We have σ α 1(S α (σ α )) σ α 1(S α (χ(a (1,1) )a (2,α) )) χ(a (1,1) )σ α 1(S α (a (2,α) )) χ(a (1,1) )Ad (rα 1) 1((S α(a (2,α) )) (1,α 1 ))χ((s α (a (2,α) )) (2,1) )

336 DINGGUO WANG AND DAOWEI LU χ(a (1,1) )Ad (rα 1) 1(S α(a (2,α)(2,α) )χ((s 1 (a (2,α)(1,1) )) χ(a (1,1) )Ad (rα 1) 1(S α(a (3,α) )χ((s 1 (a (2,1) )) χ(ε(a (1,1) )1)Ad (rα 1) 1(S α(a (2,α) ) = Ad (rα 1) 1(S α). Our next objective is to prove (3.18). It follows from (3.9) that we present (3.18) in an equivalent form, (3.19) (r α 1)S α δ α χ(a (1,1) )δ α 1(S α (a (2,α) )). We denote L α = (r α 1)S α δ α and M α = (1,1) )δ α 1(S α (a (2,α) )). χ(a From (3.11) we have α,α 1(δ 1 ) = δ α (a (1,α) ) a (2,α 1 ) +r α a (1,α) δ α 1(a (2,α 1 )), and we apply m(id S α 1) to the above equality, we get m(id S α 1)( α,α 1(δ 1 )) = m(id S α 1)( δ α (a (1,α) ) a (2,α 1 ) +r α a (1,α) δ α 1(a (2,α 1 ))), 0 = ε(δ 1 ))1 α δ α (a (1,α) )S α 1(a (2,α 1 ))+r α a (1,α) S α 1(δ α 1(a (2,α 1 ))), (r α ) 1 δ α (a (1,α) )S α 1(a (2,α 1 )) a (1,α) S α 1(δ α 1(a (2,α 1 ))). Then for any a A α 1 L α 1 = r α S α 1δ α 1 r α ε(a (1,1) )S α 1δ α 1(a (2,α 1 )) r α S α 1(a (1,α 1 ))a (2,α) S α 1δ α 1(a (3,α 1 )) (3.20) r α S α 1(a (1,α 1 ))(r α ) 1 δ α (a (2,α) )S α 1(a (3,α 1 )) Ad rα (S α 1(a (1,α 1 )))δ α (a (2,α) )S α 1(a (3,α 1 )).

ORE EXTENSIONS OF HOPF GROUP COALGEBRAS 337 Onthe otherhand, foranya A 1, we haveε1 α a (1,α)S α 1(a (2,α 1 )). The action by δ α on both sides gives (3.21) M α 1 0 δ α (a (1,α) )S α 1(a (2,α 1 ))+σ α (a (1,α) )δ α S α 1(a (2,α 1 )), χ(a (1,1) )δ α (S α 1(a (2,α 1 ))) χ(a (1,1) )ε(a (2,1) )δ α (S α 1(a (3,α 1 ))) χ(a (1,1) )σ α (S α 1(a (2,α 1 ))a (3,α) )δ α (S α 1(a (4,α 1 ))) χ(a (1,1) )σ α (S α 1(a (2,α 1 )))σ α (a (3,α) )δ α (S α 1(a (4,α 1 ))) = χ(a (1,1) )σ α (S α 1(a (2,α 1 )))δ α (a (3,α) )S α 1(a (4,α 1 )) = χ(a (1,1) )Ad rα (S α 1(a (2,α 1 )) (1,α) )χ(s α 1(a (2,α 1 )) (2,1) )δ α (a (3,α) )S α 1(a (4,α 1 )) = χ(a (1,1) )Ad rα (S α 1(a (3,α 1 )))χ(s 1 (a (2,1) ))δ α (a (4,α) )S α 1(a (5,α 1 )) = χ(ε(a (1,1) )1)Ad rα (S α 1(a (2,α 1 )))δ α (a (3,α) )S α 1(a (4,α 1 )) = Ad rα (S α 1(a (1,α 1 )))δ α (a (2,α) )S α 1(a (3,α 1 )). Comparing (3.20) and (3.21) we conclude that L α = M α. This proves both relation (3.19) and the existence of an antipode. Here we give an example from [12], and ours is a little different, where the condition crossing is not necessarily needed. Example 3.6. For n Z and α = (α ij ), β = (β ij ) GL n (k), let B n (α,β) be the algebra generalized by symbols g, x 1,...,x n satisfying the following relations: for i {1,2,...,n}, g 2 = 1, x 2 i = 0, gx i = x i g, x i x j = x j x i. ThefamilyofalgebrasD n ={B n (α,β) } (α,β) ς(gln(k)) hasastructureofς(gl n (k))- coalgebra given, for any α = (α ij ),β = (β ij ),λ = (λ ij ),γ = (γ ij ) ς(gl n (k)), and i i n, by (α,β),(λ,γ) (g) = g g, n n (α,β),(λ,γ) (x i ) = γ k,i x k 1+1 γ k,i α i,p γ p,k x k, k=1 k=1,i=1,p=1

338 DINGGUO WANG AND DAOWEI LU ε(g) = 1, ε(x i ) = 0, S (α,β) (g) = g, S (α,β) (x i ) = n k=1,p=1 β k,j α j,i gx k, where ( α i,j ) = α 1 for any α GL n (k). Now we will add n indeterminates y 1,...,y n by Ore extensions. Firstly, define σ 1,δ 1 : B n (α,β) B n (α,β) by and σ 1 (g) = g, σ 1 (x i ) = x i δ 1 (g) = 0, δ 1 (x i ) = (α 1j β 1j )g. It is easy to check that σ 1 is an endomorphism of B n (α,β) and δ 1 is a σ 1 - derivation. Thus we get the Ore extension B (α,β) n,1 = B n (α,β) [y 1 ;σ 1,δ 1 ]. Define (α,β),(λ,γ) (y 1 ) = y 1 1+g y 1, then B (α,β) n,1 [y 1 ;σ 1,δ 1 ] is the Hopf group coalgebra Ore extension of B n. Then we define σ 2,δ 2 : B (α,β) n,1 B (α,β) n,1 by and σ 2 (g) = g, σ 2 (x i ) = x i, σ 2 (y 1 ) = y 1 δ 2 (g) = 0, δ 2 (x i ) = (α 2j β 2j )g, δ 2 (y 1 ) = 0. It is easy to check that σ 1 is an endomorphism of B (α,β) n,1 and δ 2 is a σ 2 - derivation. Thus we get the Ore extension B (α,β) n,2 = B (α,β) n,1 [y 1 ;σ 1,δ 1 ]. When we define (α,β),(λ,γ) (y 2 ) = y 2 1+g y 2, B (α,β) n,2 is also an Hopf group coalgebra Ore extension of B n,1. We continue the process by n times, then we will add n indeterminates and get the Hopf group coalgebra A (α,β) n as in the example in [12]. 4. Isomophism In this section, we study the relations of two Hopf group coalgebra Ore extensions. First we need to give the following lemma. Lemma 4.1 ([1, Lemma 1.1]). Let A be a algebra, A[y;σ,δ] an Ore extension of A and i : A A[y;σ,δ] the inclusion morphism. Then for any algebra B, any algebra morphism f : A B and every element b B such that bf =

ORE EXTENSIONS OF HOPF GROUP COALGEBRAS 339 f(δ) + f(σ)b for any a A, there exists a unique algebra morphism f : A[y;σ,δ] B such that f(y) = b and the following diagram is commutative: A B i f A[y;σ,δ] f Similarly, we can generalized this lemma to the cases of π-graded algebras: Lemma4.2. LetA={A α } α π be aπ-graded algebra, R={R α =A α [y α ;σα,δ α ]} the Ore extension of A, and i : A R the inclusion morphism, where i = {i α : A α A α [y α ;σα,δ α ]}. For any π-graded algebra B = {B α } α π, f : A B algebra morphism, where f = {f α : A α B α } α π and f α is an algebra morphism. Then for any element b = {b α } α π B, if for any α π, b α f α (a α ) = f α (δ α (a α ))+f(σ α (a α ))b α for any a = {a α } A, then there exists a unique algebra morphism f = { f α } : R B such that f α (y α ) = b α and the following diagram is commutative: f α A α B i α f α A α [y α ;σ α,δ α ] Proof. The proof is simple. By Lemma 4.1 we can get the result directly. Definition 4.3. Let A = {A α } and A = {A α} be two Hopf group coalgebras, we call f : A A, where f = {f α } and f α : A α A α, the morphism of Hopf group coalgebra if for any α π, f α is algebra morphism, α,β f αβ = (f α f β ) α,β ands α f α = f α 1S α, where,, S, S arethecomultiplications and antipodes of A and A respectively. Obviously we can see that this definition generalizes the notion of morphisms of Hopf algebra, and when π = 1, it is the usual Hopf algebra morphism. In order to simplify the notation and study the isomorphism of Hopf group coalgebra-ore extensions, we introduce the following definitions. Definition 4.4. Let A = {A α } be a Hopf group coalgebra. Denote r = {r α }, σ = {σ α }. A family of mappings δ = {δ α } satisfying (3.11) is called a r-coderivation. If δ is also a σ-derivation where σ is an algebra morphism satisfying (3.9) and (3.10), then δ is called a χ, r -derivation. Notation. Denote the Hopf group coalgebra-ore extension R = {R α } α π by R = {R α = A α (χ,r α,δ α )} α π, where χ : A 1 k is a character, r = {r α } α π is a family of group-like element of A and δ is a χ,r -derivation. Now we define an isomorphism of Hopf group coalgebra-ore extensions.

340 DINGGUO WANG AND DAOWEI LU Definition 4.5. Two Hopf group coalgebra-ore extensions R = {R α = A α (χ,r α,δ α )} α π and R = {R α = A α (χ,r α,δ α )} α π of Hopf group coalgebras A and A are said to be isomorphism if there is an isomorphism of Hopf group coalgebras φ : R R such that φ(a) = A. Remark. Actually the isomorphism of Hopf group coalgebras φ is a family of isomorphism of algebras {φ α : A α A α} α π and satisfies the conditions in Definition 4.3. Definition 4.6. A χ,r -derivation δ = {δ α } is inner, where r = {r α } α π if there is a family of elements {d α } α π A such that for all a A α, δ α = σ α d α d α a and α,β (d αβ ) = d α 1+r α d β. Lemma 4.7. If δ is a χ,r -derivation, then we have ε(d 1 ) = 0, and S α (d α ) = (r α 1) 1 d α 1. Proof. ByDefinition 4.6, 1,1 (d 1 ) = d 1 1+r 1 d 1, then wehaved 1 = ε(d 1 )1+ ε(r 1 )d 1. So by ε(r 1 ) = 1 we get ε(d 1 ) = 0. And α,α 1(d 1 ) = d α 1+r α d α 1, so 0 = ε(d 1 )1 α 1 = S α (d α ) +S α (r α )d α 1 = S α (d α ) + (r α 1) 1 d α 1, then we get S α (d α ) = (r α 1) 1 d α 1 for any α π. Using the above lemma, we prove the following consequence for an inner χ, r -derivation. Proposition 4.8. Let R = {R α =A α (χ,r α,δ α )} α π be a Hopf group coalgebra- Ore extension of A. If δ = {δ α } is an inner χ,r -derivation, then R = {R α = A α (χ,r α,δ α )} α π is isomorphism to the Hopf group coalgebra-ore extension R = {R α = A α (χ,r α,0)} α π. Proof. Denote the indeterminates of {A α (χ,r α,δ α )} α π and {A α (χ,r α,0)} α π by y = {y α } α π and y = {y α} α π. Firstly we uniquely extend the inclusion morphism i = {i α } : A {A α (χ,r α,0)} α π to the algebra morphism {A α (χ,r α,δ α )} α π {A α (χ,r α,0)} α π by extending i α : A α A α (χ,r α,0)} for any α π which is denoted by ī and ī α such that the following diagram is commutative: i α A α A α (χ,r α,0) ī α A α (χ,r α,δ α )

ORE EXTENSIONS OF HOPF GROUP COALGEBRAS 341 To see this, take {y α d α } {A α (χ,r α,0)} α π and define ī α (y α ) = y α d α. Because δ = {δ α } is inner, we have for all a A α i α σ α ī α (y α )+i α δ α = σ α (y α d α )+δ α = σ α y α σ α d α +δ α = σ α y α σ αd α +σ α d α d α a = σ α y α d αa = (y α d α)a = (y α d α )i α. Then by Lemma 4.2, we complete the extension. Similarly,theinclusionmorphismj : A {A α (χ,r α,δ α )} α π canbeuniquely extended to the algebramorphism j : {A α (χ,r α,0)} α π {A α (χ,r α,δ α )} α π. By the uniqueness of extension, we get that ī is an algebra morphism and j is its inverse. It is easy to verify that ī α (A α (χ,r α,δ α )) A α (χ,r α,0), and so we have constructed an algebra isomorphism from {A α (χ,r α,δ α )} α π to {A α (χ,r α,0)} α π satisfying ī(a) = A. By the definitions of an inner χ, r -derivation and a Hopf group coalgebra- Ore extension, we have the calculation (ī α ī β ) α,β (y αβ ) = (ī α ī β )(y α 1+r α y β ) = ī(y α ) 1+r α ī β (y β ) = (y α d α) 1+r α (y β d β) = y α 1+r α y β d α 1 r α d β = y α 1+r α y β α,β d αβ = α,β (y αβ ) α,β (d αβ) = α,β (y αβ d αβ) = α,βīαβ(y αβ ). And ī α S α (y α ) = ī α ( (r α 1) 1 y α 1) = (r α 1) 1 (y α 1 d α 1) = (r α 1) 1 y α +(r 1 α 1) 1 d α 1 = (r α 1) 1 y α S α(d 1 α ) = S α(y α) S α(d α ) = S α(y α d α ) = S αīα(y α ). Thus we proves the proposition.

342 DINGGUO WANG AND DAOWEI LU Now we can introduce and prove the main result in this section. Theorem 4.9. Let R = {A α (χ,r α,δ α )} α π and R = {A α(χ,r α,δ α)} α π be two Hopf group coalgebra-ore extensions. If there exists a isomorphism of Hopf group coalgebra φ : R R such that χ = χφ 1, r = φ(r) and δ = φδφ 1 +δ, where r = {r α }, r = {r α}, δ = {δ α }, δ = {δ α} and δ is an inner χ,r -derivation of A, then R is isomorphism to R as a Hopf group coalgebra-ore extension. Proof. Denote the indeterminates of{a α (χ,r α,δ α )} α π and {A α (χ,r α,δ α )} α π by y = {y α } α π and y = {y α} α π. We will show that there exist extensions of φ and φ 1, which are denoted by φ and φ 1, such that the following diagrams are commutative: φ α A α A α A α (χ,r α,δ α ) φ α A α (χ,r α,δ α ) A φ 1 α α A α A α (χ,r α,δ α ) φ 1 α A α(χ,r α,δ α) To prove this, we first have to show σ α φ α = φ α σ α, δ α φ α = φ α δ α, σ α φ 1 = φ 1 σ α and δ αφ 1 = φ 1 δ α. In fact, by the assumption, for all a A α, we have σ αφ α χ (φ α (1,1) )φ α (2,α) χ (φ α (a (1,1) ))φ α (a (2,α)) χ(φ 1 α φ α (a (1,1) ))φ α (a (2,α)) = χ(a (1,1) )φ α (a (2,α)) φ α (χ(a (1,1) )φ α (a (2,α)) ) = φ α σ α. Similarly for δ α φ α = φ α δ α. We can prove the other two equations directly by δ α = φ α δ α φα 1. Then we have y αφ α = σ α(φ α )y α +δ α(φ α ) = φ α σ α y α +φ α δ α for all a A α and y α φα 1 (a ) = σ α (φα 1 (a ))y α +δ α (φ 1 α (a )) = (φ 1 α σ α(a ))y α +φ 1 α δ α(a ) for all a A α. So φ and φ 1 are extensions of φ and φ 1, respectively, such that φ α (y α ) = y α andφ 1 α(y α) = y α. Andbytheuniquenessoftheextensions, we obtain that φ 1 φ = id and φ φ 1 = id.

ORE EXTENSIONS OF HOPF GROUP COALGEBRAS 343 Using the assumption r = φ(r), we have ( φ α φ β ) α,β (y αβ ) = ( φ α φ β )(y α 1+r α y β ) = y α 1+φ α (r α ) y β = y α 1+r α y β = α,β (y αβ ) = α,β φ β (y αβ ). Similarly for α,β φ 1 αβ (y αβ ) = ( φ 1 α φ 1 β ) α,β (y αβ ). And Similarly for φ 1 α S α (y α ) = S αφ 1 α (y α ). Finally, it is easily see that S α φ α (y α ) = S α (y α ) = (r α 1) 1 y α 1 = φ α ( (r α 1) 1 )y α 1 = φ α ( (r α 1) 1 y α 1) = φ α S α (y α ). φ α (A α (χ,r α,δ α )) A α(χ,r α,δ α) and φ 1 αβ (A α(χ,r α,δ α)) φ α (A α (χ,r α,δ α )). So we conclude that R is isomorphism to R as a Hopf group coalgebra-ore extension. Acknowledgments. This work was supported by the National Natural Science Foundation of China (No. 11261063 and 11171183), the Shandong Provincial Natural Science Foundation of China (No. ZR2011AM013). References [1] M. Beattie, S. Dascalescu, and L. Grunenfelder, Constructing pointed Hopf algebras by Ore extensions, J. Algebra 225 (2000), no. 2, 743 770. [2] M. Beattie, S. Dascalescu, L. Grunenfelder, and C. Nastasescu, Fitness conditions, co- Frobenius Hopf algebras and quantum group, J. Algebra 200 (1998), no. 1, 312 333. [3] A. S. Hegazi, W. Morsi, and M. Mansour, Differential calculus on Hopf group coalgebras, Note Mat. 29 (2009), no. 1, 1 40. [4] S. Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, 82, Providence, RI, 1993. [5] A. Nenciu, Quasitriangular structures for a class of pointed Hopf algebras constructed by Ore extensions, Comm. Algebra 29 (2001), no. 8, 3419 3432. [6] A. N. Panov, Ore extensions of Hopf algebras, Math. Notes 74 (2003), no. 3-4, 401 410. [7] M. E. Sweedler, Hopf Algebras, Math. Lecture Notes Ser., Benjamin, New York, 1969. [8] V. G. Turaev, Homotopy field theory in dimension 3 and crossed group-categories, Preprint GT/0005291,(2000). [9], Crossed group-categories, Arab. J. Sci. Eng. Sect. C Theme Issues 33 (2008), no. 2, 483 503.

344 DINGGUO WANG AND DAOWEI LU [10] A. Van Daele, Multiplier Hopf algebras, Trans. Amer. Math. Soc. 342 (1994), no. 2, 917 932. [11] A. Virelizier, Hopf group coalgebras, J. Pure Appl. Algebra 171 (2002), no. 1, 75 122. [12] S. H. Wang, Turaev group coalgebras and twisted Drinfeld double, Indiana Univ. Math. J. 58 (2009), no. 3, 1395 1417. [13] L. Zhao and D. Lu, Ore Extension of Multiplier Hopf algebras, Comm. Algebra 40 (2012), no. 1, 248 272. [14] M. Zunino, Double construction for crossed Hopf coalgebra, J. Algebra. 278 (2004), no. 1, 43 75. [15], Yetter-Drinfeld modules for crossed structures, J. Pure Appl. Algebra 193 (2004), no. 1-3, 313 343. Dingguo Wang School of Mathematical Sciences Qufu Normal University Qufu, Shandong 273165, P. R. China E-mail address: dingguo95@126.com, dgwang@mail.qfnu.edu.cn Daowei Lu School of Mathematical Sciences Qufu Normal University Qufu, Shandong 273165, P. R. China E-mail address: ludaowei620@sina.com