aa r X i v : . [ m a t h . R A ] A p r EXTENDING STRUCTURES II: THE QUANTUM VERSION
A. L. AGORE AND G. MILITARU
Abstract.
Let A be a Hopf algebra and H a coalgebra. We shall describe and classifyup to an isomorphism all Hopf algebras E that factorize through A and H : that is E is a Hopf algebra such that A is a Hopf subalgebra of E , H is a subcoalgebra in E with 1 E ∈ H and the multiplication map A ⊗ H → E is bijective. The tool we use is anew product, we call it the unified product, in the construction of which A and H areconnected by three coalgebra maps: two actions and a generalized cocycle. Both thecrossed product of an Hopf algebra acting on an algebra and the bicrossed product oftwo Hopf algebras are special cases of the unified product. A Hopf algebra E factorizesthrough A and H if and only if E is isomorphic to a unified product of A and H . Allsuch Hopf algebras E are classified up to an isomorphism that stabilizes A and H by aSchreier type classification theorem. A coalgebra version of lazy 1-cocycles as definedby Bichon and Kassel plays the key role in the classification theorem. Introduction
Let C be a category whose objects are sets endowed with various algebraic structures( S ) and D be a category such that there exists a forgetful functor F : C → D , i.e. afunctor that forgets some of the structures ( S ). To illustrate, the following are forgetfulfunctors: F : G r → S et, F : L ie → V ec, F : H opf → C oAlg, F : H opf → A lg where G r , S et , L ie , V ec , H opf , C oAlg , A lg are the categories of all groups, sets, Liealgebras, vector spaces, Hopf algebras, coalgebras and respectively algebras. In thiscontext we formulate a general problem which may be of interest for many areas ofmathematics: Extending Structures Problem (ES):
Let F : C → D be a forgetful functor andconsider two objects C ∈ C , D ∈ D such that F ( C ) is a subobject of D in D . Describeand classify all mathematical structures ( S ) that can be defined on D such that D becomesan object of C and C is a subobject of D in the category C (the classification is up to anisomorphism that stabilizes C and a certain type of fixed quotient D/C ). The classification part of the ES-problem is a challenge for the introduction of new typesof cohomology. The ES-problem generalizes and unifies two famous and still open prob-lems in the theory of groups: the extension problem of H¨older [7] and the factorizationproblem of Ore [12]. Let us explain this. In [1] we formulated the ES-problem at thelevel of groups, corresponding to the forgetful functor F : G r → S et : if A is a group and Mathematics Subject Classification.
Key words and phrases. crossed product, bicrossed product, the factorization problem. E a set such that A ⊆ E [1, Corollary 2.10], describe all group structures ( E, · ) thatcan be defined on the set E such that A is a subgroup of ( E, · ). In order to do thatwe have introduced a new product for groups, called the unified product ([1, Theorem2.6]), such that both the crossed product (the tool for the extension problem) and thebicrossed product (the tool for the factorization problem) of two groups are special casesof it. The unified product for groups is associated to a group A and a new hidden al-gebraic structure ( H, ∗ ), connected by two actions and a generalized cocycle satisfyingsome compatibility conditions.We now take a step forward and formulate the ES-problem at the level of Hopf algebrascorresponding to the forgetful functor F : H opf → C oAlg : (H-C) Extending Structures Problem: Let A be a Hopf algebra and E a coalgebrasuch that A is a subcoalgebra of E . Describe and classify all Hopf algebra structures thatcan be defined on E such that A is a Hopf subalgebra of E . There is of course a dual version of the ES-problem corresponding to the forgetful functor F : H opf → A lg to be addressed somewhere else. If at the level of groups the ES-problemis elementary, for Hopf algebras the problem is more difficult. Indeed, let A be a groupand E a set such that A ⊆ E . For a field k we look at the extension k [ A ] ⊆ k [ E ], where k [ A ] is the group algebra that is a Hopf algebra and a subcoalgebra in the group-likecoalgebra k [ E ]. Assume now that ( E, · ) is a group structure on the set E such that A is a subgroup of ( E, · ). Thus, we obtain an extension of Hopf algebras k [ A ] ⊆ k [ E ].This extension of Hopf algebras has a remarkable property: let H ⊆ E be a systemof representatives for the right cosets of the subgroup A in the group ( E, · ) such that1 E ∈ H . Since the map u : A × H → E , u ( a, h ) = a · h is bijective, we obtain that themultiplication map k [ A ] ⊗ k [ H ] → k [ E ] , a ⊗ h a · h is bijective, i.e. the Hopf algebra k [ E ] factorizes through the Hopf subalgebra k [ A ]and the subcoalgebra k [ H ]. This is not valid for arbitrary extensions of Hopf algebras.Therefore, we have to restrict the (H-C) extending structures problem to those Hopfalgebras E that factorize through a given Hopf subalgebra A and a given subcoalgebra H : we called this the restricted (H-C) ES-problem and we shall give a completeanswer to it in the present paper. It turns out that H is not only a subcoalgebra of E but will be endowed additionally with a hidden algebraic structure that will play therole of the system of representatives for congruence in the theory of groups.The paper is organized as follows: In the first section we recall the classical constructionsof the crossed product of a Hopf algebra H acting on an algebra A and of the bicrossedproduct (double cross product in Majid’s terminology) of two Hopf algebras H and A , asthe product that we define will generalize both of them. In Section 2 we define the conceptof an extending structure of a bialgebra A consisting of a system Ω( A ) = (cid:0) H, ⊳, ⊲, f (cid:1) ,where H is a coalgebra and an unitary not necessarily associative algebra such that A and H are connected by three coalgebra maps ⊳ : H ⊗ A → H , ⊲ : H ⊗ A → A , f : H ⊗ H → A satisfying some natural normalization conditions (Definition 2.1). For a bialgebraextending structure Ω( A ) = (cid:0) H, ⊳, ⊲, f (cid:1) of A we define a product A ⋉ Ω( A ) H = A ⋉ H and call it the unified product: both the crossed product of an Hopf algebra acting on XTENDING STRUCTURES II: THE QUANTUM VERSION 3 an algebra and the bicrossed product (double cross product in Majid’s terminology) oftwo Hopf algebras are special cases of the unified product. Theorem 2.4 gives necessaryand sufficient conditions for A ⋉ H to be a bialgebra, which is precisely the Hopf algebraversion of [1, Theorem 2.6] proven for the group case that served as a model for us. Theseven compatibility conditions in Theorem 2.4 are very natural and, mutatis-mutandis,are the ones (with two reasonable deformations via the right action ⊳ ) that appear inthe construction of the crossed product and the bicrossed product of two Hopf algebras.Theorem 2.7 proves that a Hopf algebra E factorizes through a Hopf subalgebra A anda subcoalgebra H if and only if E is isomorphic to a unified product of A and H andgives the answer for the first part of the restricted (H-C) ES-problem.Section 3 is devoted to the classification part of the restricted (H-C) ES-problem. Ourview point descends from the classical classification theorem of Schreier at the level ofgroups: all extensions of an abelian group K by a group Q are classified by the secondcohomology group H ( Q, K ) [13, Theorem 7.34]. Let A be a Hopf algebra. Two Hopfalgebra extending structures Ω( A ) = (cid:0) H, ⊳, ⊲, f (cid:1) and Ω ′ ( A ) = (cid:0) H, ⊳ ′ , ⊲ ′ , f ′ (cid:1) are calledequivalent if there exists ϕ : A ⋉ ′ H → A ⋉ H a left A -module, a right H -comodule and aHopf algebra map. As in group extension theory we shall prove that any such morphism ϕ : A ⋉ ′ H → A ⋉ H is an isomorphism and the following diagram A i A / / Id A (cid:15) (cid:15) A ⊲⊳ H π H / / ϕ (cid:15) (cid:15) H Id H (cid:15) (cid:15) A i A / / A ⊲⊳ ′ H π H / / H is commutative. Theorem 3.4 shows that any such morphism ϕ : A ⋉ ′ H → A ⋉ H is uniquely determined by a coalgebra lazy 1-cocycle: i.e. a unitary coalgebra map u : H → A such that: h (1) ⊗ u ( h (2) ) = h (2) ⊗ u ( h (1) )for all h ∈ H . Corollary 3.6 is the Schreier type classification theorem for unifiedproducts: the part of the second cohomology group from the theory of groups willbe played now by a special quotient set H l,c ( H, A, ⊳ ). Also, a classification result forbicrossed product of two Hopf algebras is derived from Theorem 3.4.1.
Preliminaries
Throughout this paper, k will be a field. Unless specified otherwise, all algebras, coalge-bras, bialgebras, tensor products and homomorphisms are over k . For a coalgebra C , weuse Sweedler’s Σ-notation: ∆( c ) = c (1) ⊗ c (2) , ( I ⊗ ∆)∆( c ) = c (1) ⊗ c (2) ⊗ c (3) , etc (summa-tion understood). Let A be a bialgebra and H a coalgebra. H is called a right A -modulecoalgebra if there exists ⊳ : H ⊗ A → H a morphism of coalgebras such that ( H, ⊳ ) isa right A -module. For a k -linear map f : H ⊗ H → A we denote f ( g, h ) = f ( g ⊗ h ); f is the trivial map if f ( g, h ) = ε H ( g ) ε H ( h )1 A , for all g , h ∈ H . Similarly, the k -linearmaps ⊳ : H ⊗ A → H , ⊲ : H ⊗ A → A are the trivial actions if h ⊳ a = ε A ( a ) h andrespectively h ⊲ a = ε H ( h ) a , for all a ∈ A and h ∈ H . For further computations, the fact A. L. AGORE AND G. MILITARU that ⊳ : H ⊗ A → H , ⊲ : H ⊗ A → A and f : H ⊗ H → A are coalgebra maps can bewritten explicitly as follows:∆ H ( h ⊳ a ) = h (1) ⊳ a (1) ⊗ h (2) ⊳ a (2) , ε A ( h ⊳ a ) = ε H ( h ) ε A ( a ) (1)∆ A ( h ⊲ a ) = h (1) ⊲ a (1) ⊗ h (2) ⊲ a (2) , ε A ( h ⊲ a ) = ε H ( h ) ε A ( a ) (2)∆ A (cid:0) f ( g, h ) (cid:1) = f ( g (1) , h (1) ) ⊗ f ( g (2) , h (2) ) , ε A (cid:0) f ( g, h ) (cid:1) = ε H ( g ) ε H ( h ) (3)for all g, h ∈ H , a ∈ A . Crossed product of Hopf algebras.
The crossed product of a Hopf algebra H actingon a k -algebra A was introduced independently in [4] and [6] as a generalization of thecrossed product of groups acting on k -algebras. Let H be a Hopf algebra, A a k -algebraand two k -linear maps ⊲ : H ⊗ A → A , f : H ⊗ H → A such that h ⊲ A = ε H ( h )1 A , H ⊲ a = ah ⊲ ( ab ) = ( h (1) ⊲ a )( h (2) ⊲ b ) , f ( h, H ) = f (1 H , h ) = ε H ( h )1 A for all h ∈ H , a , b ∈ A . The crossed product A f H of A with H is the k -module A ⊗ H with the multiplication given by( a h )( c g ) := a ( h (1) ⊲ c ) f (cid:0) h (2) , g (1) (cid:1) h (3) g (2) (4)for all a , c ∈ A , h , g ∈ H , where we denoted a ⊗ h by a h . It can be proved [11, Lemma7.1.2] that A f H is an associative algebra with identity element 1 A H if and only ifthe following two compatibility conditions hold:[ g (1) ⊲ ( h (1) ⊲ a )] f (cid:0) g (2) , h (2) (cid:1) = f ( g (1) , h (1) ) (cid:0) ( g (2) h (2) ) ⊲ a (cid:1) (5) (cid:0) g (1) ⊲ f ( h (1) , l (1) ) (cid:1) f (cid:0) g (2) , h (2) l (2) (cid:1) = f ( g (1) , h (1) ) f ( g (2) h (2) , l ) (6)for all a ∈ A , g , h , l ∈ H . The first compatibility is called the twisted module conditionwhile (6) is called the cocycle condition. The crossed product A f H was studied only asan algebra extension of A , being an essential tool in Hopf-Galois extensions theory. If, inaddition, we suppose that A is also a Hopf algebra, a natural question arises: when doesthe crossed product A f H have a Hopf algebra structure with the coalgebra structuregiven by the tensor product of coalgebras? In case that ⊲ and f are coalgebra maps, as aconsequence of Theorem 2.4, we will show in Example 2.5 that A f H is a Hopf algebraif and only if the following two compatibility conditions hold: g (1) ⊗ g (2) ⊲ a = g (2) ⊗ g (1) ⊲ ag (1) h (1) ⊗ f ( g (2) , h (2) ) = g (2) h (2) ⊗ f ( g (1) , h (1) )for all g , h ∈ H and a , b ∈ A . Bicrossed product of Hopf algebras.
The bicrossed product of Hopf algebras wasintroduced by Majid in [9, Proposition 3.12] under the name of double cross product.We shall adopt the name of bicrossed product from [8, Theorem 2.3]. A matched pair of bialgebras is a system (
A, H, ⊳ , ⊲ ), where A and H are bialgebras, ⊳ : H ⊗ A → H , XTENDING STRUCTURES II: THE QUANTUM VERSION 5 ⊲ : H ⊗ A → A are coalgebra maps such that ( A, ⊲ ) is a left H -module coalgebra, ( H, ⊳ )is a right A -module coalgebra and the following compatibility conditions hold:1 H ⊳ a = ε A ( a )1 H , h ⊲ A = ε H ( h )1 A (7) g ⊲ ( ab ) = ( g (1) ⊲ a (1) ) (cid:0) ( g (2) ⊳ a (2) ) ⊲ b (cid:1) (8)( gh ) ⊳ a = (cid:0) g ⊳ ( h (1) ⊲ a (1) ) (cid:1) ( h (2) ⊳ a (2) ) (9) g (1) ⊳ a (1) ⊗ g (2) ⊲ a (2) = g (2) ⊳ a (2) ⊗ g (1) ⊲ a (1) (10)for all a , b ∈ A , g , h ∈ H . Let ( A, H, ⊳ , ⊲ ) be a matched pair of bialgebras; the bicrossedproduct A ⊲⊳ H of A with H is the k -module A ⊗ H with the multiplication given by( a ⊲⊳ h ) · ( c ⊲⊳ g ) := a ( h (1) ⊲ c (1) ) ⊲⊳ ( h (2) ⊳ c (2) ) g (11)for all a , c ∈ A , h , g ∈ H , where we denoted a ⊗ h by a ⊲⊳ h . A ⊲⊳ H is a bialgebrawith the coalgebra structure given by the tensor product of coalgebras and moreover, if A and H are Hopf algebras, then A ⊲⊳ H has an antipode given by the formula: S ( a ⊲⊳ h ) := (1 A ⊲⊳ S H ( h )) · ( S A ( a ) ⊲⊳ H ) (12)for all a ∈ A and h ∈ H [10, Theorem 7.2.2].2. Bialgebra extending structures and unified products
In this section we shall introduce the unified product for bialgebras; this will be the toolfor answering the restricted (H-C) ES-problem. First we need the following:
Definition 2.1.
Let A be a bialgebra. An extending datum of A is a system Ω( A ) = (cid:0) H, ⊳, ⊲, f (cid:1) where:( i ) H = (cid:0) H, ∆ H , ε H , H , · (cid:1) is a k -module such that (cid:0) H, ∆ H , ε H (cid:1) is a coalgebra, (cid:0) H, H , · (cid:1) is an unitary, not necessarily associative k -algebra, such that∆ H (1 H ) = 1 H ⊗ H (13)( ii ) The k -linear maps ⊳ : H ⊗ A → H , ⊲ : H ⊗ A → A , f : H ⊗ H → A are morphismsof coalgebras such that the following normalization conditions hold: h ⊲ A = ε H ( h )1 A , H ⊲ a = a, H ⊳ a = ε A ( a )1 H , h ⊳ A = h (14) f ( h, H ) = f (1 H , h ) = ε H ( h )1 A (15)for all h ∈ H , a ∈ A .Let A be a bialgebra and Ω( A ) = (cid:0) H, ⊳, ⊲, f (cid:1) an extending datum of A . We denote by A ⋉ Ω( A ) H = A ⋉ H the k -module A ⊗ H together with the multiplication:( a ⋉ h ) • ( c ⋉ g ) := a ( h (1) ⊲ c (1) ) f (cid:0) h (2) ⊳ c (2) , g (1) (cid:1) ⋉ ( h (3) ⊳ c (3) ) · g (2) (16)for all a, c ∈ A and h, g ∈ H , where we denoted a ⊗ h ∈ A ⊗ H by a ⋉ h . A. L. AGORE AND G. MILITARU
Definition 2.2.
Let A be a bialgebra and Ω( A ) = (cid:0) H, ⊳, ⊲, f (cid:1) be an extending datumof A . The object A ⋉ H introduced above is called the unified product of A and Ω( A )if A ⋉ H is a bialgebra with the multiplication given by (16), the unit 1 A ⋉ H and thecoalgebra structure given by the tensor product of coalgebras, i.e.:∆ A ⋉ H ( a ⋉ h ) = a (1) ⋉ h (1) ⊗ a (2) ⋉ h (2) (17) ε A ⋉ H ( a ⋉ h ) = ε A ( a ) ε H ( h ) (18)for all h ∈ H , a ∈ A . In this case the extending datum Ω( A ) = ( H, ⊳, ⊲, f ) is called a bialgebra extending structure of A . The maps ⊲ and ⊳ are called the actions of Ω( A )and f is called the ( ⊲ , ⊳ )- cocycle of Ω( A ). A bialgebra extending structure Ω( A ) =( H, ⊳, ⊲, f ) is called a
Hopf algebra extending structure of A if A ⋉ H has an antipode.The multiplication given by (16) has a rather complicated formula; however, for somespecific elements we obtain easier forms which will be useful for future computations. Lemma 2.3.
Let A be a bialgebra and Ω( A ) = (cid:0) H, ⊳, ⊲, f (cid:1) an extending datum of A .The following cross-relations hold: ( a ⋉ H ) • ( c ⋉ g ) = ac ⋉ g (19)( a ⋉ g ) • (1 A ⋉ h ) = af ( g (1) , h (1) ) ⋉ g (2) · h (2) (20)( a ⋉ g ) • ( b ⋉ H ) = a ( g (1) ⊲ b (1) ) ⋉ g (2) ⊳ b (2) (21) Proof.
Straightforward using the normalization conditions (13)-(15). (cid:3)
It follows from (19) that the map i A : A → A ⋉ H , i A ( a ) := a ⋉ H , for all a ∈ A , is a k -algebra map and ( a ⋉ H ) • (1 A ⋉ g ) = a ⋉ g (22)for all a ∈ A and g ∈ H . Hence the set T := { a ⋉ H | a ∈ A } ∪ { A ⋉ g | g ∈ H } is asystem of generators as an algebra for A ⋉ H and this observation will turn out to beessential in proving the next theorem which provides necessary and sufficient conditionsfor A ⋉ H to be a bialgebra: it is the Hopf algebra version of [1, Theorem 2.6] where theunified product for groups is constructed. Theorem 2.4.
Let A be a bialgebra and Ω( A ) = (cid:0) H, ⊳, ⊲, f (cid:1) an extending datum of A .The following statements are equivalent: (1) A ⋉ H is an unified product; (2) The following compatibilities hold: (2a) ∆ H : H → H ⊗ H and ε H : H → k are k -algebra maps; (2b) ( H, ⊳ ) is a right A -module structure; (2c) ( g · h ) · l = (cid:0) g ⊳ f ( h (1) , l (1) ) (cid:1) · ( h (2) · l (2) )(2d) g ⊲ ( ab ) = ( g (1) ⊲ a (1) )[( g (2) ⊳ a (2) ) ⊲ b ] XTENDING STRUCTURES II: THE QUANTUM VERSION 7 (2e) ( g · h ) ⊳ a = [ g ⊳ ( h (1) ⊲ a (1) )] · ( h (2) ⊳ a (2) )(2f) [ g (1) ⊲ ( h (1) ⊲ a (1) )] f (cid:16) g (2) ⊳ ( h (2) ⊲ a (2) ) , h (3) ⊳ a (3) (cid:17) = f ( g (1) , h (1) )[( g (2) · h (2) ) ⊲ a ](2g) (cid:16) g (1) ⊲ f ( h (1) , l (1) ) (cid:17) f (cid:16) g (2) ⊳ f ( h (2) , l (2) ) , h (3) · l (3) (cid:17) = f ( g (1) , h (1) ) f ( g (2) · h (2) , l )(2h) g (1) ⊳ a (1) ⊗ g (2) ⊲ a (2) = g (2) ⊳ a (2) ⊗ g (1) ⊲ a (1) (2i) g (1) · h (1) ⊗ f ( g (2) , h (2) ) = g (2) · h (2) ⊗ f ( g (1) , h (1) ) for all g, h, l ∈ H and a, b ∈ A . Before going into the proof of the theorem, we have a few observations on the relations(2 a ) − (2 i ) in Theorem 2.4. Although they look rather complicated at first sight, theyare in fact quite natural and can be interpreted as follows: (2a) and (2b) show that( H, ∆ H , ε H , H , · ) is a non-associative bialgebra and a right A -module coalgebra via ⊳ .(2c) measures how far ( H, H , · ) is from being an associative algebra. (2d), (2e) and (2h)are exactly, mutatis-mutandis, the compatibility conditions (7) - (10) appearing in thedefinition of a matched pair of bialgebras. (2f) and (2g) are deformations via the action ⊳ of the twisted module condition (5) and respectively of the cocycle condition (6) whichappears in the definition of the crossed product for Hopf algebras. (2i) is a symmetrycondition for the cocycle f similar to (2 h ). Both relations are trivially fulfilled if, forexample, H is cocommutative or f is the trivial cocycle. Proof.
We prove Theorem 2.4 in several steps. From (19) and (21) it is straightforwardthat 1 A ⋉ H is a unit for the algebra ( A ⋉ H, • ). Next, we prove that ε A ⋉ H given by(18) is an algebra map if and only if ε H : H → k is an algebra map. For h, g ∈ H wehave: ε A ⋉ H (cid:0) (1 A ⋉ h ) • (1 A ⋉ g ) (cid:1) ( ) = ε A ⋉ H (cid:0) f ( h (1) , g (1) ) ⋉ g (2) · h (2) (cid:1) ( ) = ε H ( h (1) ) ε H ( g (1) ) ε H ( g (2) · h (2) )= ε H ( g · h )and ε (1 A ⋉ h ) ε (1 A ⋉ g ) = ε H ( h ) ε H ( g ). Thus, if ε A ⋉ H is a k -algebra map then ε H is a k -algebra map. Conversely, suppose that ε H is a k -algebra map. Then, we have: ε A ⋉ H (cid:0) ( a ⋉ h ) • ( c ⋉ g ) (cid:1) = ε A ( a ) ε H ( h (1) ) ε A ( c (1) ) ε H ( h (2) ) ε A ( c (2) ) ε H ( g (1) ) ε H ( h (3) ) ε A ( c (3) ) ε H ( g (2) )= ε A ( a ) ε H ( h ) ε A ( c ) ε H ( g )= ε A ⋉ H ( a ⋉ h ) ε A ⋉ H ( c ⋉ g )for all a , c ∈ A and h ∈ H i.e. ε A ⋉ H is an algebra map.The next step is to prove that ∆ A ⋉ H is a k -algebra map if and only if ∆ H : H → H ⊗ H is a k -algebra map and the relations (2 h ), (2 i ) hold. Observe that ∆ A ⋉ H (1 A ⋉ H ) ( ) =1 A ⋉ H ⊗ A ⋉ H . Since T = { a ⋉ H | a ∈ A } ∪ { A ⋉ g | g ∈ H } generates A ⋉ H as A. L. AGORE AND G. MILITARU an algebra, ∆ A ⋉ H is a k -algebra map if and only if ∆ A ⋉ H ( xy ) = ∆ A ⋉ H ( x )∆ A ⋉ H ( y ) forall x, y ∈ T . First, observe that:∆ A ⋉ H (cid:0) ( a ⋉ H ) • ( b ⋉ H ) (cid:1) = ∆ A ⋉ H ( ab ⋉ H )= a (1) b (1) ⋉ H ⊗ a (2) b (2) ⋉ H = (cid:0) a (1) ⋉ H ⊗ a (2) ⋉ H (cid:1)(cid:0) b (1) ⋉ H ⊗ b (2) ⋉ H (cid:1) = ∆ A ⋉ H ( a ⋉ H )∆ A ⋉ H ( b ⋉ H )and ∆ A ⋉ H (cid:0) ( a ⋉ H ) • (1 A ⋉ g ) (cid:1) = ∆ A ⋉ H ( a ⋉ g )= a (1) ⋉ g (1) ⊗ a (2) ⋉ g (2) = (cid:0) a (1) ⋉ H ⊗ a (2) ⋉ H (cid:1)(cid:0) A ⋉ g (1) ⊗ A ⋉ g (2) (cid:1) = ∆ A ⋉ H ( a ⋉ H )∆ A ⋉ H (1 A ⋉ g )for all a, b ∈ A , g ∈ H . There are two more relations to consider; for g, h ∈ H we have:∆ A ⋉ H (cid:0) (1 A ⋉ g ) • (1 A ⋉ h ) (cid:1) ( ) = ∆ A ⋉ H (cid:0) f ( g (1) , h (1) ) ⋉ g (2) · h (2) (cid:1) ( ) = f ( g (1)(1) , h (1)(1) ) ⋉ ( g (2) · h (2) ) (1) ⊗ f ( g (1)(2) , h (1)(2) ) ⋉ ( g (2) · h (2) ) (2) = f ( g (1) , h (1) ) ⋉ ( g (3) · h (3) ) (1) ⊗ f ( g (2) , h (2) ) ⋉ ( g (3) · h (3) ) (2) and∆ A ⋉ H (1 A ⋉ g )∆ A ⋉ H (1 A ⋉ h ) = (cid:0) A ⋉ g (1) ⊗ A ⋉ g (2) (cid:1)(cid:0) A ⋉ h (1) ⊗ A ⋉ h (2) (cid:1) ( ) = f ( g (1) , h (1) ) ⋉ g (2) · h (2) ⊗ f ( g (3) , h (3) ) ⋉ g (4) · h (4) Thus ∆ A ⋉ H (cid:0) (1 A ⋉ g ) • (1 A ⋉ h ) = ∆ A ⋉ H (1 A ⋉ g )∆ A ⋉ H (1 A ⋉ h ) if and only if f ( g (1) , h (1) ) ⋉ ( g (3) · h (3) ) (1) ⊗ f ( g (2) , h (2) ) ⋉ ( g (3) · h (3) ) (2) == f ( g (1) , h (1) ) ⋉ g (2) · h (2) ⊗ f ( g (3) , h (3) ) ⋉ g (4) · h (4) We show now that this relation holds if and only if ∆ H : H → H ⊗ H is a k -algebramap and (2 i ) holds. Indeed, suppose first that the above relation holds. By applying ε A ⊗ Id ⊗ ε A ⊗ Id to it we obtain ∆ H ( g · h ) = g (1) · h (1) ⊗ g (2) · h (2) , i.e. ∆ H is a k -algebramap. Furthermore, if we apply ε A ⊗ Id ⊗ Id ⊗ ε H to it we obtain g (1) · h (1) ⊗ f ( g (2) , h (2) ) = g (2) · h (2) ⊗ f ( g (1) , h (1) ), i.e. (2 i ). Conversely, suppose that ∆ H is a k -algebra map and(2 i ) holds. We then have: f ( g (1) , h (1) ) ⋉ ( g (3) · h (3) ) (1) ⊗ f ( g (2) , h (2) ) ⋉ ( g (3) · h (3) ) (2) == f ( g (1) , h (1) ) ⋉ g (3) · h (3) ⊗ f ( g (2) , h (2) ) ⋉ g (4) · h (4) = f ( g (1) , h (1) ) ⋉ g (2)(2) · h (2)(2) ⊗ f ( g (2)(1) , h (2)(1) ) ⋉ g (3) · h (3)(2 i ) = f ( g (1) , h (1) ) ⋉ g (2)(1) · h (2)(1) ⊗ f ( g (2)(2) , h (2)(2) ) ⋉ g (3) · h (3) = f ( g (1) , h (1) ) ⋉ g (2) · h (2) ⊗ f ( g (3) , h (3) ) ⋉ g (4) · h (4)XTENDING STRUCTURES II: THE QUANTUM VERSION 9 as needed. To end with, for the last family of generators we have:∆ A ⋉ H (cid:0) (1 A ⋉ g ) • ( a ⋉ H ) (cid:1) ( ) = ∆ A ⋉ H ( g (1) ⊲ a (1) ⋉ g (2) ⊳ a (2) ) ( ) , ( ) = g (1) ⊲ a (1) ⋉ g (3) ⊳ a (3) ⊗ g (2) ⊲ a (2) ⋉ g (4) ⊳ a (4) and∆ A ⋉ H (1 A ⋉ g )∆ A ⋉ H ( a ⋉ H ) = (cid:0) A ⋉ g (1) ⊗ A ⋉ g (2) (cid:1)(cid:0) a (1) ⋉ H ⊗ a (2) ⋉ H (cid:1) ( ) = g (1) ⊲ a (1) ⋉ g (2) ⊳ a (2) ⊗ g (3) ⊲ a (3) ⋉ g (4) ⊳ a (4) Thus ∆ A ⋉ H (cid:0) (1 A ⋉ g ) • ( a ⋉ H ) (cid:1) = ∆ A ⋉ H (1 A ⋉ g )∆ A ⋉ H ( a ⋉ H ) if and only if g (1) ⊲ a (1) ⋉ g (3) ⊳ a (3) ⊗ g (2) ⊲ a (2) ⋉ g (4) ⊳ a (4) == g (1) ⊲ a (1) ⋉ g (2) ⊳ a (2) ⊗ g (3) ⊲ a (3) ⋉ g (4) ⊳ a (4) This relation is equivalent to the compatibility condition (2 h ): indeed, by applying ε A ⊗ Id ⊗ Id ⊗ ε H to it we obtain (2 h ). Conversely suppose that (2 h ) holds. Then: g (1) ⊲ a (1) ⋉ g (3) ⊳ a (3) ⊗ g (2) ⊲ a (2) ⋉ g (4) ⊳ a (4) = (2 h ) = g (1) ⊲ a (1) ⋉ g (2) ⊳ a (2) ⊗ g (3) ⊲ a (3) ⋉ g (4) ⊳ a (4) as needed.To resume, we proved until now that ∆ A ⋉ H and ε A ⋉ H are k -algebra maps if and only ifthe relations (2 a ), (2 h ), (2 i ) hold. In what follows we shall prove, in the hypothesis that∆ A ⋉ H and ε A ⋉ H are k -algebra maps, that the multiplication given by (16) is associativeif and only if the compatibility conditions (2 b )-(2 g ) hold. This will end the proof. Wemake use again of the fact that T generates A ⋉ H as an algebra. Thus • is associativeif and only if x • ( y • z ) = ( x • y ) • z , for all x , y , z ∈ T . To start with, we will prove that:( a ⋉ H ) • ( y • z ) = [( a ⋉ H ) • y ] • z for all a ∈ A and y , z ∈ T . Indeed, we have:( a ⋉ H ) • (cid:16) (1 A ⋉ g ) • ( b ⋉ H ) (cid:17) ( ) = ( a ⋉ H ) • ( g (1) ⊲ b (1) ⋉ g (2) ⊳ b (2) ) ( ) = a ( g (1) ⊲ b (1) ) ⋉ ( g (2) ⊳ b (2) )) ( ) = ( a ⋉ g ) • ( b ⋉ H )= (cid:16) ( a ⋉ H ) • (1 A ⋉ g ) (cid:17) • ( b ⋉ H ) and ( a ⋉ H ) • (cid:16) (1 A ⋉ g ) • (1 A ⋉ h ) (cid:17) ( ) = ( a ⋉ H ) • ( f ( g (1) , h (1) ) ⋉ g (2) · h (2) ) ( ) = af ( g (1) , h (1) ) ⋉ g (2) · h (2)( ) = ( a ⋉ g ) • (1 A ⋉ h ) ( ) = (cid:16) ( a ⋉ H ) • (1 A ⋉ g ) (cid:17) • (1 A ⋉ h )The other two possibilities for choosing the elements of T can also be proven by astraightforward computation. Thus • is associative if and only if (1 A ⋉ g ) • ( y • z ) =[(1 A ⋉ g ) • y ] • z , for all g ∈ H , y , z ∈ T . First we note that:(1 A ⋉ g ) • (cid:16) ( a ⋉ H ) • (1 A ⋉ h ) (cid:17) = (1 A ⋉ g ) • ( a ⋉ h )= ( g (1) ⊲ a (1) ) f ( g (2) ⊳ a (2) , h (1) ) ⋉ ( g (3) ⊳ a (3) ) · h (2)( ) = ( g (1) ⊲ a (1) ⋉ g (2) ⊳ a (2) ) • (1 A ⋉ h ) ( ) = (cid:16) (1 A ⋉ g ) • ( a ⋉ H ) (cid:17) • (1 A ⋉ h )On the other hand:(1 A ⋉ g ) • (cid:16) ( b ⋉ H ) • ( c ⋉ H ) (cid:17) = (1 A ⋉ g ) • ( bc ⋉ H ) ( ) = g (1) ⊲ ( b (1) c (1) ) ⋉ g (2) ⊳ ( b (2) c (2) )and (cid:16) (1 A ⋉ g ) • ( b ⋉ H ) (cid:17) • ( c ⋉ H ) ( ) = ( g (1) ⊲ b (1) ⋉ g (2) ⊳ b (2) ) • ( c ⋉ H ) ( ) = ( g (1) ⊲ b (1) ) (cid:0) ( g (2) ⊳ b (2) ) ⊲ c (1) (cid:1) ⋉ ( g (3) ⊳ b (3) ) ⊳ c (2) Hence (1 A ⋉ g ) • (cid:16) ( b ⋉ H ) • ( c ⋉ H ) (cid:17) = (cid:16) (1 A ⋉ g ) • ( b ⋉ H ) (cid:17) • ( c ⋉ H ) if and only if g (1) ⊲ ( b (1) c (1) ) ⋉ g (2) ⊳ ( b (2) c (2) ) = ( g (1) ⊲ b (1) ) (cid:0) ( g (2) ⊳ b (2) ) ⊲ c (1) (cid:1) ⋉ ( g (3) ⊳ b (3) ) ⊳ c (2) (23)for all b, c ∈ A and g ∈ H . We show now that the relation (23) is equivalent to thecompatibility conditions (2 b ) and (2 d ). Indeed, by applying ε A ⊗ Id and respectively Id ⊗ ε H in (23) we obtain relations (2 b ) respectively (2 d ). Conversely, suppose thatrelations (2 b ) and (2 d ) hold. We then have:( g (1) ⊲ b (1) ) (cid:0) ( g (2) ⊳ b (2) ) ⊲ c (1) (cid:1) ⋉ ( g (3) ⊳ b (3) ) ⊳ c (2) == ( g (1)(1) ⊲ b (1)(1) ) (cid:0) ( g (1)(2) ⊳ b (1)(2) ) ⊲ c (1) (cid:1) ⋉ ( g (2) ⊳ b (2) ) ⊳ c (2)(2 d ) = g (1) ⊲ ( b (1) c (1) ) ⊗ ( g (2) ⊳ b (2) ) ⊳ c (2)(2 b ) = g (1) ⊲ ( b (1) c (1) ) ⊗ g (2) ⊳ ( b (2) c (2) ) XTENDING STRUCTURES II: THE QUANTUM VERSION 11 i.e. (23) holds. Now we deal with the last two cases. Since ⊲ is a coalgebra map weobtain:(1 A ⋉ g ) • (cid:16) (1 A ⋉ h ) • ( a ⋉ H ) (cid:17) ( ) = (1 A ⋉ g ) • ( h (1) ⊲ a (1) ⋉ h (2) ⊳ a (2) ) ( ) = (cid:16) g (1) ⊲ ( h (1) ⊲ a (1) ) (cid:17) f ( g (2) ⊳ ( h (2) ⊲ a (2) ) , h (4) ⊳ a (4) ) ⋉ [ g (3) ⊳ ( h (3) ⊲ a (3) )] · ( h (5) ⊳ a (5) )and (cid:16) (1 A ⋉ g ) • (1 A ⋉ h ) (cid:17) • ( a ⋉ H ) ( ) = (cid:0) f ( g (1) , h (1) ) ⋉ g (2) · h (2) (cid:1) • ( a ⋉ H ) ( ) = f ( g (1) , h (1) )[( g (2) · h (2) ) ⊲ a (1) ] ⋉ ( g (3) · h (3) ) ⊳ a (2) Thus (1 A ⋉ g ) • (cid:16) (1 A ⋉ h ) • ( a ⋉ H ) (cid:17) = (cid:16) (1 A ⋉ g ) • (1 A ⋉ h ) (cid:17) • ( a ⋉ H ) if and only if (cid:16) g (1) ⊲ ( h (1) ⊲ a (1) ) (cid:17) f (cid:16) g (2) ⊳ ( h (2) ⊲ a (2) ) , h (4) ⊳ a (4) (cid:17) ⋉ [ g (3) ⊳ ( h (3) ⊲ a (3) )] · ( h (5) ⊳ a (5) )= f ( g (1) , h (1) )[( g (2) · h (2) ) ⊲ a (1) ] ⋉ ( g (3) · h (3) ) ⊳ a (2) We shall prove, using (2 h ), that this relation is equivalent to the compatibility conditions(2 e ) and (2 f ). Indeed, by applying Id ⊗ ε H and respectively ε A ⊗ Id to it we obtain (2 e )and respectively (2 f ). Conversely, suppose that relations (2 e ) respectively (2 f ) hold.We denote LHS the left hand side of the above relation. We have:LHS = (cid:0) g (1) ⊲ ( h (1) ⊲ a (1) ) (cid:1) f (cid:0) g (2) ⊳ ( h (2) ⊲ a (2) ) , h (3)(2) ⊳ a (3)(2) (cid:1) ⋉ [ g (3) ⊳ ( h (3)(1) ⊲ a (3)(1) )] · ( h (4) ⊳ a (4) ) (2 h ) = (cid:16) g (1) ⊲ ( h (1) ⊲ a (1) ) (cid:17) f ( g (2) ⊳ ( h (2) ⊲ a (2) ) , h (3) ⊳ a (3) ) ⋉ [ g (3) ⊳ ( h (4) ⊲ a (4) )] · ( h (5) ⊳ a (5) ) (2 f ) = f ( g (1) , h (1) )[( g (2) · h (2) ) ⊲ a (1) ] ⊗ [ g (3) ⊳ ( h (3) ⊲ a (2) )] · ( h (4) ⊳ a (3) ) (2 e ) = f ( g (1) , h (1) )[( g (2) · h (2) ) ⊲ a (1) ] ⊗ ( g (3) · h (3) ) ⊳ a (2) as needed. Only one associativity relation remains to be verified:(1 A ⋉ g ) • (cid:16) (1 A ⋉ h ) • (1 A ⋉ l ) (cid:17) ( ) = (1 A ⋉ g ) • ( f ( h (1) , l (1) ) ⋉ h (2) · l (2) ) ( ) = (cid:0) g (1) ⊲ f ( h (1) , l (1) ) (cid:1) f (cid:0) g (2) ⊳ f ( h (2) , l (2) ) , h (4) · l (4) (cid:1) ⋉ (cid:0) g (3) ⊳ f ( h (3) , l (3) ) (cid:1) · ( h (5) · l (5) )and (cid:16) (1 A ⋉ g ) • (1 A ⋉ h ) (cid:17) • (1 A ⋉ l ) ( ) = (cid:0) f ( g (1) , h (1) ) ⋉ g (2) · h (2) (cid:1) • (1 A ⋉ l ) ( ) = f ( g (1) , h (1) ) f (cid:0) g (2) · h (2) , l (1) (cid:1) ⋉ ( g (3) · h (3) ) · l (2)2 A. L. AGORE AND G. MILITARU Hence (1 A ⋉ g ) • (cid:16) (1 A ⋉ h ) • (1 A ⋉ l ) (cid:17) = (cid:16) (1 A ⋉ g ) • (1 A ⋉ h ) (cid:17) • (1 A ⋉ l ) if and only if (cid:0) g (1) ⊲ f ( h (1) , l (1) ) (cid:1) f (cid:16) g (2) ⊳ f ( h (2) , l (2) ) , h (4) · l (4) (cid:17) ⊗ (cid:0) g (3) ⊳ f ( h (3) , l (3) ) (cid:1) · ( h (5) · l (5) ) == f ( g (1) , h (1) ) f (cid:0) g (2) · h (2) , l (1) (cid:1) ⊗ ( g (3) · h (3) ) · l (2) for all g, h, l ∈ H . We shall prove, using (2 i ), that this relation is equivalent to thecompatibility conditions (2 c ) and (2 g ). Indeed, by applying Id ⊗ ε H and respectively ε A ⊗ Id to it we obtain (2 c ) and respectively (2 g ). Conversely, suppose that relations(2 c ) and (2 g ) hold and denote LHS ′ the left hand side of the above relation. Then:LHS ′ = (cid:0) g (1) ⊲ f ( h (1) , l (1) ) (cid:1) f (cid:16) g (2) ⊳ f ( h (2) , l (2) ) , h (3)(2) · l (3)(2) (cid:17) ⊗ (cid:0) g (3) ⊳ f ( h (3)(1) , l (3)(1) ) (cid:1) · ( h (4) · l (4) ) (2 i ) = (cid:0) g (1) ⊲ f ( h (1) , l (1) ) (cid:1) f (cid:16) g (2) ⊳ f ( h (2) , l (2) ) , h (3) · l (3) (cid:17) ⊗ (cid:0) g (3) ⊳ f ( h (4) , l (4) ) (cid:1) · ( h (5) · l (5) ) (2 g ) = f ( g (1) , h (1) ) f ( g (2) · h (2) , l (1) ) ⊗ (cid:0) g (3) ⊳ f ( h (3) , l (2) ) (cid:1) · ( h (4) · l (3) )= f ( g (1) , h (1) ) f ( g (2) · h (2) , l (1) ) ⊗ (cid:0) g (3) ⊳ f ( h (3)(1) , l (2)(1) ) (cid:1) · ( h (3)(2) · l (2)(2) ) (2 c ) = f ( g (1) , h (1) ) f ( g (2) · h (2) , l (1) ) ⊗ ( g (3) · h (3) ) · l (2) as needed and the proof is now finished. (cid:3) Examples 2.5.
1. Let A be a bialgebra and Ω( A ) = (cid:0) H, ⊳, ⊲, f (cid:1) an extending datum of A such that the cocycle f is trivial, that is f ( g, h ) = ε H ( g ) ε H ( h )1 A , for all g , h ∈ H .Then Ω( A ) = (cid:0) H, ⊳, ⊲, f (cid:1) is a bialgebra extending structure of A if and only if H is abialgebra and ( A, H, ⊳, ⊲ ) is a matched pair of bialgebras. In this case, the associatedunified product A ⋉ H = A ⊲⊳ H is the bicrossed product of bialgebras constructed in(11).Conversely, a matched pair of bialgebras can be deformed using a coalgebra lazy cocy-cle in order to obtain a bialgebra extending structure as follows. Let (
A, H, ⊳, ⊲ ) be amatched pair of bialgebras such that A has antipode S A and u : H → A a coalgebra lazy1-cocycle in the sense of Definition 3.3 such that h ⊳ u ( g ) = hε H ( g ), for all h ∈ H and g ∈ G . Then Ω( A ) = (cid:0) H, ⊳, ⊲ ′ , f ′ (cid:1) is a bialgebra extending structure of A , where ⊲ ′ and f ′ are given by h ⊲ ′ c = u ( h (1) )( h (2) ⊲ c (1) ) S A (cid:16) u (cid:0) h (3) ⊳ c (2) (cid:1)(cid:17) f ′ ( h, g ) = u ( h (1) )( h (2) ⊲ u ( g (1) )) S A (cid:16) u (cid:0) h (3) g (2) (cid:1)(cid:17) for all h , g ∈ H and c ∈ A .2. Let A be a bialgebra and Ω( A ) = (cid:0) H, ⊳, ⊲, f (cid:1) an extending datum of A such that theaction ⊳ is trivial, that is h ⊳ a = ε A ( a ) h , for all h ∈ H and a ∈ A .Then Ω( A ) = (cid:0) H, ⊳, ⊲, f (cid:1) is a bialgebra extending structure of A if and only if H is anusual bialgebra and the following compatibility conditions are fulfilled: XTENDING STRUCTURES II: THE QUANTUM VERSION 13 (a) The twisted module condition (5) and the cocycle condition (6) hold;(b) g ⊲ ( ab ) = ( g (1) ⊲ a )( g (2) ⊲ b )(c) g (1) ⊗ g (2) ⊲ a = g (2) ⊗ g (1) ⊲ a (d) g (1) h (1) ⊗ f ( g (2) , h (2) ) = g (2) h (2) ⊗ f ( g (1) , h (1) )for all g , h ∈ H and a , b ∈ A .In this case, the associated unified product A ⋉ H = A f H is the crossed productconstructed in (4). In particular, if A is a bialgebra, the crossed product A f H is abialgebra with the coalgebra structure given by the tensor product of coalgebras if andonly if the compatibility conditions (c) and (d) above hold.Let A be a bialgebra and Ω( A ) = (cid:0) H, ⊳, ⊲, f (cid:1) a bialgebra extending structure of A .Then i A : A → A ⋉ H , i A ( a ) = a ⋉ H , for all a ∈ A is an injective bialgebra map, i H : H → A ⋉ H , i H ( h ) = 1 A ⊗ h , for all h ∈ H is an injective coalgebra map and u : A ⊗ H → A ⋉ H, u ( a ⊗ h ) = i A ( a ) • i H ( h ) = ( a ⋉ H ) • (1 A ⋉ h ) = a ⋉ h for all a ∈ A and h ∈ H is bijective, i.e. the unified product A ⋉ H factorizes through A and H . The next theorem shows the converse of this remark: any bialgebra E thatfactorizes through a subbialgebra of A and a subcoalgebra H is isomorphic to a unifiedproduct. In order to avoid complicated computations we use the following elementaryremark: Lemma 2.6.
Let E be a bialgebra, L a coalgebra and u : L → E an isomorphism ofcoalgebras. Then there exists a unique algebra structure on L such that u : L → E is anisomorphism of bialgebras given by: l · l ′ := u − ( u ( l ) u ( l ′ )) , L := u − (1 E ) for all l , l ′ ∈ L . Furthermore, if E has an antipode S E , then L is a Hopf algebra withthe antipode S L := u − ◦ S E ◦ u .Proof. Straightforward: the algebra structure on L is obtained by transfering the algebrastructure from E via the isomorphism of coalgebras u . The multiplication on L is acoalgebra map since it is a composition of coalgebra maps. (cid:3) Theorem 2.7.
Let E be a bialgebra, A ⊆ E a subbialgebra, H ⊆ E a subcoalgebra suchthat E ∈ H and the multiplication map u : A ⊗ H → E , u ( a ⊗ h ) = ah , for all a ∈ A , h ∈ H is bijective.Then, there exists Ω( A ) = ( H, ⊳, ⊲, f ) a bialgebra extending structure of A such that u : A ⋉ H → E , u ( a ⋉ h ) = ah is an isomorphism of bialgebras. Furthermore, if E is aHopf algebra then A ⋉ H is a Hopf algebra.Proof. Since E is a bialgebra, the multiplication m E : E ⊗ E → E is a coalgebramap. Thus u : A ⊗ H → E is in fact an isomorphism of coalgebras, with its inverse u − : E → A ⊗ H which is also a coalgebra map. The k -linear map µ : H ⊗ A → A ⊗ H, µ ( h ⊗ a ) := u − ( ha ) for all h ∈ H and a ∈ A is a coalgebra map as a composition of coalgebra maps. Wedefine the actions ⊲ , ⊳ by the formulas: ⊲ : H ⊗ A → A, ⊲ := ( Id ⊗ ε H ) ◦ µ (24) ⊳ : H ⊗ A → H, ⊳ := ( ε A ⊗ Id ) ◦ µ (25)They are coalgebra maps as compositions of coalgebra maps. Moreover, the normaliza-tion conditions (14) and (15) are trivially fulfilled. More explicitly, ⊲ and ⊳ are given asfollows: let h ∈ H and c ∈ A . Since u is a bijective map, there exists an unique element P j α j ⊗ l j ∈ A ⊗ H such that hc = P j α j l j . Then: h ⊲ c = X j α j ε H ( l j ) , h ⊳ c = X j ε A ( α j ) l j Next we construct the coalgebra maps f : H ⊗ H → A and · : H ⊗ H → H . The k -linearmap ν : H ⊗ H → A ⊗ H, ν ( h ⊗ g ) := u − ( hg )for all h , g ∈ H is a coalgebra map as a composition of coalgebra maps. We define: f : H ⊗ H → A, f := ( Id ⊗ ε H ) ◦ ν (26) · : H ⊗ H → H, · := ( ε A ⊗ Id ) ◦ ν (27)They are coalgebra maps as compositions of coalgebra maps. The normalization condi-tions 1 E · h = h · E = h and f ( h, E ) = f (1 E , h ) = ε H ( h )1 A , for all h ∈ H are triviallyfulfilled.In order to prove that Ω( A ) = ( H, ⊳, ⊲, f, · ) is a bialgebra extending structure of A weuse Lemma 2.6 and then Theorem 2.4: the unique algebra structure that can be definedon A ⊗ H such that u becomes an isomorphism of bialgebras is given by:( a ⊗ h ) • ( c ⊗ g ) = u − (cid:0) u ( a ⊗ h ) u ( c ⊗ g ) (cid:1) = u − ( ahcg )This algebra structure on A ⊗ H coincides with the one given by (16) on a unified productif and only if u − ( ahcg ) = a ( h (1) ⊲ c (1) ) f ( h (2) ⊳ c (2) , g (1) ) ⊗ ( h (3) ⊳ c (3) ) · g (2) Since u is a bijective map the above formula holds if and only if: hcg = ( h (1) ⊲ c (1) ) f ( h (2) ⊳ c (2) , g (1) ) (cid:0) ( h (3) ⊳ c (3) ) · g (2) (cid:1) (28)holds for all c ∈ A and h, g ∈ H . Therefore, the proof is finished if we prove that therelation (28) holds in the bialgebra E . Let c ∈ A and h , g ∈ H . Then there exists anunique element P nj =1 α j ⊗ l j ∈ A ⊗ H such that: hc = n X j =1 α j l j (29) XTENDING STRUCTURES II: THE QUANTUM VERSION 15
Hence h ⊲ c = P nj =1 ε H ( l j ) α j and h ⊳ c = P nj =1 ε A ( α j ) l j . Moreover, for any j = 1 , · · · , n there exists an unique element P mi =1 A ji ⊗ Z i ∈ A ⊗ H such that: l j g = m X i =1 A ji Z i (30)Using relations (26) and (27) we obtain: f ( l j , g ) = m X i =1 ε H ( Z i ) A ji , l j · g = m X i =1 ε A ( A ji ) Z i (31)and hcg = m,n X i,j =1 α j A ji Z i (32)In what follows we use the fact that m E , ⊲ and ⊳ are coalgebra maps. For example, byapplying ∆ to the relation (29) we obtain: h (1) ⊲ c (1) ⊗ h (2) ⊳ c (2) ⊗ h (3) ⊳ c (3) = n X j =1 ε H ( l j (1) ) α j (1) ⊗ ε A ( α j (2) ) l j (2) ⊗ ε A ( α j (3) ) l j (3) = n X j =1 ε H ( l j (1) ) α j ⊗ l j (2) ⊗ l j (3) = n X j =1 α j ⊗ l j (1) ⊗ l j (2) Thus, we have: h (1) ⊲ c (1) ⊗ h (2) ⊳ c (2) ⊗ h (3) ⊳ c (3) = n X j =1 α j ⊗ l j (1) ⊗ l j (2) (33)Moreover, by applying ∆ to the relation (30) and using the relation (31) we obtain: f (cid:0) l j (1) , g (1) (cid:1) ⊗ l j (1) · g (2) = m X i =1 ε H ( Z i (1) ) A ji (1) ⊗ ε A ( A ji (2) ) Z i (2) = m X i =1 A ji ⊗ Z i (34)We denote by RHS the right hand side of (28). Then:RHS ( ) = n X j =1 α j f (cid:0) l j (1) , g (1) (cid:1) l j (2) · g (2)( ) = m,n X i,j =1 α j A ji Z i ( ) = hcg Thus the relation (28) holds true and the proof is now finished since u − (1 E ) = 1 A ⊗ E .We use Theorem 2.4 in order to obtain that Ω( A ) = ( H, ⊳, ⊲, f, · ) is a bialgebra extending structure of A . Moreover, if E is a Hopf algebra then A ⋉ H is also a Hopf algebra withthe antipode given by S A ⋉ H = u − ◦ S E ◦ u according to Lemma 2.6. (cid:3) Next we construct an antipode for the unified product A ⋉ H . Proposition 2.8.
Let A be a Hopf algebra with an antipode S A and Ω( A ) = ( H, ⊳, ⊲, f ) a bialgebra extending structure of A such that there exists an antimorphism of coalgebras S H : H → H such that h (1) · S H ( h (2) ) = S H ( h (1) ) · h (2) = ε H ( h )1 H (35) for all h ∈ H . Then the unified product A ⋉ H is a Hopf algebra with the antipode S : A ⋉ H → A ⋉ H given by: S ( a ⋉ g ) := (cid:16) S A [ f (cid:0) S H ( g (2) ) , g (3) (cid:1) ] ⋉ S H ( g (1) ) (cid:17) • (cid:0) S A ( a ) ⋉ H (cid:1) (36) for all a ∈ A and g ∈ H . XTENDING STRUCTURES II: THE QUANTUM VERSION 17
Proof.
Let a ⋉ g ∈ A ⋉ H . Since the multiplication • on A ⋉ H is associative we have: S ( a (1) ⋉ g (1) ) • ( a (2) ⋉ g (2) ) == (cid:16) S A [ f (cid:0) S H ( g (2) ) , g (3) (cid:1) ] ⋉ S H ( g (1) ) (cid:17) • (cid:0) S A ( a (1) ) ⋉ H (cid:1) • ( a (2) ⋉ g (4) ) ( ) = (cid:16) S A [ f (cid:0) S H ( g (2) ) , g (3) (cid:1) ] ⋉ S H ( g (1) ) (cid:17) • (cid:0) S A ( a (1) ) a (2) ⋉ g (4) (cid:1) = ε A ( a ) (cid:16) S A [ f (cid:0) S H ( g (2) ) , g (3) (cid:1) ] ⋉ S H ( g (1) ) (cid:17) • (1 A ⋉ g (4) ) ( ) = ε A ( a ) S A (cid:16) f (cid:0) S H ( g (2) ) , g (3) (cid:1)(cid:17) f (cid:0) S H ( g (1) ) (1) , g (4)(1) (cid:1) ⋉ S H ( g (1) ) (2) · g (4)(2) = ε A ( a ) S A (cid:16) f (cid:0) S H ( g (1)(2) ) , g (2) (cid:1)(cid:17) f (cid:0) S H ( g (1)(1) ) (1) , g (3)(1) (cid:1) ⋉ S H ( g (1)(1) ) (2) · g (3)(2) = ε A ( a ) S A (cid:16) f (cid:0) S H ( g (1) ) (1) , g (2) (cid:1)(cid:17) f (cid:0) S H ( g (1) ) (2)(1) , g (3)(1) (cid:1) ⋉ S H ( g (1) ) (2)(2) · g (3)(2) = ε A ( a ) S A (cid:16) f (cid:0) S H ( g (1) ) (1) , g (2) (cid:1)(cid:17) f (cid:0) S H ( g (1) ) (2) , g (3)(1) (cid:1) ⋉ S H ( g (1) ) (3) · g (3)(2) = ε A ( a ) S A (cid:16) f (cid:0) S H ( g (1) ) (1) , g (2)(1) (cid:1)(cid:17) f (cid:0) S H ( g (1) ) (2) , g (2)(2)(1) (cid:1) ⋉ S H ( g (1) ) (3) · g (2)(2)(2) = ε A ( a ) S A (cid:16) f (cid:0) S H ( g (1) ) (1) , g (2)(1) (cid:1)(cid:17) f (cid:0) S H ( g (1) ) (2) , g (2)(2) (cid:1) ⋉ S H ( g (1) ) (3) · g (2)(3) = ε A ( a ) S A (cid:16) f (cid:0) S H ( g (1) ) (1)(1) , g (2)(1)(1) (cid:1)(cid:17) f (cid:0) S H ( g (1) ) (1)(2) , g (2)(1)(2) (cid:1) ⋉ S H ( g (1) ) (2) · g (2)(2)( ) = ε A ( a ) S A (cid:16) f (cid:0) S H ( g (1) ) (1) , g (2)(1) (cid:1) (1) (cid:17) f (cid:0) S H ( g (1) ) (1) , g (2)(1) (cid:1) (2) ⋉ S H ( g (1) ) (2) · g (2)(2) = ε A ( a ) ε A (cid:16) f (cid:0) S H ( g (1) ) (1) , g (2)(1) (cid:1)(cid:17) A ⋉ S H ( g (1) ) (2) · g (2)(2)( ) = ε A ( a ) ε H (cid:16) S H ( g (1) ) (1) (cid:17) ε H (cid:0) g (2)(1) (cid:1) ⋉ S H ( g (1) ) (2) · g (2)(2) = ε A ( a )1 A ⋉ S H ( g (1) ) · g (2) = ε A ( a ) ε H ( g )1 A ⋉ H Thus S is a left inverse of the identity in the convolution algebra Hom( A ⋉ H, A ⋉ H ).By similar computations one can show that S is also a right inverse of the identity, thusis an antipode of A ⋉ H . (cid:3) In Proposition 2.8 we imposed the condition for S H to be a coalgebra antimorphismbecause the algebra structure on H is not an associative one and for this reason a k -linear map S H which satisfies the antipode condition (35) is not necessarily a coalgebraantimorphism as in the classical case of Hopf algebras.3. The classification of unified products
In this section we prove the classification theorem for unified products: as a special case,a classification theorem for bicrossed products of Hopf algebras is obtained. Our viewpoint is inspired from Schreier’s classification theorem for extensions of an abelian group K by a group Q [13, Theorem 7.34]: they are classified by the second cohomology group H ( Q, K ). Let ϕ : G → G ′ be a morphism between two extensions of a group K by agroup Q , i.e. ϕ is a morphism of groups such that the diagram k [ K ] i / / Id (cid:15) (cid:15) k [ G ] π / / ϕ (cid:15) (cid:15) k [ Q ] Id (cid:15) (cid:15) k [ K ] i ′ / / k [ G ′ ] π ′ / / k [ Q ]is commutative (we wrote the diagram for the induced morphism for group algebras).Then ϕ is an isomorphism [13, Theorem 7.32]. Now, the left hand square of the diagramis commutative if and only if ϕ is a left k [ K ]-module map while the right hand squareof the diagram is commutative if and only if ϕ is a morphism of right k [ Q ]-comodules.This motivates the way of considering the classification of unified products up to anisomorphism of Hopf algebras that is also a left A -module map and a right H -comodulemap.Let Ω( A ) = (cid:0) H, ⊳, ⊲, f (cid:1) be a bialgebra extending structure of A . The unified product A ⋉ H is a right H -comodule via the coaction a ⋉ h a ⋉ h (1) ⊗ h (2) , for all a ∈ A and h ∈ H and a left A -module via the restriction of scalars map i A : A → A ⋉ H .From now on the Hopf algebra structure on A and the coalgebra structure on H will beset. First, we need the following. Lemma 3.1.
Let A be a Hopf algebra, Ω( A ) = (cid:0) H, ⊳, ⊲, f (cid:1) and Ω ′ ( A ) = (cid:0) H, ⊳ ′ , ⊲ ′ , f ′ (cid:1) two Hopf algebra extending structures of A . Then a k -linear map ϕ : A ⋉ H → A ⋉ ′ H is a left A -module, a right H -comodule and a coalgebra morphism if and only if thereexists a unique morphism of coalgebras u : H → A such that h (1) ⊗ u ( h (2) ) = h (2) ⊗ u ( h (1) ) (37) for all h ∈ H and ϕ is given by ϕ ( a ⋉ h ) = au ( h (1) ) ⋉ ′ h (2) (38) for all a ∈ A and h ∈ H . Furthermore, any such a morphism ϕ : A ⋉ H → A ⋉ ′ H is anisomorphism with the inverse given by ψ : A ⋉ ′ H → A ⋉ H, ψ ( a ⋉ ′ h ) = aS A (cid:0) u ( h (1) ) (cid:1) ⋉ h (2) for all a ∈ A and h ∈ H .Proof. Let ϕ : A ⋉ H → A ⋉ ′ H be a left A -module, a right H -comodule and a coalgebramorphism. We shall adopt the notation ϕ (1 A ⋉ h ) = P h A ⊗ h H ∈ A ⊗ H , for all h ∈ H .Since ϕ is a left A -module map we have ϕ ( a ⋉ h ) = aϕ (1 A ⋉ h ) = a X h A ⊗ h H for all a ∈ A and h ∈ H . As ϕ is also a right H -comodule map we have: X ah A ⊗ ( h H ) (1) ⊗ ( h H ) (2) = ϕ ( a ⋉ h (1) ) ⊗ h (2)XTENDING STRUCTURES II: THE QUANTUM VERSION 19 By applying ε H on the second position of the above identity we obtain: ϕ ( a ⋉ h ) = X a ( h (1) ) A ε H (( h (1) ) H ) ⊗ h (2) for all a ∈ A and h ∈ H . Now, if we define u : H → A by: u ( h ) = ( Id ⊗ ε H ) ◦ ϕ (1 A ⋉ h ) = X h A ε H ( h H )for all h ∈ H , it follows that (38) holds. We shall prove now that ϕ given by (38) is acoalgebra map if and only if u is a coalgebra map and (37) holds. First we observe that ε A ⋉ ′ H ◦ ϕ = ε A ⋉ H if and only if ε A ◦ u = ε H . Now, the fact that ϕ is comultiplicative isequivalent to: u ( h (1) ) (1) ⊗ h (2) ⊗ u ( h (1) ) (2) ⊗ h (3) = u ( h (1) ) ⊗ h (2) ⊗ u ( h (3) ) ⊗ h (4) (39)for all h ∈ H . By applying Id ⊗ ε H ⊗ Id ⊗ ε H to this relation we obtain that u is acoalgebra map; using this fact and then applying ε A ⊗ Id ⊗ Id ⊗ ε H in relation (39) weobtain relation (37). Conversely, if u is a coalgebra map such that relation (37) holds,then (39) follows straightforward, i.e. ϕ is a coalgebra map. The fact that ψ is an inversefor φ is also straightforward. (cid:3) Remark 3.2.
At this point we should remark the perfect similarity with the theory ofextensions from the groups case. If ϕ : A ⋉ H → A ⋉ ′ H is a left A -module, a right H -comodule and a coalgebra morphism between two unified products then the followingdiagram A i A / / Id A (cid:15) (cid:15) A ⊲⊳ H π H / / ϕ (cid:15) (cid:15) H Id H (cid:15) (cid:15) A i A / / A ⊲⊳ ′ H π H / / H is commutative and ϕ is an isomorphism.For A = k and a Hopf algebra H the group H l ( H, k ) of all unitary algebra maps u : H → k satisfying the compatibility condition (40) below was called in [3, Definition1.1] the first lazy cohomology group of H with coefficients in k . We shall now define thecoalgebra version of lazy 1-cocyles. Definition 3.3.
Let A be a Hopf algebra and H a coalgebra, unitary not necessarilyassociative algebra. A morphism of coalgebras u : H → A is called a coalgebra lazy -cocyle if u (1 H ) = 1 A and the following compatibility holds: h (1) ⊗ u ( h (2) ) = h (2) ⊗ u ( h (1) ) (40)for all h ∈ H . We denote by H l,c ( H, A ) the group of all coalgebra lazy 1-cocyles of H with coefficients in A . H l,c ( H, A ) is a group with respect to the convolution product. We have to prove that if u and v ∈ H l,c ( H, A ), then u ∗ v ∈ H l,c ( H, A ). Indeed, is straightforward to prove that u ∗ v satisfy (40). Let us show that u ∗ v is a morphism of coalgebras. First, if we apply v on the first position in (40) we obtain v ( h (1) ) ⊗ u ( h (2) ) = v ( h (2) ) ⊗ u ( h (1) ), for all h ∈ H .Using this relation we obtain:∆ A (cid:0) u ( h (1) ) v ( h (2) ) (cid:1) = u ( h (1) ) v ( h (3) ) ⊗ u ( h (2) ) v ( h (4) )= u ( h (1) ) v ( h (2) ) ⊗ u ( h (3) ) v ( h (4) )= u ∗ v ( h (1) ) ⊗ u ∗ v ( h (2) )for all h ∈ H , hence u ∗ v is also a coalgebra map.The main theorem of this section now follows: Theorem 3.4.
Let A be a Hopf algebra, Ω( A ) = (cid:0) H, ⊳, ⊲, f (cid:1) and Ω ′ ( A ) = (cid:0) H, ⊳ ′ , ⊲ ′ , f ′ (cid:1) two Hopf algebra extending structures of A . Then there exists ϕ : A ⋉ ′ H → A ⋉ H a left A -module, a right H -comodule and a Hopf algebra map if and only if ⊳ ′ = ⊳ and thereexists a coalgebra lazy -cocyle u ∈ H l,c ( H, A ) such that: h ⊲ ′ c = u ( h (1) )( h (2) ⊲ c (1) ) S A (cid:16) u (cid:0) h (3) ⊳ c (2) (cid:1)(cid:17) (41) f ′ ( h, g ) = u ( h (1) )( h (2) ⊲ u ( g (1) )) f ( h (3) ⊳ u ( g (2) ) , g (3) ) S A (cid:16) u (cid:0) h (4) · ′ g (4) (cid:1)(cid:17) (42) h · ′ g = (cid:0) h ⊳ u ( g (1) ) (cid:1) · g (2) (43) for all h , g ∈ H and c ∈ A . In this case ϕ is given by (38) and it is an isomorphism.Proof. We already proved in Lemma 3.1 that ϕ : A ⋉ ′ H → A ⋉ H is a left A -module, aright H -comodule and a coalgebra map if and only if ϕ ( a ⋉ ′ h ) = au ( h (1) ) ⋉ h (2) , for all a ∈ A , h ∈ H and for a unique coalgebra map u : H → A such that the (40) holds. Ofcourse, ϕ (1 A ⋉ ′ H ) = 1 A ⋉ H if and only if u is unitary. Moreover, as u is a morphismof coalgebras it is invertible in convolution with the inverse u − = S A ◦ u .In what follows we shall prove, in the hypothesis that ϕ is a coalgebra map and u isunitary, that ϕ is an algebra map (thus a map of bialgebras) if and only if ⊳ ′ = ⊳ andthe compatibility conditions (41) - (43) hold. By a straightforward computation we canshow that ϕ is an algebra map if and only if( C ) ( h (1) ⊲ ′ c (1) ) f ′ (cid:0) h (2) ⊳ ′ c (2) , g (1) (cid:1) u (cid:0) ( h (3) ⊳ ′ c (3) ) · ′ g (2) (cid:1) ⋉ ( h (4) ⊳ ′ c (4) ) · ′ g (3) == u ( h (1) ) (cid:0) h (2) ⊲ c (1) u ( g (1) ) (cid:1) f (cid:0) h (3) ⊳ c (2) u ( g (2) ) , g (4) (cid:1) ⋉ (cid:0) h (4) ⊳ c (3) u ( g (3) ) (cid:1) · g (5) for all h , g ∈ H and c ∈ A . We shall prove that the compatibility ( C ) is equivalent to(41) - (43).Indeed, by considering g = 1 H in ( C ) and then by applying ε A ⊗ Id we obtain h⊳ ′ c = h⊳c ,for all h ∈ H and c ∈ A . If we consider again g = 1 H we obtain, after applying first Id ⊗ ε H and then inverting u , that (41) holds. Relation (43) is obtained by considering c = 1 A in ( C ), applying Id ⊗ ε H and finally inverting u . To end with, relation (43)follows by considering c = 1 A and by applying ε A ⊗ Id in ( C ).Conversely, suppose that h ⊳ ′ c = h ⊳ c , for all h ∈ H and c ∈ A and there exists acoalgebra lazy 1-cocyle u such that relations (41) - (43) are fulfilled. We then have (we XTENDING STRUCTURES II: THE QUANTUM VERSION 21 denote by
LHS the left hand side of ( C )): LHS ( ) − ( ) = u ( h (1)(1) )( h (1)(2) ⊲ c (1)(1) ) S A (cid:16) u (cid:0) h (1)(3) ⊳ c (1)(2) (cid:1)(cid:17) u ( h (2)(1) ⊳ c (2)(1) ) (cid:0) ( h (2)(2) ⊳ c (2)(2) ) ⊲ u ( g (1)(1) ) (cid:1) f (cid:0) ( h (2)(3) ⊳ c (2)(3) ) ⊳ u ( g (1)(2) ) , g (1)(3) (cid:1) S A (cid:16) u (cid:0) h (2)(4) ⊳ c (2)(4) (cid:1) · ′ g (1)(4) (cid:17) u (cid:0) ( h (3) ⊳ c (3) ) · ′ g (2) (cid:1) ⋉ ( h (4) ⊳ c (4) ) · ′ g (3) = u ( h (1) )( h (2) ⊲ c (1) ) S A (cid:16) u (cid:0) h (3) ⊳ c (2) (cid:1)(cid:17) u ( h (4) ⊳ c (3) ) (cid:0) ( h (5) ⊳ c (4) ) ⊲ u ( g (1) ) (cid:1) f (cid:0) ( h (6) ⊳ c (5) ) ⊳ u ( g (2) ) , g (3) (cid:1) S A (cid:16) u (cid:0) h (7) ⊳ c (6) (cid:1) · ′ g (4) (cid:17) u (cid:0) ( h (8) ⊳ c (7) ) · ′ g (5) (cid:1) ⋉ ( h (9) ⊳ c (8) ) · ′ g (6) = u ( h (1) )( h (2) ⊲ c (1) ) (cid:0) ( h (3) ⊳ c (2) ) ⊲ u ( g (1) ) (cid:1) f (cid:0) ( h (4) ⊳ c (3) ) ⊳ u ( g (2) ) , g (3) (cid:1) ⋉ ( h (5) ⊳ c (4) ) · ′ g (4)(2 d ) = u ( h (1) ) (cid:16) h (2) ⊲ (cid:0) c (1) u ( g (1) ) (cid:1)(cid:17) f (cid:0) ( h (3) ⊳ c (2) ) ⊳ u ( g (2) ) , g (3) (cid:1) ⋉ ( h (4) ⊳ c (3) ) · ′ g (4)( ) = u ( h (1) ) (cid:16) h (2) ⊲ (cid:0) c (1) u ( g (1) ) (cid:1)(cid:17) f (cid:0) ( h (3) ⊳ c (2) ) ⊳ u ( g (2) ) , g (3) (cid:1) ⋉ ( h (4) ⊳ c (3) u ( g (4) )) · g (5)( ) = u ( h (1) ) (cid:16) h (2) ⊲ (cid:0) c (1) u ( g (1) ) (cid:1)(cid:17) f (cid:0) ( h (3) ⊳ c (2) ) ⊳ u ( g (2) ) , g (4) (cid:1) ⋉ ( h (4) ⊳ c (3) u ( g (3) )) · g (5) where the third equality holds by using the antipode conditions and the fact that u is acoalgebra map. Thus ( C ) holds and the proof is now finished. (cid:3) Even if for the classification problem we only set the Hopf algebra structure of A andthe coalgebra structure of H , Theorem 3.4 tells us that we can set also the coalgebramap ⊳ : H ⊗ A → H . We shall phrase Theorem 3.4 as a description of the skeleton forthe category C ( A, H, ⊳ ) defined below.Let A be a Hopf algebra, H a coalgebra with a fixed group-like element 1 H ∈ H and ⊳ : H ⊗ A → H a morphism of coalgebras. Let E S ( A, H, ⊳ ) be the set of all triples( · , ⊲, f ) such that (cid:0) ( H, X , · ) , ⊳, ⊲, f (cid:1) is a Hopf algebra extending structure of A . Thenext definition is the Hopf algebra version for unified product [13, Definition 7.31] givenfor extensions of groups. Definition 3.5.
Two elements ( · , ⊲, f ), ( · ′ , ⊲ ′ , f ′ ) of E S ( A, H, ⊳ ) are called cohomologous and we denote this by ( · , ⊲, f ) ≈ ( · ′ , ⊲ ′ , f ′ ) if there exists a coalgebra lazy 1-cocyle u ∈ H l,c ( H, A ) such that the compatibility conditions (41) - (43) are fulfilled.It follows from Theorem 3.4 that ( · , ⊲, f ) ≈ ( · ′ , ⊲ ′ , f ′ ) if and only if there exists ϕ : A ⋉ ′ H → A ⋉ H a left A -module, a right H -comodule and a Hopf algebra map. Moreover, from Lemma 3.1 we obtain that any such map ϕ : A ⋉ ′ H → A ⋉ H is an isomorphism,thus ≈ is an equivalence relation on the set E S ( A, H, ⊳ ). We denote by H l,c ( H, A, ⊳ ) thequotient set E S ( A, H, ⊳ ) / ≈ .Let C ( A, H, ⊳ ) be the category whose class of objects is the set E S ( A, H, ⊳ ). A morphism ϕ : (cid:0) · , ⊲, f (cid:1) → (cid:0) · , ⊲ ′ , f ′ (cid:1) in C ( A, H, ⊳ ) is a Hopf algebra morphism ϕ : A ⋉ H → A ⋉ ′ H that is a left A -module and a right H -comodule map. Thus we obtain the categoricalversion of Theorem 3.4: Corollary 3.6. (Schreier theorem for unified products)
Let A be a Hopf algebra, H a coalgebra with a group-like element H and ⊳ : H ⊗ A → H a morphism of coalgebras.There exists a bijection between the set of objects of the skeleton of the category C ( A, H, ⊳ ) and the quotient set H l,c ( H, A, ⊳ ) . H l,c ( H, A, ⊳ ) is for the classification of the unified products the counterpart of the secondcohomology group for the classification of an extension of an abelian group by a group[13, Theorem 7.34].We can apply Theorem 3.4 to obtain classification theorems for various special cases ofthe unified products: for instance, Doi’s results on the classification of crossed products([5]) is obtain as a special case if we let ⊳ ′ = ⊳ be the trivial actions. Now, we shallindicate the classification of bicrossed product of Hopf algebras. Corollary 3.7. (Schreier theorem for bicrossed products)
Let A and H be twoHopf algebras and (cid:0) A, H, ⊳, ⊲ (cid:1) , (cid:0) A, H, ⊳ ′ , ⊲ ′ (cid:1) two matched pairs of Hopf algebras. Then A ⊲⊳ H ∼ = A ⊲⊳ ′ H (isomorphism of Hopf algebras, left A -modules and right H -comodules)if and only if ⊳ ′ = ⊳ and there exists a coalgebra lazy -cocyle u ∈ H l,c ( H, A ) such that: h ⊲ ′ c = u ( h (1) )( h (2) ⊲ c (1) ) S A (cid:16) u (cid:0) h (3) ⊳ c (2) (cid:1)(cid:17) u ( h (1) )( h (2) ⊲ u ( g (1) )) S A (cid:16) u (cid:0) h (3) g (2) (cid:1)(cid:17) = ε H ( g ) ε H ( h )1 A h ⊳ u ( g ) = h ε H ( g ) for all h , g ∈ H and c ∈ A .Proof. We apply Theorem 3.4 for the case when f and f ′ are the trivial cocycles. Asthe multiplication on the algebra H is the same (i.e. · = · ′ ), the condition (43) inTheorem 3.4 takes the equivalent form h ⊳ u ( g ) = hε H ( g ), for all h , g ∈ H . (cid:3) The construction of unified products is a challenging problem considering the number ofcompatibilities that need to be fulfilled. In particular, an example of an unified productwhich is neither a crossed product nor a bicrossed product is interesting in the picture.We provide such an example below: it is a Hopf algebra k [ A ] ⋉ k [ S ] ∼ = k [ A ], where A n is the alternating group on a set with n elements and S is a set with thirty elements. Example 3.8.
Let G be a group and ( X, X ) a pointed set. We consider the group Hopfalgebra A := k [ G ] and the group-like coalgebra H := k [ X ]. We note that coalgebra mor-phisms between two group-like coalgebras are in one to one correspondence with the mapsbetween the corresponding sets. Thus, any bialgebra extending structure ( k [ X ] , ⊳ , ⊲ , f ) XTENDING STRUCTURES II: THE QUANTUM VERSION 23 of the Hopf algebra k [ G ] is induced by an extending structure ( X, ⊳ ′ , ⊲ ′ , f ′ ) of the group G in the sense of [1, Definition 2.3]. Moreover there exists a canonical isomorphism ofbialgebras k [ G ] ⋉ k [ X ] ∼ = k [ G ⋉ X ]where G ⋉ X is the unified product at the level of groups (see [1] for further details). Thisgeneralizes the fact that a bicrossed product of two group Hopf algebras is isomorphicto the group Hopf algebra of the bicrossed product of the corresponding groups [8,Example 1, pg. 207]. The same type of isomorphism holds also for crossed products ofHopf algebras between two group Hopf algebras.Now, let A be the alternating group on a set with six elements. A is the simple groupof smallest order that cannot be written as a bicrossed product of two proper subgroups([14]). Being a simple group it can not be written neither as a crossed product of twoproper subgroups. On the other hand, A can be written as an unified product betweenany of its subgroups and an extending structure. For instance, we can write A ∼ = A ⋉ S for an extending structure (cid:0) S, S , α, β, f, ∗ , i (cid:1) of A , where S is a set of representativesfor the right cosets of A in A with 30 elements such that 1 ∈ S . Thus there exists anexample of an unified product for Hopf algebras k [ A ] ⋉ k [ S ] ∼ = k [ A ] which is neither acrossed product nor a bicrossed product of two Hopf algebras.Two general methods for constructing unified products are proposed in [2]. One of themconstructs an unified product starting with a minimal set of data: a Hopf algebra A , aunitary not necessarily associative bialgebra H which is a right A -module coalgebra anda unitary coalgebra map γ : H → A satisfying four technical compatibility conditions([2, Theorem 2.9]). 4. Acknowledgment
A.L. Agore is ”Aspirant” Fellow of the Fund for Scientific Research-Flanders (Belgium)(F.W.O. Vlaanderen). G. Militaru was supported from CNCSIS grant 24/28.09.07 ofPN II ”Groups, quantum groups, corings and representation theory”.
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