aa r X i v : . [ m a t h . QA ] M a y ON THE COHERENT HOPF 2-ALGEBRAS
XIAO HAN
Abstract.
We construct a coherent Hopf 2-algebra as quantization of a coherent 2-group, which consists of two Hopf coquasigroups and a coassociator. For this construc-tive method, if we replace Hopf coquasigroups by Hopf algebras, we can construct astrict Hoft 2-algebra, which is a quantisation of 2-group. We also study the crossedcomodule of Hopf algebras, which is shown to be a strict Hopf 2-algebra under someconditions. As an example, a quasi coassociative Hopf coquasigroup is employed to builda special coherent Hopf 2-algebra with nontrivial coassociator. Following this we studyfunctions on Cayler algebra basis.
Contents
1. Introduction 12. Algebraic preliminaries 22.1. Algebras, coalgebras and all that 22.2. Hopf algebroids 53. Coherent 2-group 94. Coherent Hopf-2-algebras 105. Crossed comodule of Hopf coquasigroups 146. Quasi coassociative Hopf coquasigroup 197. Finite dimensional coherent Hopf 2-algebra and example 25References 291.
Introduction
The study of Higher group theory and quantum group theory become more and moreimportant in various branch of mathematics and physics, such as topological field theoryand quantum gravity. In the present paper, the main idea of constructing a coherentquantum 2-group (or coherent Hopf 2-algebra) is to make a quantization on the 2-arrowscorresponding to the 2-category.In [9] and [5], the researchers constructed a strict quantum 2-group, while here weconsider to construct a generalization of them, all the quantum groups correspondingto the objects and morphisms are no more coassociative, and thus there will be a corre-sponding coherent condition. For noncoassociative quantum group, Hopf coquasigroup [8]usually offers some new properties and interesting examples, such as functions on Cayleralgebras, we are motivated to build the coherent Hopf 2-algebra by Hopf coquasigroups.Moreover, the nonassociativity of Cayler algebras are controlled by a 3-cocycle cobound-ary corresponding to a 2-cochain, this fact will play an improtant role in satisfying thecoherent condition of a coherent Hopf 2-algebra. We choose a special quantization forthe 2-structure, then the definition of coherent Hopf 2-algebra will be composed of twoHopf coquasigroups, which corresponds the “quantum” object and morphisms. or a coherent 2-group, since all the morphisms are invertible, there is a groupoidstructure on basis of the composition of morphisms. Therefore, a quantum groupoid orHopf algebroid will naturally exist in the coherent Hopf 2-algebra. These facts result intwo different structures for the quantum 2-arrows, on one hand, it is a Hopf coquasigroup,which corresponds to the ‘horizontal’ coproduct (or tensor coproduct); on the other handit is also a Hopf algebroid, which corresponds to the ‘vertical’ coproduct (or morphismscocomposition). These two kinds of coproducts also satisfy the interchange law. Moreover,the antipode of the Hopf coquasigroup preserves the coproduct of Hopf algebroid whilethe antipode of the Hopf algebroid is even a coalgebra map, which break the common factthat antipode is usually an anti-coalgebra map. The coherent condition will be describedby a coassociator, which satisfies the “3-cocycle” condition. When we consider Hopfalgebras instead of Hopf coquasigroups with trivial coassociator, we will get a strict Hopf2-algebra.For a strict 2-group, there is an equivalent definition, called crossed module of groups.In “quantum case” [5], the researchers construct a crossed comodule of Hopf algebra as astrict quantum 2-group. Here we show that under some conditions, a crossed comoduleof Hopf algebra is a strict Hopf 2-algebra with its coassociator to be trivial. Severalexamples are also put up here, which can be characterised by the corresponding bialgebramorphism.We also make a generalisation for crossed comodule of Hopf algebra, i.e. crossed co-module of Hopf coquasigroup, by replacing the pair of Hopf algebras with a special pairof Hopf coquasigroups. We show that if a Hopf coquasigroup is quasi coassociative, onecan construct a special crossed comodule of Hopf coquasigroup and furthermore a coas-sociator, which can build a coherent Hopf 2-algebra. An example is a Hopf coquasigroupwhich consists of functions on unit Cayler algebra basis. This Hopf coquasigroup is quasicoassociative and we will give all the structure maps precisely. Finally, we show that thecoassociator is indeed controlled by a 3-coboundary cocycle corresponding to a 2-cochain.The paper is organised as follow: In § § § § §
6, we will first give the definition of quasi coassociative Hopf coquasigroup,and then construct a crossed comodule of Hopf coquasigroup and futhermore a coherentHopf 2-algebra. In §
7, the finite dimensional coherent Hopf 2-algebra is discussed, andthrough an investigation into the dual pairing, we make clear of why quasi coassociativeHopf coquasigroup is the quantization of quasiassociative quasi group; we also consideran example built by functions on Cayler algebra basis.2.
Algebraic preliminaries
At first we recall some materials about Hopf coquasigroups and corresponding modulesand comodules. We also review the more general notions of rings and corings over analgebra as well as the associated notion of bialgebroid.2.1.
Algebras, coalgebras and all that.
To be definite we work over the field C of com-plex numbers but in the following this could be substituted by any field k . Algebras areassumed to be unital and associative with morphisms of algebras taken to be unital, andco-algebras are assumed to be counital, but not necessary coassociative with morphismof co-algebras taken to be co-unital. For the coproduct of a coalgebra ∆ : H → H ⊗ H we use the Sweedler notation ∆( h ) = h (1) ⊗ h (2) (sum understood), and its iterations: n = (id ⊗ ∆ H ) ◦ ∆ n − H : h h (1) ⊗ h (2) ⊗ · · · ⊗ h ( n +1) . However, this n -th interatecoproduct is not unique, if the coproduct is not coassociative.In order to give a coherent Hopf 2-algebra, which is a weaker version of Hopf 2-algebra,we need a more general algebra structure, that is a Hopf coquasi-group [8]. Definition 2.1. A Hopf coquasigroup H is an unital associate algebra, equiped withcounital algebra homomorphisms ∆ : H → H ⊗ H , ǫ : H → C , and linear map S H : H → H such that( m ⊗ H )( S H ⊗ H ⊗ H )( H ⊗ ∆)∆ = 1 ⊗ H = ( m ⊗ H )( H ⊗ S H ⊗ H )( H ⊗ ∆)∆ (2.1)( H ⊗ m )( H ⊗ S H ⊗ H )(∆ ⊗ H )∆ = H ⊗ H ⊗ m )( H ⊗ H ⊗ S H )(∆ ⊗ H )∆ . (2.2)A morphism between two Hopf coquasigroups is an algebra map f : H → G , such thatfor any h ∈ H , f ( h ) (1) ⊗ f ( h ) (2) = f ( h (1) ) ⊗ f ( h (2) ) and ǫ G ( f ( h )) = ǫ H ( h ). Remark . Hopf coquasigroup is a generalisation of Hopf algebra, for which the coprod-uct is not necessary coassociative. As a result, we can not use the Sweedler index notionas Hopf algebra (but we still use h (1) ⊗ h (2) as the image of coproduct), so in general wedon’t have: h (1)(1) ⊗ h (1)(2) ⊗ h (2) = h (1) ⊗ h (2) ⊗ h (3) = h (1) ⊗ h (2)(1) ⊗ h (2)(2) . It is given in[8] that the linear map S H (we also call it antipode) of H has similar property as theantipode of a Hopf algebra. That is: • h (1) S H ( h (2) ) = ǫ ( h ) = S H ( h (1) ) h (2) , • S H ( hh ′ ) = S H ( h ′ ) S H ( h ), • S H ( h ) (1) ⊗ S H ( h ) (2) = S H ( h (2) ) ⊗ S H ( h (1) ),for any h, h ′ ∈ H .Given a Hopf coquasigroup H , define a linear map β : H → H ⊗ H ⊗ H by β ( h ) := h (1)(1) S H ( h (2) ) (1)(1) ⊗ h (1)(2)(1) S H ( h (2) ) (1)(2) ⊗ h (1)(2)(2) S H ( h (2) ) (2) (2.3)for any h ∈ H . We can see that β ⋆ ((∆ H ⊗ H ) ◦ ∆ H ) = ( H ⊗ ∆ H ) ◦ ∆ H , (2.4)where ‘ ⋆ ’ is convolution product corresponding to the coproduct of Hopf coquasigroup.More precisely, it can be written as h (1)(1)(1) S H ( h (1)(2) ) (1)(1) h (2)(1)(1) ⊗ h (1)(1)(2)(1) S H ( h (1)(2) ) (1)(2) h (2)(1)(2) ⊗ h (1)(1)(2)(2) S H ( h (1)(2) ) (2) h (2)(2) = h (1) ⊗ h (1)(2) ⊗ h (2)(2) , which can be derived from the definition of Hopf coquasigroup. For any h ∈ H we willalways denote the image of β by β ( h ) = h ˆ1 ⊗ h ˆ2 ⊗ h ˆ3 . A Hopf coquasigroup H is coassociative if and only if β ( h ) = ǫ ( h )1 H ⊗ H ⊗ H if andonly if H is a Hopf algebra.Given a Hopf coquasigroup H , a left H -comodule is a vector space V carrying a left H -coaction, that is with a C -linear map δ V : V → H ⊗ V such that( H ⊗ δ V ) ◦ δ V = (∆ ⊗ V ) ◦ δ V , ( ǫ ⊗ V ) ◦ δ V = V . (2.5)In Sweedler notation, v δ V ( v ) = v ( − ⊗ v (0) , and the left H -comodule properties read, v ( − ⊗ v ( − ⊗ v (0) = v ( − ⊗ v (0)( − ⊗ v (0)(0) ,ǫ ( v ( − ) v (0) = v , or all v ∈ V . The C -vector space tensor product V ⊗ W of two H -comodules is a H -comodule with the left tensor product H -coaction δ V ⊗ W : V ⊗ W −→ H ⊗ V ⊗ W, v ⊗ w v ( − w ( − ⊗ v (0) ⊗ w (0) . (2.6)A H -comodule map ψ : V → W between two H -comodules is a C -linear map ψ : V → W which is H -equivariant (or H -colinear), that is, δ W ◦ ψ = ( H ⊗ ψ ) ◦ δ V .In particular, a left H -comodule algebra is an algebra A , which is a left H -comodulesuch that the multiplication and unit of A are morphisms of H -comodules. This isequivalent to requiring the coaction δ : A → H ⊗ A to be a morphism of unital algebras(where H ⊗ A has the usual tensor product algebra structure). Corresponding morphismsare H -comodule maps which are also algebra maps.In the same way, a left H -comodule coalgebra is a coalgebra C , which is a left H -comodule and such that the coproduct and the counit of C are morphisms of H -comodules.Explicitly, this means that, for each c ∈ C , c ( − ⊗ c (0)(1) ⊗ c (0)(2) = c (1)( − c (2)( − ⊗ c (1)(0) ⊗ c (2)(0) ,ǫ C ( c ) = c ( − ǫ C ( c (0) ) . Corresponding morphisms are H -comodule maps which are also coalgebra maps. Clearly,there are right A -modules and left H -comodule versions of the above ones. Definition 2.3. A coassociative pair ( A, B, φ ) consist of a Hopf coquasigroup B and aHopf algebra A , together with a Hopf coquasigroup morphism φ : B → A , such that φ ( b (1)(1) ) ⊗ b (1)(2) ⊗ b (2) = φ ( b (1) ) ⊗ b (2)(1) ⊗ b (2)(2) b (1)(1) ⊗ φ ( b (1)(2) ) ⊗ b (2) = b (1) ⊗ φ ( b (2)(1) ) ⊗ b (2)(2) b (1)(1) ⊗ b (1)(2) ⊗ φ ( b (2) ) = b (1) ⊗ b (2)(1) ⊗ φ ( b (2)(2) ) . (2.7)Clearly, for any morphism φ between two Hopf algebras B and A , ( A, B, φ ) is a coas-sociative pair.We know for a Hopf coquasigroup, the n-th iterate coproducts ∆ n are not always equal,when n ≥ n can be equal to 0, and ∆ is identity map). However, given a coassociativepair ( A, B, φ ), there is an interesting property:
Proposition 2.4.
Let ( A, B, φ ) be a coassociative pair of a Hopf algebra A and a Hopfcoquasigroup B , and ∆ nI , ∆ nJ be both n-th iterate coproducts on B . Let ∆ nI ( b ) = b I ⊗ b I ⊗· · · ⊗ b I n +1 and ∆ nJ ( b ) = b J ⊗ b J ⊗ · · · ⊗ b J n +1 . If b I ⊗ b I ⊗ · · · ⊗ ǫ B ( b I m ) ⊗ · · · ⊗ b I n +1 = b J ⊗ b J ⊗ · · · ⊗ ǫ B ( b J m ) ⊗ · · · ⊗ b J n +1 (2.8) for ≤ m ≤ n + 1 , then we have b I ⊗ b I ⊗ · · · ⊗ φ ( b I m ) ⊗ · · · ⊗ b I n +1 = b J ⊗ b J ⊗ · · · ⊗ φ ( b J m ) ⊗ · · · ⊗ b J n +1 . Proof.
We can prove this proposition inductively. For n = 2, this is obvious by thedefinition of coassociative pair. Now we consider the case for n ≥
3. We can see bothsides of equation (2.8) are equal to the image of a ( n − n − K , andit can be written as ∆ n − K = (∆ pK ′ ⊗ ∆ qK ′′ ) ◦ ∆ for some iterate coproducts ∆ pK ′ , ∆ qK ′′ with p + q = n −
2. Assume this proposition is correct for n = N −
1. We have two cases forthe index of I m and J m :The first case is that the first index of I m and J m are the same (where first index meansthe first Sweedler index on the left, for example the first index of b (2)(2)(1) is 2). When m ≤ ( p + 2), the first index has to be 1. In this case, we can see ∆ nI = (∆ p +1 I ⊗ ∆ qK ′′ ) ◦ ∆and ∆ nJ = (∆ p +1 J ⊗ ∆ qK ′′ ) ◦ ∆, for some ( p + 1)-th iterate coproducts ∆ p +1 I and ∆ p +1 J . Then e can apply the hypotheses for the terms, whose first index is 1. For m ≥ p + 3, thesituation is the similar.The second case is that the first index of I m and J m are different, and assume the firstindex of I m is 1 and J m is 2. In this case m has to equal to p +2, and ∆ nI = (∆ p +1 E ⊗ ∆ qK ′′ ) ◦ ∆and ∆ nJ = (∆ pK ′ ⊗ ∆ q +1 F ) ◦ ∆ for some iterate ( p + 1)-th coproduct ∆ p +1 E with ( B ⊗ p ⊗ ǫ B ) ◦ ∆ p +1 E = ∆ pK ′ and iterate ( q + 1)-th coproduct ∆ q +1 F with ( ǫ B ⊗ B ⊗ q ) ◦ ∆ q +1 F = ∆ qK ′′ . Define∆ p +1 G := (∆ pK ′ ⊗ B ) ◦ ∆ and ∆ q +1 H := ( B ⊗ ∆ qK ′′ ) ◦ ∆, then we can see b I ⊗ b I ⊗ · · · ⊗ φ ( b I m ) ⊗ b I m +1 ⊗ · · · ⊗ b I n +1 = b (1) E ⊗ b (1) E ⊗ · · · ⊗ φ ( b (1) Ep +2 ) ⊗ b (2) K ′′ ⊗ · · · ⊗ b (2) K ′′ q +1 = b (1) G ⊗ b (1) G ⊗ · · · ⊗ φ ( b (1) Gp +2 ) ⊗ b (2) K ′′ ⊗ · · · ⊗ b (2) K ′′ q +1 = b (1)(1) K ′ ⊗ b (1)(1) K ′ ⊗ · · · ⊗ b (1)(1) K ′ p +1 ⊗ φ ( b (1)(2) ) ⊗ b (2) K ′′ ⊗ · · · ⊗ b (2) K ′′ q +1 = b (1) K ′ ⊗ b (1) K ′ ⊗ · · · ⊗ b (1) K ′ p +1 ⊗ φ ( b (2)(1) ) ⊗ b (2)(2) K ′′ ⊗ · · · ⊗ b (2)(2) K ′′ q +1 = b (1) K ′ ⊗ b (1) K ′ ⊗ · · · ⊗ b (1) K ′ p +1 ⊗ φ ( b (2) H ) ⊗ b (2) H ⊗ · · · ⊗ b (2) Hq +2 = b (1) K ′ ⊗ b (1) K ′ ⊗ · · · ⊗ b (1) K ′ p +1 ⊗ φ ( b (2) F ) ⊗ b (2) F ⊗ · · · ⊗ b (2) Fq +2 = b J ⊗ b J ⊗ · · · ⊗ φ ( b J m ) ⊗ b J m +1 ⊗ · · · ⊗ b J n +1 , where b (1) E ⊗ b (1) E ⊗ · · · ⊗ φ ( b (1) Ep +2 ) := ∆ p +1 E ( b (1) ) and similar for the rest. And the2nd, 6th step use the hypotheses for n ≤ N −
1, and the 4th step use the definition ofcoassociate pair. (cid:3)
From this proposition, we can make a generalisation by using the proposition twice: If b I ⊗ b I ⊗ · · · ⊗ ǫ B ( b I m ) ⊗ · · · ⊗ ǫ B ( b I m ′ ) ⊗ · · · ⊗ b I n +1 = b J ⊗ b J ⊗ · · · ⊗ ǫ B ( b J m ) ⊗ · · · ⊗ ǫ B ( b J m ′ ) ⊗ · · · ⊗ b J n +1 for 1 ≤ m < m ′ ≤ n + 1. Then we have b I ⊗ b I ⊗ · · · ⊗ φ ( b I m ) ⊗ · · · ⊗ φ ( b I m ′ ) ⊗ · · · ⊗ b I n +1 = b J ⊗ b J ⊗ · · · ⊗ φ ( b J m ) ⊗ · · · ⊗ φ ( b J m ′ ) ⊗ · · · ⊗ b J n +1 . There is a dual version of Hopf coquasigroup, which is Hopf quasigroup [8]:
Definition 2.5. A Hopf quasigroup A is a coascociative coalgebra with a coproduct∆ : A → A ⊗ A and counit ǫ : A → k , together with a unital and possibly nonassociativealgebra structure, such that the coproduct and counit are algebra map. Moreover, thereis an linear map (antipode) S A : A → A such that m ( A ⊗ m )( S A ⊗ A ⊗ A )(∆ ⊗ A ) = ǫ ⊗ A = m ( A ⊗ m )( A ⊗ S A ⊗ A )(∆ ⊗ A ) (2.9) m ( m ⊗ A )( A ⊗ S A ⊗ A )( A ⊗ ∆) = A ⊗ ǫ = m ( m ⊗ A )( A ⊗ A ⊗ S A )( A ⊗ ∆) . (2.10)A Hopf quasigroup is a Hopf algebra, if and only if it is associative.2.2. Hopf algebroids.
In the following, we will give an introduction to Hopf algebroid.For an algebra B a B -ring is a triple ( A, µ, η ). Here A is a B -bimodule with B -bimodulemaps µ : A ⊗ B A → A and η : B → A , satisfying the following associativity µ ◦ ( µ ⊗ B A ) = µ ◦ ( A ⊗ B µ ) (2.11)and unit conditions, µ ◦ ( η ⊗ B A ) = A = µ ◦ ( A ⊗ B η ) . (2.12) morphism of B -rings f : ( A, µ, η ) → ( A ′ , µ ′ , η ′ ) is an B -bimodule map f : A → A ′ ,such that f ◦ µ = µ ′ ◦ ( f ⊗ B f ) and f ◦ η = η ′ . Here for any B -bimodule M , the balancedtensor product M ⊗ B M is given by M ⊗ B M := M ⊗ M/ h m ⊗ bm ′ − mb ⊗ m ′ i m,m ′ ∈ M, b ∈ B . From [3, Lemma 2.2] there is a bijective correspondence between B -rings ( A, µ, η ) andalgebra morphisms η : B → A . Starting with a B -ring ( A, µ, η ), one obtains a multipli-cation map A ⊗ A → A by composing the canonical surjection A ⊗ A → A ⊗ B A withthe map µ . Conversely, starting with an algebra map η : B → A , a B -bilinear associa-tive multiplication µ : A ⊗ B A → A is obtained from the universality of the coequaliser A ⊗ A → A ⊗ B A which identifies an element ar ⊗ a ′ with a ⊗ ra ′ .Dually, for an algebra B a B -coring is a triple ( C, ∆ , ǫ ). Here C is a B -bimodule with B -bimodule maps ∆ : C → C ⊗ B C and ǫ : C → B , satisfying the following coassociativityand counit conditions,(∆ ⊗ B C ) ◦ ∆ = ( C ⊗ B ∆) ◦ ∆ , ( ǫ ⊗ B C ) ◦ ∆ = C = ( C ⊗ B ǫ ) ◦ ∆ . (2.13)A morphism of B -corings f : ( C, ∆ , ǫ ) → ( C ′ , ∆ ′ , ǫ ′ ) is a B -bimodule map f : C → C ′ ,such that ∆ ′ ◦ f = ( f ⊗ B f ) ◦ ∆ and ǫ ′ ◦ f = ǫ . Definition 2.6.
Given an algebra B , a left B -bialgebroid L consists of an ( B ⊗ B op )-ringtogether with a B -coring structures on the same vector space L with mutual compatibilityconditions. From what said above, an ( B ⊗ B op )-ring L is the same as an algebra map η : B ⊗ B op → L . Equivalently, one may consider the restrictions s := η ( · ⊗ B B ) : B → L and t := η (1 B ⊗ B · ) : B op → L which are algebra maps with commuting ranges in L , called the source and the target map of the ( B ⊗ B op )-ring L . Thus a ( B ⊗ B op )-ring is the same as a triple ( L , s, t ) with L an algebra and s : B → L and t : B op → L both algebra maps with commuting range.Thus, for a left B -bialgebroid L the compatibility conditions are required to be(i) The bimodule structures in the B -coring ( L , ∆ , ǫ ) are related to those of the B ⊗ B op -ring ( L , s, t ) via b ⊲ a ⊳ b ′ := s ( b ) t ( b ′ ) a for b, b ′ ∈ B, a ∈ L . (2.14)(ii) Considering L as a B -bimodule as in (2.14), the coproduct ∆ corestricts to analgebra map from L to L × B L := n X j a j ⊗ B a ′ j | X j a j t ( b ) ⊗ B a ′ j = X j a j ⊗ B a ′ j s ( b ) , for all b ∈ B o , (2.15)where L × B L is an algebra via component-wise multiplication.(iii) The counit ǫ : L → B is a left character on the B -ring ( L , s, t ), that is it satisfiesthe properties, for b ∈ B and a, a ′ ∈ L ,(1) ǫ (1 L ) = 1 B , (unitality)(2) ǫ ( s ( b ) a ) = bǫ ( a ), (left B -linearity)(3) ǫ ( as ( ǫ ( a ′ ))) = ǫ ( aa ′ ) = ǫ ( at ( ǫ ( a ′ ))), (associativity) .Similarly, we have the definition of right bialgebroid: Definition 2.7.
Given an algebra B , a right B -bialgebroid R consists of an ( B ⊗ B op )-ringtogether with a B -coring structures on the same vector space R with mutual compatibility onditions. From what said above, an ( B ⊗ B op )-ring R is the same as an algebra map η : B ⊗ B op → R . Equivalently, one may consider the restrictions s := η ( · ⊗ B B ) : B → R and t := η (1 B ⊗ B · ) : B op → R which are algebra maps with commuting ranges in R , called the source and the target map of the ( B ⊗ B op )-ring R . Thus a ( B ⊗ B op )-ring is the same as a triple ( R , s, t ) with R an algebra and s : B → R and t : B op → R both algebra maps with commuting range.Thus, for a right B -bialgebroid R the compatibility conditions are required to be(i) The bimodule structures in the B -coring ( R , ∆ , ǫ ) are related to those of the B ⊗ B op -ring ( R , s, t ) via b ⊲ a ⊳ b ′ := as ( b ′ ) t ( b ) for b, b ′ ∈ B, a ∈ R . (2.16)(ii) Considering R as a B -bimodule as in (2.16), the coproduct ∆ corestricts to analgebra map from R to R × B R := n X j a j ⊗ B a ′ j | X j s ( b ) a j ⊗ B a ′ j = X j a j ⊗ B t ( b ) a ′ j , for all b ∈ B o , (2.17)where R × B R is an algebra via component-wise multiplication.(iii) The counit ǫ : R → B is a right character on the B -ring ( R , s, t ), that is it satisfiesthe properties, for b ∈ B and a, a ′ ∈ R ,(1) ǫ (1 R ) = 1 B , (unitality)(2) ǫ ( as ( b )) = ǫ ( a ) b , (right B -linearity)(3) ǫ ( s ( ǫ ( a )) a ′ ) = ǫ ( aa ′ ) = ǫ ( t ( ǫ ( a )) a ′ ), (associativity) . Remark . Given a left B -bialgebroid L and s L , t L be the corresponding source andtarget map. If the image of s L and t L belong to the centre of H (this imply B is a com-mutative algebra, since the source map is injective), we can construct a right bialgebroidwith the same underlying k algebra L and B -coring structure on L , but a new sourceand target map s R := t L , t R := s L . Indeed, they have a same bimodule structure on L , since r ⊲ b ⊳ r ′ = s L ( r ) t L ( r ′ ) b = bt R ( r ) s R ( r ′ ). With the same bimodule structure andthe same coproduct and counit, one can get a same B -coring. By using the assumptionthat the image of s L and t L belong to the centre of H , we can check all the conditions ofbeing a right bialgebroid can be satisfied. Similarly, under the same assumption, a rightbialgebroid can induce a left bialgebroid. Since the image of B belongs to the center of H ,we can also see ǫ is an algebra map, indeed, ǫ ( bb ′ ) = ǫ ( bs ( ǫ ( b ′ ))) = ǫ ( s ( ǫ ( b ′ )) b ) = ǫ ( b ) ǫ ( b ′ ).To make a proper definition of ‘quantum’ groupoid, a left or right bialgebroid is notsufficient, since we still need to have the antipode, which play the role of the inverse of‘quantum groupoid’. Thus we have the definition of Hopf algebroid [3]: Definition 2.9.
Given two algebra B and C , a Hopf algebroid ( H L , H R , S ) consists of aleft B -bialgebroid ( H L , s L , t L , ∆ L , ǫ L ) and a right C -bialgebroid ( H R , s R , t R , ∆ R , ǫ R ), suchthat their underlying algebra H is the same. The antipode S : H → H is a linear map.Let µ L : H ⊗ s L H → H be the B -ring ( H, s L ) product induced by s L (where the tensorproduct ⊗ s L means: hs L ( b ) ⊗ s L h ′ = h ⊗ s L s L ( b ) h ), and µ R : H ⊗ s R H → H be the C -ring( H, s R ) product induced by s R , such that all the structure above satisfy the followingaxioms:(i) s L ◦ ǫ L ◦ t R = t R , t L ◦ ǫ L ◦ s R = s R , s R ◦ ǫ R ◦ t L = t L and t R ◦ ǫ R ◦ s L = s L .(ii) (∆ L ⊗ C H ) ◦ ∆ R = ( H ⊗ B ∆ R ) ◦ ∆ L and (∆ R ⊗ B H ) ◦ ∆ L = ( H ⊗ C ∆ L ) ◦ ∆ R .(iii) For b ∈ B , c ∈ C and h ∈ H , S ( t L ( b ) ht R ( c )) = s R ( c ) S ( h ) s L ( b ). iv) µ L ◦ ( S ⊗ B H ) ◦ ∆ L = s R ◦ ǫ R and µ R ◦ ( H ⊗ C S ) ◦ ∆ R = s L ◦ ǫ L . Remark . We can see axiom (i) make the coproduct ∆ L (∆ R resp.) to be a C -bimodulemap ( B -bimodule map resp.), so that (ii) is well defined. The axiom (ii) make H to beboth a H L - H R bicomodule and a H R - H L bicomodule, since the regular coaction ∆ L and∆ R commute.In order to let axiom (iv) well defined, we need axiom (iii), where S ⊗ B H : H ⊗ B H → H ⊗ s L H maps the tensor product ⊗ B into a different tensor product ⊗ s L , so that µ L make sense. Remark . In particular, given a Hopf algebroid as above ( H L , H R , S ) such that(1) B = C ;(2) s L = r R and t L = s R , with their images belong to the center of H ;(3) The coproduct and counit of H L coincide with the coproduct and counit of H R .With the help of Remark 2.8, we know the left and right bialgebroids structure compatiblewith each other. In other words, the right bialgebroid H R is constructed from H L as inRemark 2.8. Therefore, axioms (i) and (ii) are satisfied automatically. The axiom (iii)asserts that S ◦ s L = t L , and S ◦ t L = s L . We use ∆ and ǫ to denote the coproduct andcounit for both H L and H R , s to denote s L = t R , and t to denote t L = s R . So (iv) canbe written as: µ L ◦ ( S ⊗ B H ) ◦ ∆ = t ◦ ǫ, (2.18)and µ R ◦ ( H ⊗ B S ) ◦ ∆ = s ◦ ǫ. (2.19)A Hopf algebroid of this kind is denoted by ( H, s, t, ∆ , ǫ, S ).From now on we will only consider the left bialgebroids, Hopf algebroids, whose under-lying algebra B is commutative and the image of source and target maps belongs to thecenter. With the help of Remark 2.11 we have the definition of central Hopf algebroids,which is a simplification of the Hopf algebroid in Definition 2.9. Definition 2.12. A central Hopf algebroid is a left bialgebroid ( H, s, t, ∆ , ǫ ) over analgebra B , whose image of source and target maps belong to the center of H , togetherwith a linear map S : H → H , such that:(1) For any h ∈ H and b, b ′ ∈ B , S ( t ( b ) hs ( b ′ )) = t ( b ′ ) S ( h ) s ( b ) . (2.20)(2) µ L ◦ ( S ⊗ B H ) ◦ ∆ = t ◦ ǫ and µ R ◦ ( H ⊗ B S ) ◦ ∆ = s ◦ ǫ ,where µ L : H ⊗ s H → H is the B -ring ( H, s ) product induced by s , and µ R : H ⊗ t H → H is the B -ring ( H, t ) product induced by t .Let C be a B -coring, we denote ∗ : B Hom B ( C, B ) × B Hom B ( C, B ) → A the convolutionproduct ( f ∗ g )( c ) := f ( c (1) ) g ( c (2) ), where c (1) ⊗ B c (2) is the image of the coproduct of thecoring, and B Hom B ( C, B ) is the vector space of B -bimodule maps. In [2] we know B Hom B ( C, B ) is a B -ring with unit ǫ : C → B .In the following we will use lower Sweedler notation for the coproduct of Hopf coquasi-group (include Hopf algebra) and upper notation for Hopf algebroid. And whenever wesay Hopf algebroid, we mean central Hopf algebroid. . Coherent 2-group
Before talking about coherent 2-algebra, in this section we will first give an introductionto coherent 2-group [1], [4]. In next section we will see coherent Hopf 2-algebra is aquantisation of coherent 2-group.
Definition 3.1. A coherent 2-group is a monoidal category ( G, ⊗ , I, α, r, l ), where ⊗ : G ⊗ G → G is the multiplication functor, I is the unit, α : ⊗ ◦ ( ⊗ × G ) = ⇒ ⊗ ◦ ( G × ⊗ ) isthe associator, r g : g ⊗ I → g and l g : I ⊗ g → g are the right and left unitor ( α , r and l arenatural equivalence), together with a additional functor ι : G → G , natural equivalences i g : g ⊗ ι ( g ) → I and e g : ι ( g ) ⊗ g → I , such that all the morphisms are invertible andobjects are weakly invertible. In other words, the following diagrams commute.(1) ( g ⊗ h ) ⊗ ( k ⊗ l ) α g,h,k ⊗ l , , ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ (( g ⊗ h ) ⊗ k ) ⊗ l α g ⊗ h,k,l ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ α g,h,k ⊗ id l % % ❑❑❑❑❑❑❑❑❑❑❑ g ⊗ ( h ⊗ ( k ⊗ l ))( g ⊗ ( h ⊗ k )) ⊗ l α g,h ⊗ k,l / / g ⊗ (( h ⊗ k ) ⊗ l ) id g ⊗ α h,k,l sssssssssss (2) ( g ⊗ I ) ⊗ h r g ⊗ id h $ $ ■■■■■■■■■ α g,I,h / / g ⊗ ( I ⊗ h ) id g ⊗ l h z z ✉✉✉✉✉✉✉✉✉ g ⊗ h (3) ( g ⊗ ι ( g )) ⊗ g α g,ι ( g ) ,g (cid:15) (cid:15) i g ⊗ id g / / I ⊗ g l g / / g id g (cid:15) (cid:15) g ⊗ ( ι ( g ) ⊗ g ) id g ⊗ e g / / g ⊗ I r g / / g A strict 2-group is a coherent 2-group, such that all the natural transformation α , l , r , i and e are identity.There are several equivalent definitions of strict 2-group, one is called crossed module: Definition 3.2. A crossed module ( M, N, ψ, γ ) consists of two groups M , N togetherwith a group morphism ψ : M → N and a group morphism γ : N → Aut( M ) such that,denoting γ n : M → M for every n ∈ N , the following conditions are satisfied:(1) ψ ( γ n ( m )) = nψ ( m ) n − , for any n ∈ N and m ∈ M ;(2) γ ψ ( m ) ( m ′ ) = mm ′ m − , for any m, m ′ ∈ M .Another one can be given by strict 2-category: Definition 3.3.
A strict 2-group is a strict 2-categroy, with only one object and all1-arrows and 2-arrows are invertible.The equivalence of Definition 3.2 and Definition 3.3 can be found in [10].In general the objects of a coherent 2-group can be any unital set with a binary op-eration. However, in this paper we are interested in a more restrict case that both theobjects and morphisms are quasigroups: efinition 3.4. A quasigroup is a set G with a product and identity, for each element g there is a inverse g − ∈ G , such that g − ( gh ) = h and ( hg − ) g = h for any h ∈ G .As a spectial case of coherent 2-group, we are more interested in the case that theobjects of the corresponding monoidal category is a quasigroup, such that l, r, i, e areidentity natural transformations, and for any objects g, h , there are also some restrictionon the natural transformation α : α ,g,h = α g, ,h = α g,h, = id gh (3.1) α g,g − ,h = α g − ,g,h = id h = α h,g,g − = α h,g − ,g (3.2)We can see the morphisms and their composition form a groupoid. Moreover, the coherent2-group also satisfy the following property:(i) The morphisms and their tensor product form a quasigroup, because of the natu-rality of α , which we will explain later.(ii) By the definition of monoidal category, the identity map from the objects to themorphism preserve the tensor product, i.e. id g ⊗ id h = id g ⊗ h , for any objects g, h .(iii) Similarly, the source and target map of morphisms also preserve the tensor productby the definition of monoidal category.(iv) The composition and tensor product of morphisms satisfy the interchange rule:( φ ⊗ φ ) ◦ ( ψ ⊗ ψ ) = ( φ ◦ ψ ) ⊗ ( φ ◦ ψ ) . (3.3)For (i), let φ : g → h and ψ : k → l be two morphisms, we know the inverse of φ (in thesense of tensor product inverse with unit to be id ) is φ − : g − → h − . By the naturalityof α we have ψ = α h,h − ,l ◦ (( φ ⊗ φ − ) ⊗ ψ ) = ( φ ⊗ ( φ − ⊗ ψ )) ◦ α g,g − ,k = φ ⊗ ( φ − ⊗ ψ ) . Similarly, we also have ψ = ( ψ ⊗ φ ) ⊗ φ − . By the same method, we can check ( id ⊗ φ ) ⊗ ψ = φ ⊗ ψ = φ ⊗ ( ψ ⊗ id ). Thus the morphisms with their tensor product form aquasigroup.For a quasigroup, the multiplicative associator β : G → G is defined by g ( hk ) = β ( g, h, k )( gh ) k, (3.4)for any g, h, k ∈ G .The group of associative elememts N ( G ) is given by N ( G ) = { a ∈ G | ( ag ) h = a ( gh ) , g ( ah ) = ( ga ) h, ( gh ) a = g ( ha ) , ∀ g, h ∈ G } , which is called associative elements or ‘nucleus’ [8]. A quasi group is call quasiassociative ,if β have their image in N ( G ) and uN ( G ) u − ⊆ N ( G ).4. Coherent Hopf-2-algebras
In last section we explain a special case of coherent 2-group, whose morphisms andobjects form a quasigroup, since usually we are more interested in a more strict case thatall the objects have strict inverse and the unit object of the monoidal category is alsostrict. Under this condition, we have a more interesting property on the associator. Sobase on the idea of 2-arrows quantilisation we can construct a coherent quantum 2-group,which is also called coherent Hopf 2-algebra. efinition 4.1. A coherent Hopf 2-algebra consists of a commutative Hopf coquasigroup( B, m B , B , ∆ B , ǫ B , S B ) and a Hopf coquasigroup ( H, m, H , N , ǫ H , S H ), together with acentral Hopf algebroid ( H, m, H , ∆ , ǫ, S ) over B . Moreover, there is an algebra map(called coassociator) α : H → B ⊗ B ⊗ B . Denote the image of α by α ( h ) =: h ˜1 ⊗ h ˜2 ⊗ h ˜3 for any h ∈ H , and Sweedler notation for both the coproduct of Hopf coquasigroup andHopf algebroid, N ( h ) =: h (1) ⊗ h (2) , ∆( h ) =: h (1) ⊗ h (2) , such that all the structure aboveneed to satisfy the following axioms:(i) The underlying algebra of the Hopf coquasigroup ( H, m, H , N , ǫ H , S H ) and theHopf algebroid ( H, m, H , ∆ , ǫ, S ) coincide with each other.(ii) ǫ : H → B is a morphism of Hopf coquasigroups.(iii) s, t : B → H are morphisms of Hopf coquasigroups.(iv) The two coproducts ∆ and N have cocomutation relation:(∆ ⊗ ∆) ◦ N = ( H ⊗ τ ⊗ H ) ◦ ( N ⊗ B N ) ◦ ∆ , (4.1)where τ : H ⊗ H → H ⊗ H is given by τ ( h ⊗ g ) := g ⊗ h .(v) α ◦ t = (∆ B ⊗ B ) ◦ ∆ B , α ◦ s = ( B ⊗ ∆ B ) ◦ ∆ B (4.2)(vi) ǫ B ( h ˜1 )1 B ⊗ h ˜2 ⊗ h ˜3 = 1 B ⊗ ǫ ( h (1) ) ⊗ ǫ ( h (2) ) h ˜1 ⊗ ǫ B ( h ˜2 )1 B ⊗ h ˜3 = ǫ ( h (1) ) ⊗ B ⊗ ǫ ( h (2) ) h ˜1 ⊗ h ˜2 ⊗ ǫ B ( h ˜3 )1 B = ǫ ( h (1) ) ⊗ ǫ ( h (2) ) ⊗ B . (4.3)(vii) ( h ˜1 S B ( h ˜2 ) ⊗ h ˜3 = S B ( h ˜1 ) h ˜2 ⊗ h ˜3 = 1 B ⊗ ǫ ( h ) h ˜1 ⊗ S B ( h ˜2 ) h ˜3 = h ˜1 ⊗ h ˜2 S B ( h ˜3 ) = ǫ ( h ) ⊗ B . (4.4)(viii) Let ∗ be the convolution product corresponding to the Hopf algebroid coproduct,we have(( s ⊗ s ⊗ s ) ◦ α ) ∗ (( N ⊗ H ) ◦ N ) = (( H ⊗ N ) ◦ N ) ∗ (( t ⊗ t ⊗ t ) ◦ α ) (4.5)More precisely, s ( h (1) ˜1 ) h (2)(1)(1) ⊗ s ( h (1) ˜2 ) h (2)(1)(2) ⊗ s ( h (1) ˜3 ) h (2)(2) = h (1)(1) t ( h (2) ˜1 ) ⊗ h (1)(2)(1) t ( h (2) ˜2 ) ⊗ h (1)(2)(2) t ( h (2) ˜3 ) , (ix) The 3-cocycle condition:(( ǫ ⊗ α ) ◦ N ) ∗ (( B ⊗ ∆ B ⊗ B ) ◦ α ) ∗ (( α ⊗ ǫ ) ◦ N ) = (( B ⊗ B ⊗ ∆ B ) ◦ α ) ∗ ((∆ B ⊗ B ⊗ B ) ◦ α ) . (4.6)More precisely, ǫ ( h (1)(1) ) h (2) ˜1 h (3)(1) ˜1 ⊗ h (1)(2) ˜1 h (2) ˜2 (1) h (3)(1) ˜2 ⊗ h (1)(2) ˜2 h (2) ˜2 (2) h (3)(1) ˜3 ⊗ h (1)(2) ˜3 h (2) ˜3 ǫ ( h (3)(2) )= h (1) ˜1 h (2) ˜1 (1) ⊗ h (1) ˜2 h (2) ˜1 (2) ⊗ h (1) ˜3 (1) h (2) ˜2 ⊗ h (1) ˜3 (2) h (2) ˜3 . A coherent Hopf 2-algebra is called a strict Hopf 2-algebra , if H and B are coassociative( H and B are Hopf algebras), and α = ( ǫ ⊗ ǫ ⊗ ǫ ) ◦ ( N ⊗ H ) ◦ N . ow let’s explain why definition 3.1 is a quantisation of the coherent 2-group, whoseobjects is a quasigroup. First the morphisms and their composition form a groupoid,which corresponds to a Hopf algebroid, and the tensor product of objects and morphismsform a quasigroup, which corresponds to a Hopf coquasigroup.By the definition of monoidal category, we can see axiom (ii), (iii) are (iv) are natural,since the source and target map from objects to morphisms preserve the tensor product,and the identity map from objects to morphisms also preserve the tensor product. Theinterchange law corresponds to condition (iv). The source and target of the morphism α g,h,k is ( gh ) k and g ( hk ), which corresponds to condition (v). Since α ,g,h = α g, ,h = α g,h, = id gh , we have condition (vi). Because α g,g − ,h = α g − ,g,h = id h and α h,g,g − = α h,g − ,g = id h , we have the corresponding (vii). The naturality of α corresponds tocondition (viii), the pentagon corresponds to condition (ix). Here we still call definition4.1 coherent Hopf 2-algebra even though it only due to a special case of coherent 2-group. Remark . For a strict Hopf 2-algebra, we can see the morphisms s, t and ǫ are mor-phisms of Hopf algebras, and (v), (vi), (vii) are automatically satisfied. For (viii), wehave (( s ⊗ s ⊗ s ) ◦ α ) ∗ (( N ⊗ H ) ◦ N )( h )= s ( ǫ ( h (1)(1) )) h (2)(1) ⊗ s ( ǫ ( h (1)(2) )) h (2)(2) ⊗ s ( ǫ ( h (1)(3) )) h (2)(3) =( N ⊗ B ) ◦ N ( h )= h (1)(1) t ( ǫ ( h (2)(1) )) ⊗ h (1)(2) t ( ǫ ( h (2)(2) )) ⊗ h (1)(3) t ( ǫ ( h (2)(3) ))=(( H ⊗ N ) ◦ N ) ∗ (( t ⊗ t ⊗ t ) ◦ α )( b ) . For (ix) we can see the left hand side is ǫ ( h (1)(1) ) h (2) ˜1 h (3)(1) ˜1 ⊗ h (1)(2) ˜1 h (2) ˜2 (1) h (3)(1) ˜2 ⊗ h (1)(2) ˜2 h (2) ˜2 (2) h (3)(1) ˜3 ⊗ h (1)(2) ˜3 h (2) ˜3 ǫ ( h (3)(2) )= ǫ ( h (1)(1) ) ǫ ( h (2)(1) ) ǫ ( h (3)(1) ) ⊗ ǫ ( h (1)(2) ) ǫ ( h (2)(2) ) ǫ ( h (3)(2) ) ⊗ ǫ ( h (1)(3) ) ǫ ( h (2)(3) ) ǫ ( h (3)(3) ) ⊗ ǫ ( h (1)(4) ) ǫ ( h (2)(4) ) ǫ ( h (3)(4) ) , while the right hand side is h (1) ˜1 h (2) ˜1 (1) ⊗ h (1) ˜2 h (2) ˜1 (2) ⊗ h (1) ˜3 (1) h (2) ˜2 ⊗ h (1) ˜3 (2) h (2) ˜3 = ǫ ( h (1)(1) ) ǫ ( h (2)(1) ) ⊗ ǫ ( h (1)(2) ) ǫ ( h (2)(2) ) ⊗ ǫ ( h (1)(3) ) ǫ ( h (2)(3) ) ⊗ ǫ ( h (1)(4) ) ǫ ( h (2)(4) ) , using the fact that ǫ ( h (1) ) ǫ ( h (2) ) = ǫ ( s ( ǫ ( h (1) ))) ǫ ( h (2) ) = ǫ ( s ( ǫ ( h (1) )) h (2) ) = ǫ ( h ), we get theleft and right hand side of (ix) are equal. Remark . (1) In condition (iv), N ⊗ B N : H ⊗ B H → ( H ⊗ H ) ⊗ B ⊗ B ( H ⊗ H ) is welldefined since H ⊗ H has B ⊗ B -bimodule structure: ( b ⊗ b ′ ) ⊲ ( h ⊗ h ′ ) = s ( b ) h ⊗ s ( b ′ ) h ′ and ( h ⊗ h ′ ) ⊳ ( b ⊗ b ′ ) = t ( b ) h ⊗ t ( b ′ ) h ′ , for any b ⊗ b ′ ∈ B ⊗ B and h ⊗ h ′ ∈ H ⊗ H .Indeed, for any b ∈ B and h , h ′ ∈ H we have( N ⊗ B N )( h ⊗ B b ⊲ h ′ ) =( N ⊗ B N )( h ⊗ B s ( b ) h ′ )=( h (1) ⊗ h (2) ) ⊗ B ⊗ B ( s ( b ) (1) g (1) ⊗ s ( b ) (2) g (2) )=( h (1) ⊗ h (2) ) ⊗ B ⊗ B ( s ( b (1) ) g (1) ⊗ s ( b (2) ) g (2) )=( t ( b (1) ) h (1) ⊗ t ( b (2) ) h (2) ) ⊗ B ⊗ B ( g (1) ⊗ g (2) )=( t ( b ) (1) h (1) ⊗ t ( b ) (2) h (2) ) ⊗ B ⊗ B ( g (1) ⊗ g (2) )=( N ⊗ B N )( h ⊳ b ⊗ B h ′ ) , here the 2nd step use that N is an algebra map, and the 3rd step use the fact that s is coalgebra map. Clearly, H ⊗ τ ⊗ H : h ⊗ h ′ ⊗ B ⊗ B g ⊗ g ′ ( h ⊗ B g ) ⊗ ( h ′ ⊗ B g ′ )is also well defined for any h ⊗ h ′ , g ⊗ g ′ ∈ H ⊗ H . Concretely, (iv) can be writtenas h (1)(1) ⊗ B h (1)(2) ⊗ h (2)(1) ⊗ B h (2)(2) = h (1)(1) ⊗ B h (2)(1) ⊗ h (1)(2) ⊗ B h (2)(2) , (4.7)for any h ∈ H .(3) By using condition (v) and s, t being bialgebra maps, (viii) is well defined overthe balanced tensor product ⊗ B , since ( s ⊗ s ⊗ s ) ◦ α ◦ t = ( N ⊗ H ) ◦ N ◦ s and( t ⊗ t ⊗ t ) ◦ α ◦ s = ( H ⊗ N ) ◦ N ◦ t .(4) The left hand side of (4.6) is well defined since:(( ǫ ⊗ α ) ◦ N )( t ( b )) = ǫ ( t ( b (1) )) ⊗ α ( t ( b (2) )) = b (1) ⊗ b (2)(1)(1) ⊗ b (2)(1)(2) ⊗ b (2)(2) =( B ⊗ ∆ B ⊗ B ) ◦ α ( s ( b )) . ( B ⊗ ∆ B ⊗ B ) ◦ α ( t ( b )) = b (1)(1) ⊗ b (1)(2)(1) ⊗ b (1)(2)(2) ⊗ b (2) = α ( s ( b (1) )) ⊗ ǫ ( s ( b (2) ))=(( α ⊗ ǫ ) ◦ N )( s ( b )) . The right hand side of (4.6) is also well defined, indeed,(( B ⊗ B ⊗ ∆ B ) ◦ α )( t ( b )) = b (1)(1) ⊗ b (1)(2) ⊗ b (2)(1) ⊗ b (2)(2) = ((∆ B ⊗ B ⊗ B ) ◦ α )( s ( b )) . Proposition 4.4.
Given a coherent Hopf 2-algebra as in definition 4.1, the antipodeshave the following property: (i) ∆ ◦ S H = ( S H ⊗ B S H ) ◦ ∆ . (ii) S is a coalgebra map on ( H, N , ǫ H ) . In other words, N ◦ S = ( S ⊗ S ) ◦ N and ǫ H ◦ S = ǫ H . (iii) If H is commutative, S ◦ S H = S H ◦ S .Proof. For (i), let h ∈ H , S H ( h (1) ) ⊗ B S H ( h (2) )=( S H ( h (1)(1)(1) ) ⊗ B S H ( h (1)(1)(2) ))(∆( h (1)(2) S H ( h (2) )))=( S H ( h (1)(1)(1) ) ⊗ B S H ( h (1)(1)(2) ))( h (1)(2)(1) ⊗ B h (1)(2)(2) )(( S H ( h (2) )) (1) ⊗ B ( S H ( h (2) )) (2) )=( S H ( h (1)(1)(1) ) ⊗ B S H ( h (1)(2)(1) ))( h (1)(1)(2) ⊗ B h (1)(2)(2) )(( S H ( h (2) )) (1) ⊗ B ( S H ( h (2) )) (2) )=( ǫ H ( h (1)(1) ) ⊗ B ǫ H ( h (1)(2) ))(( S H ( h (2) )) (1) ⊗ B ( S H ( h (2) )) (2) )= ǫ B ( ǫ ( s ( ǫ ( h (1)(1) )) h (1)(2) ))(( S H ( h (2) )) (1) ⊗ B ( S H ( h (2) )) (2) )= ǫ H ( h (1) )(( S H ( h (2) )) (1) ⊗ B ( S H ( h (2) )) (2) )=( S H ( h )) (1) ⊗ B ( S H ( h )) (2) For (ii), on one hand we have( S ( h (1)(1) ) ⊗ S ( h (1)(2) ))( h (2)(1) ⊗ h (2)(2) )( S ( h (3) ) (1) ⊗ S ( h (3) ) (2) )=( S ( h (1)(1) ) ⊗ S ( h (1)(2) ))( h (2)(1)(1) ⊗ h (2)(1)(2) )( S ( h (2)(2) ) (1) ⊗ S ( h (2)(2) ) (2) )=( S ( h (1)(1) ) ⊗ S ( h (1)(2) ))( N ( h (2)(1) S ( h (2)(2) )))=( S ( h (1)(1) ) ⊗ S ( h (1)(2) ))( s ( ǫ ( h (2) )) (1) ⊗ s ( ǫ ( h (2) )) (2) )=( S ( h (1)(1) ) ⊗ S ( h (1)(2) ))( s ( ǫ ( h (2)(1) )) ⊗ s ( ǫ ( h (2)(2) )))= S ( h (1)(1) t ( ǫ ( h (2)(1) ))) ⊗ S ( h (1)(2) t ( ǫ ( h (2)(2) )))= S ( h (1)(1) t ( ǫ ( h (1)(2) ))) ⊗ S ( h (2)(1) t ( ǫ ( h (2)(2) )))= S ( h (1) ) ⊗ S ( h (2) ) n other hand we have( S ( h (1)(1) ) ⊗ S ( h (1)(2) ))( h (2)(1) ⊗ h (2)(2) )( S ( h (3) ) (1) ⊗ S ( h (3) ) (2) )=( S ( h (1)(1)(1) ) ⊗ S ( h (1)(1)(2) ))( h (1)(2)(1) ⊗ h (1)(2)(2) )( S ( h (2) ) (1) ⊗ S ( h (2) ) (2) )=( S ( h (1)(1)(1) ) ⊗ S ( h (1)(2)(1) ))( h (1)(1)(2) ⊗ h (1)(2)(2) )( S ( h (2) ) (1) ⊗ S ( h (2) ) (2) )= t ( ǫ ( h (1)(1) )) ⊗ t ( ǫ ( h (1)(2) ))(( S ( h (2) )) (1) ⊗ S ( h (2) ) (2) )=( t ( ǫ ( h (1) )) S ( h (2) )) (1) ⊗ ( t ( ǫ ( h (1) )) S ( h (2) )) (2) = N ( S ( s ( ǫ ( h (1) )) h (2) ))= N ( S ( h )) , and ǫ H ( S ( h )) = ǫ H ( S ( h (1) )) ǫ H ( h (2) ) = ǫ H ( S ( h (1) ) h (2) ) = ǫ H ( t ( ǫ ( h ))) = ǫ H ( h ) . For (iii), on one hand we have S ( S H ( h (1) )) S H ( h (2) ) S H ( S ( h (3) )) = S ( S H ( h (1) )) S H ( h (2) S ( h (3) ))= S ( S H ( h (1) )) S H ( s ( ǫ ( h (2) ))) = S ( S H ( h (1) ))) s ( S B ( ǫ ( h (2) )))= S ( S H ( h (1) )) t ( S B ( ǫ ( h (2) )))) = S ( S H ( h (1) ) S H ( t ( ǫ ( h (2) ))))= S ( S H ( h (1) t ( ǫ ( h (2) )))) = S ( S H ( h ))where the second step use H is commutative. On the other hand we have S ( S H ( h (1) )) S H ( h (2) ) S H ( S ( h (3) )) = S (( S H ( h (1) )) (1) )( S H ( h (1) )) (2) S H ( S ( h (2) ))= t ( ǫ ( S H ( h (1) ))) S H ( S ( h (2) )) = S H ( t ( ǫ ( h (1) ))) S H ( S ( h (2) ))= S H ( S ( s ( ǫ ( h (1) )) h (2) )) = S H ( S ( h )) , where the first step use (i) of this Proposition. (cid:3) Remark 4.1. S H ⊗ B S H is well defined, since for any b ∈ B and h, h ′ ∈ H we have( S H ⊗ B S H )( h ⊗ B b ⊲ h ′ ) =( S H ⊗ B S H )( h ⊗ B s ( b ) h ′ )= S H ( h ) ⊗ B S H ( s ( b ) h ′ ) = S H ( h ) ⊗ B S H ( h ′ ) S H ( s ( b ))= S H ( h ) ⊗ B S H ( h ′ ) s ( S B ( b )) = S H ( h ) ⊗ B s ( S B ( b )) S H ( h ′ )= t ( S B ( b )) S H ( h ) ⊗ B S H ( h ′ ) = S H ( h ) t ( S B ( b )) ⊗ B S H ( h ′ )= S H ( h ) S H ( t ( b )) ⊗ B S H ( h ′ ) = S H ( h ⊳ b ) ⊗ B S H ( h ′ )=( S H ⊗ B S H )( h ⊳ b ⊗ B h ′ ) , where the 4th and 8th steps use the fact that s, t are Hopf algebra map; the 5th and 7thsteps use the fact that the image of s, t belongs to the centre of H .5. Crossed comodule of Hopf coquasigroups
We know a strict 2-group is equivalent to a crossed module, so it is natural to constructa quantum 2-group in terms of a crossed comodule of Hopf algebra [5]. In this section wecan show that if the base algebra is commutative, a crossed comodule of Hopf algebra isa strict Hopf 2-algebra. Moreover, we will make a generalisation of it in terms of Hopfcoquasigroup, which corresponds to coherent Hopf 2-algebra.
Definition 5.1. A crossed comodule of Hopf coquasigroup consists of a coassociative pair( A, B, φ ) , such that the following conditions are satisfied: A is a left B comodule coalgebra and left B comodule algebra, that is:(i) A is a left comodule of B with coaction δ : A → B ⊗ A , here we use theSweedler index notation: δ ( a ) = a ( − ⊗ a (0) ;(ii) For any a ∈ A , a ( − ⊗ a (0)(1) ⊗ a (0)(2) = a (1)( − a (2)( − ⊗ a (1)(0) ⊗ a (2)(0) ; (5.1)(iii) For any a ∈ A , ǫ A ( a ) = a ( − ǫ A ( a (0) ); (5.2)(iv) δ is an algebra map.(2) For any b ∈ B , φ ( b ) ( − ⊗ φ ( b ) (0) = b (1)(1) S B ( b (2) ) ⊗ φ ( b (1)(2) ) = b (1) S B ( b (2)(2) ) ⊗ φ ( b (2)(1) ); (5.3)(3) For any a ∈ A , φ ( a ( − ) ⊗ a (0) = a (1) S A ( a (3) ) ⊗ a (2) . (5.4)If B is coassociative we call the crossed comodule of Hopf coquasigroup a crossedcomodule of Hopf algebra . Lemma 5.2.
Let ( A, B, φ, δ ) be a crossed comodule of Hopf coquasigroup, if B is commu-tative, then the tensor product H := A ⊗ B is a Hopf coquasigroup, with tensor productmultiplication, and unit A ⊗ B . The coproduct is defined by N ( a ⊗ b ) := a (1) ⊗ a (2)( − b (1) ⊗ a (2)(0) ⊗ b (2) , for any a ⊗ b ∈ A ⊗ B , the counit is defined by ǫ H ( a ⊗ b ) := ǫ A ( a ) ǫ B ( b ) . Theantipode is given by S H ( a ⊗ b ) := S A ( a (0) ) ⊗ S B ( a ( − b ) . Moreover, if B is coassociative,then H is a Hopf algebra.Proof. A ⊗ B is clearly an unital algebra. Now we show it is also a Hopf coquasigroup:(( id H ⊗ ǫ H ) ◦ N )( a ⊗ b ) = a (1) ⊗ a (2)( − b (1) ǫ A ( a (2)(0) ) ǫ B ( b (2) )= a (1) ⊗ a (2)( − bǫ A ( a (2)(0) )= a (1) ⊗ ǫ A ( a (2) ) b = a ⊗ b, where the 3rd step use the fact that A is comodule coalgebra,(( ǫ H ⊗ id H ) ◦ N )( a ⊗ b ) = ǫ A ( a (1) ) ǫ B ( a (2)( − b (1) ) a (2)(0) ⊗ b (2) = a ⊗ b. Now we show H is also a bialgebra: N ( aa ′ ⊗ bb ′ ) = a (1) a ′ (1) ⊗ a (2)( − a ′ (2)( − b (1) b ′ (1) ⊗ a (2)(0) a ′ (2)(0) ⊗ b (2) b ′ (2) = a (1) a ′ (1) ⊗ a (2)( − b (1) a ′ (2)( − b ′ (1) ⊗ a (2)(0) a ′ (2)(0) ⊗ b (2) b ′ (2) = N ( a ⊗ b ) N ( a ′ ⊗ b ′ ) , ere we use B is commutative algebra in the 2nd step. So N and ǫ H are clearly algebramap, thus H is a bialgebra. Now check the antipode S H for h = a ⊗ b , h (1)(1) ⊗ S H ( h (1)(2) ) h (2) = a (1)(1) ⊗ a (1)(2)( − a (2)( − b (1)(1) ⊗ S A ( a (1)(2)(0)(0) ) a (2)(0) ⊗ S B ( a (2)( − b (1)(2) ) S B ( a (1)(2)(0)( − ) b (2) = a (1)(1) ⊗ a (1)(2)( − a (2)( − b ⊗ S A ( a (1)(2)(0)(0) ) a (2)(0) ⊗ S B ( a (1)(2)(0)( − a (2)( − )= a (1)(1) ⊗ a (1)(2)( − a (2)( − b ⊗ S A ( a (1)(2)(0) ) a (2)(0) ⊗ S B ( a (1)(2)( − a (2)( − )= a (1)(1) ⊗ S A ( a (1)(2) ) ( − a (2)( − b ⊗ S A ( a (1)(2) ) (0) a (2)(0) ⊗ S B ( S A ( a (1)(2) ) ( − a (2)( − )= a ⊗ b ⊗ A ⊗ B where the second step use B is commutative, the fourth step use the fact that a ( − ⊗ S A ( a (0) ) = S A ( a ) ( − ⊗ S A ( a ) (0) , indeed, a ( − ⊗ S A ( a (0) )= a (1)(1)( − a (1)(2)( − S A ( a (2) ) ( − ⊗ S A ( a (1)(1)(0) ) a (1)(2)(0) S A ( a (2) ) (0) = a (1)( − S A ( a (2) ) ( − ⊗ S A ( a (1)(0)(1) ) a (1)(0)(2) S A ( a (2) ) (0) = a (1)( − S A ( a (2) ) ( − ⊗ ǫ A ( a (1)(0) ) S A ( a (2) ) (0) = S A ( a ) ( − ⊗ S A ( a ) (0) , where the second and third steps use the comodule coalgebra property. The rest axiomsof Hopf coquasigroup are similar. Thus H is a Hopf coquasigroup.When B is coassociative, for any a ⊗ b ∈ A ⊗ B , and we also have(( H ⊗ N ) ◦ N )( a ⊗ b ) =( H ⊗ N )( a (1) ⊗ a (2)( − b (1) ⊗ a (2)(0) ⊗ b (2) )= a (1) ⊗ a (2)( − b (1) ⊗ a (2)(0)(1) ⊗ a (2)(0)(2)( − b (2) ⊗ a (2)(0)(2)(0) ⊗ b (3) = a (1) ⊗ a (2)( − a (3)( − b (1) ⊗ a (2)(0) ⊗ a (3)(0)( − b (2) ⊗ a (3)(0)(0) ⊗ b (3) = a (1) ⊗ a (2)( − a (3)( − b (1) ⊗ a (2)(0) ⊗ a (3)( − b (2) ⊗ a (3)(0) ⊗ b (3) =( N ⊗ H )( a (1) ⊗ a (2)( − b (1) ⊗ a (2)(0) ⊗ b (2) )=(( N ⊗ H ) ◦ N )( a ⊗ b ) , where the 3rd step use the fact that A is a comodule coalgebra and the 4th step use thefact that A is a left B comodule. So ( H, N , ǫ H ) is coassociative. (cid:3) From the proof above we can also see that even if A is a Hopf coquasigroup, we canalso get a Hopf coquasigroup A ⊗ B , with the same coproduct, counit and antipole. Lemma 5.3.
Let ( A, B, φ, δ ) be a crossed comodule of Hopf coquasigroup. If B is com-mutative and the image of φ belongs to the center of A , then H = A ⊗ B is a central Hopfalgebroid over B , such that the source, target and counit (of bialgebroid) are bialgebramap.Proof. We can see H is a tensor product algebra. The source and target map s, t : B → H is given by s ( b ) := φ ( b (1) ) ⊗ b (2) , and t ( b ) := 1 A ⊗ b , for any b ∈ B . The counit map ǫ : H → B is defined to be ǫ ( a ⊗ b ) := ǫ A ( a ) b , and the left bialgebroid coproduct is defined to be∆( a ⊗ b ) := ( a (1) ⊗ B ) ⊗ B ( a (2) ⊗ b ). And the antipode given by S ( a ⊗ b ) := S A ( a ) φ ( b (1) ) ⊗ b (2) .Now we show all the structure above form a left bialgebroid structure on H . First wecan see that s, t are algebra maps, so H is a B ⊗ B -ring. Now we show H is a B -coring.Here the B -bimodule structure on H is given by b ′ ⊲ ( a ⊗ b ) ⊳ b ′′ = s ( b ′ ) t ( b ′′ )( a ⊗ b ) for ⊗ b ∈ H , b ′ , b ′′ ∈ B . So we have ǫ ( b ′ ⊲ ( a ⊗ b ) ⊳ b ′′ ) = ǫ ( s ( b ′ ) t ( b ′′ )( a ⊗ b )) = ǫ A ( φ ( b ′ (1) ) a ) b ′ (2) b ′′ b = ǫ B ( b ′ (1) ) ǫ A ( a ) b ′ (2) b ′′ b = b ′ ǫ ( a ⊗ b ) b ′′ , where we use φ is a bialgebra map in the 3rd step. Clearly, ǫ is an algebra map from A ⊗ B to B . We also have( ǫ ⊗ ǫ )( N ( a ⊗ b )) =( ǫ ⊗ ǫ )( a (1) ⊗ a (2)( − b (1) ⊗ a (2)(0) ⊗ b (2) )= ǫ A ( a (1) ) a (2)( − b (1) ⊗ ǫ A ( a (2)(0) ) b (2) = a ( − b (1) ⊗ ǫ A ( a (0) ) b (2) = ǫ A ( a ) b (1) ⊗ b (2) =∆ B ( ǫ ( a ⊗ b )) , where the 3rd step use that A is comodule algebra. So we can see that ǫ is a bialgebramap from A ⊗ B to B . We can also show s and t are also bialgebra maps, since we have N ( s ( b )) = N ( φ ( b (1) ) ⊗ b (2) )= φ ( b (1) ) (1) ⊗ φ ( b (1) ) (2)( − b (2)(1) ⊗ φ ( b (1) ) (2)(0) ⊗ b (2)(2) = φ ( b (1)(1) ) ⊗ φ ( b (1)(2) ) ( − b (2)(1) ⊗ φ ( b (1)(2) ) (0) ⊗ b (2)(2) = φ ( b (1)(1) ) ⊗ φ ( b (1)(2)(1) ) ( − b (1)(2)(2) ⊗ φ ( b (1)(2)(1) ) (0) ⊗ b (2)(2) = φ ( b (1)(1) ) ⊗ b (1)(2)(1)(1)(1) S B ( b (1)(2)(1)(2) ) b (1)(2)(2) ⊗ φ ( b (1)(2)(1)(1)(2) ) ⊗ b (2)(2) = φ ( b (1)(1) ) ⊗ b (1)(2)(1) ⊗ φ ( b (1)(2)(2) ) ⊗ b (2)(2) = φ ( b (1)(1) ) ⊗ b (1)(2) ⊗ φ ( b (2)(1) ) ⊗ b (2)(2) =( s ⊗ s )(∆ B ( b ))where the 4th step use the axiom of coassociative pairing, the 5th step use (5.3). We alsohave N ( t ( b )) = N (1 ⊗ b ) = 1 ⊗ b (1) ⊗ ⊗ b (2) = ( t ⊗ t )(∆ B ( b )) , for any b ∈ B . So s and t are algebra maps. We can also show ∆ is a B -bimodule map:∆( b ′ ⊲ ( a ⊗ b )) =∆( φ ( b ′ (1) ) a ⊗ b ′ (2) b )=( φ ( b ′ (1) ) (1) a (1) ⊗ ⊗ B ( φ ( b ′ (1) ) (2) a (2) ⊗ b ′ (2) b )=( φ ( b ′ (1)(1) ) a (1) ⊗ ⊗ B ( φ ( b ′ (1)(2) ) a (2) ⊗ b ′ (2) b )=( φ ( b ′ (1) ) a (1) ⊗ ⊗ B ( φ ( b ′ (2)(1) ) a (2) ⊗ b ′ (2)(2) b )=( φ ( b ′ (1) ) a (1) ⊗ ⊗ B s ( b ′ (2) )( a (2) ⊗ b )=( φ ( b ′ (1) ) a (1) ⊗ t ( b ′ (2) ) ⊗ B ( a (2) ⊗ b )=( φ ( b ′ (1) ) a (1) ⊗ b ′ (2) ) ⊗ B ( a (2) ⊗ b )= b ′ ⊲ ∆( a ⊗ b ) , where the fourth step use the axiom of coassociative pairing. We also have∆(( a ⊗ b ) ⊳ b ′ ) =∆( a ⊗ bb ′ ) = ( a (1) ⊗ ⊗ B ( a (2) ⊗ bb ′ ) = ∆( a ⊗ b ) ⊳ b ′ for any a ⊗ b ∈ A ⊗ B and b ′ ∈ B . ∆ is clearly coassociative, and we also have( H ⊗ B ǫ ) ◦ ∆( a ⊗ b ) = a ⊗ b ⊗ B H , and ( ǫ ⊗ B H ) ◦ ∆( a ⊗ b ) = ǫ A ( a (1) ) ⊗ ⊗ B ( a (2) ⊗ b ) = 1 H ⊗ B a ⊗ b y straightforward compution. Up to now we already shown that H is a B -coring. Clearly,∆ is also an algebra map from H to H × B H . Given a ⊗ b, a ′ ⊗ b ′ ∈ H , we have ǫ (( a ⊗ b )( a ′ ⊗ b ′ )) = ǫ A ( aa ′ ) bb ′ , and ǫ (( a ⊗ b ) t ( ǫ ( a ′ ⊗ b ′ ))) = ǫ ( a ⊗ bǫ A ( a ′ ) b ′ ) = ǫ A ( aa ′ ) bb ′ .We also have ǫ (( a ⊗ b ) s ( ǫ ( a ′ ⊗ b ′ ))) = ǫ (( a ⊗ b )( φ ( ǫ A ( a ′ ) b ′ (1) ) ⊗ b ′ (2) ))= ǫ A ( aa ′ ) bb ′ , thus ǫ is a left character and H is therefore a left bialgebroid. Since φ belong to thecenter of A , we can check that S ( t ( b ′ )( a ⊗ b ) s ( b ′′ )) = t ( b ′′ ) S ( a ⊗ b ) s ( b ′ ) for any a ⊗ b ∈ H and b ′ , b ′′ ∈ B : S ( t ( b ′ )( a ⊗ b ) s ( b ′′ )) = S ( aφ ( b ′′ (1) ) ⊗ b ′ bb ′′ (2) )= S A ( aφ ( b ′′ (1) )) φ ( b ′ (1) b (1) b ′′ (2)(1) ) ⊗ b ′ (2) b (2) b ′′ (2)(2) = S A ( a ) φ ( b (1) ) φ ( b ′ (1) ) ⊗ b ′′ b (2) b ′ (2) = t ( b ′′ ) S ( a ⊗ b ) s ( b ′ ) , where the 3rd step use B is commutative and its image of φ belongs to the center of A .Now we can see that S ( a (1) ⊗ a (2) ⊗ b ) = S A ( a (1) ) a (2) ⊗ b = ( t ◦ ǫ )( a ⊗ b ) , and ( a (1) ⊗ S ( a (2) ⊗ b ) = a (1) S A ( a (2) ) φ ( b (1) ) ⊗ b (2) = ( s ◦ ǫ )( a ⊗ b )So H is a Hopf algebroid. (cid:3) Lemma 5.4.
For A , B and H = A ⊗ B as above, we have (∆ ⊗ ∆) ◦ N = ( H ⊗ τ ⊗ H ) ◦ ( N ⊗ B N ) ◦ ∆ . Proof.
Let h = a ⊗ b ∈ H , on one hand we have(∆ ⊗ ∆) ◦ N ( h ) = a (1) ⊗ ⊗ B a (2) ⊗ a (3)( − b (1) ⊗ a (3)(0)(1) ⊗ ⊗ B a (3)(0)(2) ⊗ b (2) , on the other hand( H ⊗ τ ⊗ H ) ◦ ( N ⊗ B N ) ◦ (∆( h ))= a (1) ⊗ a (2)( − ⊗ B a (3) ⊗ a (4)( − b (1) ⊗ a (2)(0) ⊗ ⊗ B a (4)(0) ⊗ b (2) = a (1) ⊗ ⊗ B φ ( a (2)( − ) a (3) ⊗ a (2)( − a (4)( − b (1) ⊗ a (2)(0) ⊗ ⊗ B a (4)(0) ⊗ b (2) = a (1) ⊗ ⊗ B φ ( a (2)( − ) a (3) ⊗ a (2)(0)( − a (4)( − b (1) ⊗ a (2)(0)(0) ⊗ ⊗ B a (4)(0) ⊗ b (2) = a (1) ⊗ ⊗ B a (2)(1) S A ( a (2)(3) ) a (3) ⊗ a (2)(2)( − a (4)( − b (1) ⊗ a (2)(2)(0) ⊗ ⊗ B a (4)(0) ⊗ b (2) = a (1) ⊗ ⊗ B a (2) ⊗ a (3)( − a (4)( − b (1) ⊗ a (3)(0) ⊗ ⊗ B a (4)(0) ⊗ b (2) = a (1) ⊗ ⊗ B a (2) ⊗ a (3)( − b (1) ⊗ a (3)(0)(1) ⊗ ⊗ B a (3)(0)(2) ⊗ b (2) , where the second step use the balance tensor product over B , the fourth step use (5.4),the last step use the that A is comodule coalgebra of B . (cid:3) Since for strict Hopf 2-algebra all the axioms of coassociator are trivial, we can conclude:
Theorem 5.5.
Let ( A, B, φ, δ ) be a crossed comodule of Hopf algebra, if B is commutativeand the image of φ belongs to the center of B , then H = A ⊗ B is a strict Hopf 2-algebra ith the structure maps given by: N ( a ⊗ b ) = a (1) ⊗ a (2)( − b (1) ⊗ a (2)(0) ⊗ b (2) ,ǫ H ( a ⊗ b ) = ǫ A ( a ) ǫ B ( b ) ,S H ( a ⊗ b ) = S A ( a (0) ) ⊗ S B ( a ( − b ) ,s ( b ) = φ ( b (1) ) ⊗ b (2) ,t ( b ) =1 ⊗ b, ∆( a ⊗ b ) = a (1) ⊗ ⊗ B a (2) ⊗ b,ǫ ( a ⊗ b ) = ǫ A ( a ) b,S ( a ⊗ b ) = S A ( a ) φ ( b (1) ) ⊗ b (2) . Here are some examples of crossed comodule of Hopf algebras:
Example . Let φ : B → A be a surjective morphism of Hopf algebra, A is commutative,such that for ∀ i ∈ I := ker ( φ ), i (1) S B ( i (3) ) ⊗ i (2) ∈ B ⊗ I . Thus we can define δ : A → B ⊗ A by δ ([ a ]) := a (1) S B ( a (3) ) ⊗ [ a (2) ], where [ a ] denote the image of φ . We can see that A is acomdule coalgebra and comodule algebra of B , since A is commutative. Moreover, (5.3)and (5.4) are also satisfied. Therefore ( A, B, φ, δ ) is a crossed comodule of Hopf algebra.
Example . Let
G ֒ → H ։ E be a short exact sequence of Hopf algebras with injection i : G → H , sujection π : H → E and B is commutative, such that h (1) ⊗ π ( h (2) ) = h (2) ⊗ π ( h (1) ) for ∀ h ∈ H . For any k ∈ G , we can see k (1) S H ( k (3) ) ⊗ k (2) ∈ i ( G ) ⊗ H , since k (1) S H ( k (3) ) ⊗ k (2) ∈ ker ( π ⊗ H ). Therefore, we can define a coaction δ : H → G ⊗ H by δ ( h ) := h (1) S H ( h (3) ) ⊗ h (2) (here we coincide G and its image of i ). We can see the H isa G -comodule algebra and comodule coalgebra. (5.3) and (5.4) are also satisfied. Thus( H, G, i, δ ) is a crossed module of Hopf algebra.
Example . Let A , B be two Hopf algebras and A is cocommutative, such that A is a comodule alebra and comoudle coalgebra of B . Define φ : B → A by φ ( b ) := ǫ B A . Clearly, φ is a Hopf algebra map, (5.3) and (5.4) are also satisfied, since A iscocommutative. Therefore ( A, B, φ, δ ) is a crossed comodule of Hopf algebra.
Example . Let A , B be two cocomutative Hopf algebras, and φ : B → A be a Hopfalgebra map. Define δ : A → B ⊗ A by δ ( a ) := 1 B ⊗ a . Clearly, A is a comodule algebraand comodule coalgebra of B , and (5.3) and (5.4) are also satisfied, since A and B arecocommutative. Therefore ( A, B, φ, δ ) is a crossed comodule of Hopf algebra.6.
Quasi coassociative Hopf coquasigroup
In this section we will construct a crossed comodule of Hopf coquasigroup as a general-isation of Exmaple 5.6, and then construct a coherent Hopf 2-algebra. First we define aquasi coassociative Hopf coquasigroup, which can be viewed as quantum quasiassociativequasigroup.
Definition 6.1.
Let (
C, B, φ ) be a coassociative pair, we call the Hopf coquasigroup B quasi coassociative corresponding to ( C, B, φ ), if: • φ : B → C is sujective morphism of Hopf coquasigroup. • For ∀ i ∈ I := ker ( φ ), ( i (1)(2) ⊗ i (1)(1) S B ( i (2) ) ∈ I B ⊗ B,i (2)(1) ⊗ i (1) S B ( i (2)(2) ) ∈ I B ⊗ B. (6.1) I ⊆ ker ( β ), where β : B → B ⊗ B ⊗ B is the coassociator (2.3).If B is a quasi coassociative, by (6.1) there is a linear map Ad : C → B ⊗ C , Ad ([ c ]) := c (1) S B ( c (3) ) ⊗ [ c (2) ] := c (1)(1) S B ( c (2) ) ⊗ [ c (2)(1) ] = c (1) S B ( c (2)(2) ) ⊗ [ c (1)(2) ], where [ c ] := φ ( c ) ∈ C ,since φ is surjective. We can see in the following that Ad is a comodule map and C is acomodule coalgebra of B . For any b ∈ B , there is an important result of Proposition 2.4: b (1)(1) S B ( b (1)(3) ) b (2) ⊗ [ b (1)(2) ] = b (1)(1)(1) S B ( b (1)(2) ) b (2) ⊗ [ b (1)(1)(2) ] = b (1) ⊗ [ b (2) ] . (6.2)Similarly, S B ( b (1) ) b (2)(1) S B ( b (2)(3) ) ⊗ [ b (2)(2) ] = S B ( b (1) ) ⊗ b (2) . (6.3)Since I ∈ ker ( φ ), there is a linear map ˜ β : C → B ⊗ B ⊗ B given by ˜ β ([ b ]) := β ( b ), whichis denoted by β ( b ) = b ˆ1 ⊗ b ˆ2 ⊗ b ˆ3 . Lemma 6.2.
Let B be a quasi coassociative Hopf coquasigroup corresponding to ( C, B, φ ) .If B is commutative, then the Hopf coquasigroup B and Hopf algebra C together with themaps Ad : C → B ⊗ C and the quotient map φ : B → C is a crossed comodule of Hopfcoquasigroup.Proof. We first prove Ad is a comodule map: c (1)(1) S B ( c (3)(2) ) ⊗ c (1)(2) S B ( c (3)(1) ) ⊗ [ c (2) ] = c (1) S B ( c (3) ) ⊗ c (2)(1) S B ( c (2)(3) ) ⊗ [ c (2)(2) ] , (6.4)which is sufficient to show c (1)(1)(1) S B ( c (1)(3)(2) ) c (2) ⊗ c (1)(1)(2) S B ( c (1)(3)(1) ) ⊗ [ c (1)(2) ]= c (1)(1) S B ( c (1)(3) ) c (2) ⊗ c (1)(2)(1) S B ( c (1)(2)(3) ) ⊗ [ c (1)(2)(2) ] . On one hand we have c (1)(1)(1) S B ( c (1)(3)(2) ) c (2) ⊗ c (1)(1)(2) S B ( c (1)(3)(1) ) ⊗ [ c (1)(2) ]= c (1)(1)(1) S B ( c (1)(3) ) (1) c (2)(1)(1) ⊗ c (1)(1)(2) S B ( c (1)(3) ) (2) c (2)(1)(2) S B ( c (2)(2) ) ⊗ [ c (1)(2) ]= c (1)(1)(1)(1) S B ( c (1)(1)(3) ) (1) c (1)(2)(1) ⊗ c (1)(1)(1)(2) S B ( c (1)(1)(3) ) (2) c (1)(2)(2) S B ( c (2) ) ⊗ [ c (1)(1)(2) ]= c (1)(1)(1) ⊗ c (1)(1)(2) S B ( c (2) ) ⊗ [ c (1)(2) ]= c (1)(1) ⊗ c (1)(2) S B ( c (2)(2) ) ⊗ [ c (2)(1) ] , where the first step use the definition of Hopf coquasigroup, the second and last step useProposition (2.4), the third step use (6.2). On the other hand we have c (1)(1) S B ( c (1)(3) ) c (2) ⊗ c (1)(2)(1) S B ( c (1)(2)(3) ) ⊗ [ c (1)(2)(2) ]= c (1) ⊗ c (2)(1) S B ( c (2)(3) ) ⊗ [ c (2)(2) ]= c (1)(1)(1) ⊗ c (1)(1)(2) S B ( c (1)(2) ) c (2)(1) S B ( c (2)(3) ) ⊗ [ c (2)(2) ]= c (1)(1) ⊗ c (1)(2) S B ( c (2)(1) ) c (2)(2)(1) S B ( c (2)(2)(3) ) ⊗ [ c (2)(2)(2) ]= c (1)(1) ⊗ c (1)(2) S B ( c (2)(2) ) ⊗ [ c (2)(1) ] , where the first and last step use (6.2) and (6.3), the second step use the definition of Hopfcoquasigroup, the third step use Proposition (2.4). So we have c (1)(1) S B ( c (3)(2) ) ⊗ c (1)(2) S B ( c (3)(1) ) ⊗ [ c (2) ]= c (1)(1)(1)(1) S B ( c (1)(1)(3)(2) ) c (1)(2) S B ( c (2) ) ⊗ c (1)(1)(1)(2) S B ( c (1)(1)(3)(1) ) ⊗ [ c (1)(1)(2) ]= c (1)(1)(1) S B ( c (1)(1)(3) ) c (1)(2) S B ( c (2) ) ⊗ c (1)(1)(2)(1) S B ( c (1)(1)(2)(3) ) ⊗ [ c (1)(1)(2)(2) ]= c (1) S B ( c (3) ) ⊗ c (2)(1) S B ( c (2)(3) ) ⊗ [ c (2)(2) ] , here the first and last step use the definition of Hopf coquasigroup, and second stepuses the relation we just proved above. We can see that Ad is an algebra map, since B is commutative. Now let’s show Ad is a comodule coalgebra map: On one hand[ c ] ( − ⊗ [ c ] (0)(1) ⊗ [ c ] (0)(2) = c (1) S B ( c (3) ) ⊗ [ c (2)(1) ] ⊗ [ c (2)(2) ] . On the other hand [ c ] (1)( − [ c ] (2)( − ⊗ [ c ] (1)(0) ⊗ [ c ] (2)(0) = c (1)(1) S B ( c (1)(3) ) c (2)(1) S B ( c (2)(3) ) ⊗ [ c (1)(2) ] ⊗ [ c (2)(2) ]= c (1) S B ( c (3) ) ⊗ [ c (2)(1) ] ⊗ [ c (2)(2) ] , where the last step uses Proposition 2.4. And ǫ C ([ c ]) = ǫ B ( c ) = c (1) S B ( c (3) ) ǫ B ( c (2) ) = [ c ] ( − ǫ C ([ c ] (0) ) . (5.3) and (5.4) are given by the construction of Ad . (cid:3) Now we want to construct a coherent 2-group in terms of the crossed comodule (
C, B, φ, Ad )we just considered above. And in the following we always assume B to be commutative.Compare to Definition 4.1, the first Hopf coquasigroup is B . And the second Hopf co-quasigroup is H := C ⊗ B , with canonical unit and factorwise multiplication. And thecoproduct, counit and antipode are defined in the following: N ([ c ] ⊗ b ) := [ c ] (1) ⊗ [ c ] (2)( − b (1) ⊗ [ c ] (2)(0) ⊗ b (2) (6.5) ǫ H ([ c ] ⊗ b ) := ǫ B ( c ) ǫ B ( b ) (6.6) S H ([ c ] ⊗ b ) := [ S B ( c (1)(2) )] ⊗ S B ( c (1) ) c (2)(2) S B ( b ) = S C ( c (0) ) ⊗ S B ( c ( − b ) (6.7)By Lemma 5.2 and Lemma 6.2, we have H = C ⊗ B is a Hopf coquasigroup.Then we constructure a Hopf algebroid structure on H by Lemma 5.3 with the sourceand target maps s, t : B → H given by s ( b ) := [ b (1) ] ⊗ b (2) , and t ( b ) := 1 C ⊗ b, (6.8)for any b ∈ B . The Hopf algebroid coproduct is given by∆([ c ] ⊗ b ) := ([ c (1) ] ⊗ B ) ⊗ B ([ c (2) ] ⊗ b ) , (6.9)and counit is given by ǫ ([ c ] ⊗ b ) := ǫ B ( c ) b. (6.10)The antipode is S ([ c ] ⊗ b ) := [ S B ( c ) b (1) ] ⊗ b (2) . (6.11)Using Lemma 5.4, we can also get the cocomutation relation of coproducts:(∆ ⊗ ∆) ◦ N = ( H ⊗ τ ⊗ H ) ◦ ( N ⊗ B N ) ◦ ∆ . Now let’s deal with the coassociator α : H → B ⊗ B ⊗ B , which is given by α ([ c ] ⊗ b ) := β ( c )( b (1)(1) ⊗ b (1)(2) ⊗ b (2) ) = c ˆ1 b (1)(1) ⊗ c ˆ2 b (1)(2) ⊗ c ˆ3 b (2) . (6.12)This is well define, since B is quasi coassociative with I ⊆ ker ( β ). By using (2.4) we cancheck condition (v): α ( t ( b )) = b (1)(1) ⊗ b (1)(2) ⊗ b (2) nd α ( s ( b )) =( b (1) ) ˆ1 b (2)(1)(1) ⊗ ( b (1) ) ˆ2 b (2)(1)(2) ⊗ ( b (1) ) ˆ3 b (2)(2) = b (1) ⊗ b (2)(1) ⊗ b (2)(2) . For condition (vi) of Definition 3.1, we can see ǫ B ( h ˜1 )1 B ⊗ h ˜2 ⊗ h ˜3 = 1 B ⊗ c (1)(1) S B ( c (2) ) (1) b (1) ⊗ c (1)(2) S B ( c (2) ) (2) b (2) = 1 B ⊗ ǫ ( h (1) ) ⊗ ǫ ( h (2) ) , where h = [ c ] ⊗ b , and the rest of condition (vi) can be checked similarly. For condition(vii) we can see h ˜1 S B ( h ˜2 ) ⊗ h ˜3 = c (1)(1) S B ( c (2) ) (1)(1) S B ( c (1)(2)(1) S B ( c (2) ) (1)(2) ) ⊗ c (1)(2)(2) S B ( c (2) ) (2) b = c (1)(1) S B ( c (1)(2)(1) ) S B ( c (2) ) (1)(1) S B ( S B ( c (2) ) (1)(2) ) ⊗ c (1)(2)(2) S B ( c (2) ) (2) b =1 B ⊗ ǫ ( h ) , where we use the fact that B is commutative and the rest of (vii) are the same. Now let’scheck (viii) and (ix). Lemma 6.3.
For any h ∈ H , we have s ( h (1) ˜1 ) h (2)(1)(1) ⊗ s ( h (1) ˜2 ) h (2)(1)(2) ⊗ s ( h (1) ˜3 ) h (2)(2) = h (1)(1) t ( h (2) ˜1 ) ⊗ h (1)(2)(1) t ( h (2) ˜2 ) ⊗ h (1)(2)(2) t ( h (2) ˜3 ) . Proof.
Let h = [ c ] ⊗ b , the left hand side of the equation above is: s (( c (1) ) ˆ1 )[ c (2)(1)(1) ] ⊗ [ c (2)(1)(2) ] ( − [ c (2)(2) ] ( − b (1)(1) ⊗ s (( c (1) ) ˆ2 )[ c (2)(1)(2) ] (0) ⊗ [ c (2)(2) ] ( − b (1)(2) ⊗ s (( c (1) ) ˆ3 )[ c (2)(2) ] (0) ⊗ b (2) , while the right hand side of the equation is[ c (1)(1) ] ⊗ [ c (1)(2) ] ( − ( c (2) ) ˆ1 b (1)(1) ⊗ [ c (1)(2) ] (0)(1) ⊗ [ c (1)(2) ] (0)(2)( − ( c (2) ) ˆ2 b (1)(2) ⊗ [ c (1)(2) ] (0)(2)(0) ⊗ ( c (2) ) ˆ3 b (2) . So it is sufficient to show s (( c (1) ) ˆ1 )[ c (2)(1)(1) ] ⊗ [ c (2)(1)(2) ] ( − [ c (2)(2) ] ( − ⊗ s (( c (1) ) ˆ2 )[ c (2)(1)(2) ] (0) ⊗ [ c (2)(2) ] ( − ⊗ s (( c (1) ) ˆ3 )[ c (2)(2) ] (0) ⊗ c (1)(1) ] ⊗ [ c (1)(2) ] ( − ( c (2) ) ˆ1 ⊗ [ c (1)(2) ] (0)(1) ⊗ [ c (1)(2) ] (0)(2)( − ( c (2) ) ˆ2 ⊗ [ c (1)(2) ] (0)(2)(0) ⊗ ( c (2) ) ˆ3 . By the definition of Hopf coquasigroup, this is equivalent to s (( c (1)(1)(1) ) ˆ1 )[ c (1)(1)(2)(1)(1) ] ⊗ [ c (1)(1)(2)(1)(2) ] ( − [ c (1)(1)(2)(2) ] ( − c (1)(2)(1)(1) S B ( c (2) ) (1)(1) ⊗ s (( c (1)(1)(1) ) ˆ2 )[ c (1)(1)(2)(1)(2) ] (0) ⊗ [ c (1)(1)(2)(2) ] ( − c (1)(2)(1)(2) S B ( c (2) ) (1)(2) ⊗ s (( c (1)(1)(1) ) ˆ3 )[ c (1)(1)(2)(2) ] (0) ⊗ c (1)(2)(2) S B ( c (2) ) (2) =[ c (1)(1)(1)(1) ] ⊗ [ c (1)(1)(1)(2) ] ( − ( c (1)(1)(2) ) ˆ1 c (1)(2)(1)(1) S B ( c (2) ) (1)(1) ⊗ [ c (1)(1)(1)(2) ] (0)(1) ⊗ [ c (1)(1)(1)(2) ] (0)(2)( − ( c (1)(1)(2) ) ˆ2 c (1)(2)(1)(2) S B ( c (2) ) (1)(2) ⊗ [ c (1)(1)(1)(2) ] (0)(2)(0) ⊗ ( c (1)(1)(2) ) ˆ3 c (1)(2)(2) S B ( c (2) ) (2) . hus it is sufficient to show s (( c (1)(1) ) ˆ1 )[ c (1)(2)(1)(1) ] ⊗ [ c (1)(2)(1)(2) ] ( − [ c (1)(2)(2) ] ( − c (2)(1)(1) ⊗ s (( c (1)(1) ) ˆ2 )[ c (1)(2)(1)(2) ] (0) ⊗ [ c (1)(2)(2) ] ( − c (2)(1)(2) ⊗ s (( c (1)(1) ) ˆ3 )[ c (1)(2)(2) ] (0) ⊗ c (2)(2) =[ c (1)(1)(1) ] ⊗ [ c (1)(1)(2) ] ( − ( c (1)(2) ) ˆ1 c (2)(1)(1) ⊗ [ c (1)(1)(2) ] (0)(1) ⊗ [ c (1)(1)(2) ] (0)(2)( − ( c (1)(2) ) ˆ2 c (2)(1)(2) ⊗ [ c (1)(1)(2) ] (0)(2)(0) ⊗ ( c (1)(2) ) ˆ3 c (2)(2) . The left hand side is s (( c (1)(1) ) ˆ1 )[ c (1)(2)(1)(1) ] ⊗ [ c (1)(2)(1)(2) ] ( − [ c (1)(2)(2) ] ( − c (2)(1)(1) ⊗ s (( c (1)(1) ) ˆ2 )[ c (1)(2)(1)(2) ] (0) ⊗ [ c (1)(2)(2) ] ( − c (2)(1)(2) ⊗ s (( c (1)(1) ) ˆ3 )[ c (1)(2)(2) ] (0) ⊗ c (2)(2) = s (( c (1) ) ˆ1 )[ c (2)(1)(1) ] ⊗ [ c (2)(1)(2) ] ( − [ c (2)(2)(1)(1) ] ( − c (2)(2)(1)(2)(1) ⊗ s (( c (1) ) ˆ2 )[ c (2)(1)(2) ] (0) ⊗ [ c (2)(2)(1)(1) ] ( − c (2)(2)(1)(2)(2) ⊗ s (( c (1) ) ˆ3 )[ c (2)(2)(1)(1) ] (0) ⊗ c (2)(2)(2) = s (( c (1) ) ˆ1 )[ c (2)(1)(1)(1) ] ⊗ [ c (2)(1)(1)(2) ] ( − c (2)(1)(2)(1) ⊗ s (( c (1) ) ˆ2 )[ c (2)(1)(1)(2) ] (0) ⊗ c (2)(1)(2)(2) ⊗ s (( c (1) ) ˆ3 )[ c (2)(2)(1) ] ⊗ c (2)(2)(2) = s (( c (1) ) ˆ1 )[ c (2)(1)(1) ] ⊗ [ c (2)(1)(2)(1)(1) ] ( − c (2)(1)(2)(1)(2) ⊗ s (( c (1) ) ˆ2 )[ c (2)(1)(2)(1)(1) ] (0) ⊗ c (2)(1)(2)(2) ⊗ s (( c (1) ) ˆ3 )[ c (2)(2)(1) ] ⊗ c (2)(2)(2) = s (( c (1) ) ˆ1 )[ c (2)(1)(1) ] ⊗ c (2)(1)(2)(1)(1) ⊗ s (( c (1) ) ˆ2 )[ c (2)(1)(2)(1)(2) ] ⊗ c (2)(1)(2)(2) ⊗ s (( c (1) ) ˆ3 )[ c (2)(2)(1) ] ⊗ c (2)(2)(2) =[(( c (1) ) ˆ1 ) (1) c (2)(1)(1)(1) ] ⊗ (( c (1) ) ˆ1 ) (2) c (2)(1)(1)(2) ⊗ [(( c (1) ) ˆ2 ) (1) c (2)(1)(2)(1) ] ⊗ (( c (1) ) ˆ2 ) (2) c (2)(1)(2)(2) ⊗ [(( c (1) ) ˆ3 ) (1) c (2)(2)(1) ] ⊗ (( c (1) ) ˆ3 ) (2) c (2)(2)(2) =[ c (1)(1) ] ⊗ c (1)(2) ⊗ [ c (2)(1)(1) ] ⊗ c (2)(1)(2) ⊗ [ c (2)(2)(1) ] ⊗ c (2)(2)(2) where the 1st, 3rd, 5th step use Proposition 2.4, the 2nd, 4th step use (6.2), the last stepuse (2.4). The right hand side is:[ c (1)(1)(1) ] ⊗ [ c (1)(1)(2) ] ( − ( c (1)(2) ) ˆ1 c (2)(1)(1) ⊗ [ c (1)(1)(2) ] (0)(1) ⊗ [ c (1)(1)(2) ] (0)(2)( − ( c (1)(2) ) ˆ2 c (2)(1)(2) ⊗ [ c (1)(1)(2) ] (0)(2)(0) ⊗ ( c (1)(2) ) ˆ3 c (2)(2) =[ c (1)(1) ] ⊗ [ c (1)(2) ] ( − ( c (2)(1) ) ˆ1 c (2)(2)(1)(1) ⊗ [ c (1)(2) ] (0)(1) ⊗ [ c (1)(2) ] (0)(2)( − ( c (2)(1) ) ˆ2 c (2)(2)(1)(2) ⊗ [ c (1)(2) ] (0)(2)(0) ⊗ ( c (2)(1) ) ˆ3 c (2)(2)(2) =[ c (1)(1) ] ⊗ [ c (1)(2) ] ( − c (2)(1) ⊗ [ c (1)(2) ] (0)(1) ⊗ [ c (1)(2) ] (0)(2)( − c (2)(2)(1) ⊗ [ c (1)(2) ] (0)(2)(0) ⊗ c (2)(2)(2) =[ c (1) ] ⊗ [ c (2)(1)(1) ] ( − c (2)(1)(2) ⊗ [ c (2)(1)(1) ] (0)(1) ⊗ [ c (2)(1)(1) ] (0)(2)( − c (2)(2)(1) ⊗ [ c (2)(1)(1) ] (0)(2)(0) ⊗ c (2)(2)(2) =[ c (1) ] ⊗ c (2)(1)(1) ⊗ [ c (2)(1)(2)(1) ] ⊗ [ c (2)(1)(2)(2) ] ( − c (2)(2)(1) ⊗ [ c (2)(1)(2)(2) ] (0) ⊗ c (2)(2)(2) =[ c (1) ] ⊗ c (2)(1) ⊗ [ c (2)(2)(1) ] ⊗ [ c (2)(2)(2)(1)(1) ] ( − c (2)(2)(2)(1)(2) ⊗ [ c (2)(2)(2)(1)(1) ] (0) ⊗ c (2)(2)(2)(2) =[ c (1) ] ⊗ c (2)(1) ⊗ [ c (2)(2)(1) ] ⊗ c (2)(2)(2)(1)(1) ⊗ [ c (2)(2)(2)(1)(2) ] ⊗ c (2)(2)(2)(2) =[ c (1)(1) ] ⊗ c (1)(2) ⊗ [ c (2)(1)(1) ] ⊗ c (2)(1)(2) ⊗ [ c (2)(2)(1) ] ⊗ c (2)(2)(2) , where the 1st, 3rd, 5th and last step use Proposition 2.4, the 2nd step use (2.4), the 4thand 6th step use (6.2). (cid:3) emma 6.4. α : H → B ⊗ B ⊗ B satisfy the 3-cocycle condition: h (1) ˜1 h (2) ˜1 (1) ⊗ h (1) ˜2 h (2) ˜1 (2) ⊗ h (1) ˜3 (1) h (2) ˜2 ⊗ h (1) ˜3 (2) h (2) ˜3 = ǫ ( h (1)(1) ) h (2) ˜1 h (3)(1) ˜1 ⊗ h (1)(2) ˜1 h (2) ˜2 (1) h (3)(1) ˜2 ⊗ h (1)(2) ˜2 h (2) ˜2 (2) h (3)(1) ˜3 ⊗ h (1)(2) ˜3 h (2) ˜3 ǫ ( h (3)(2) ) . for any h ∈ H .Proof. Let h = [ c ] ⊗ b , the left hand side is( c (1) ) ˆ1 ( c (2) ) ˆ1 (1) b (1)(1)(1) ⊗ ( c (1) ) ˆ2 ( c (2) ) ˆ1 (2) b (1)(1)(2) ⊗ ( c (1) ) ˆ3 (1) ( c (2) ) ˆ2 b (1)(2) ⊗ ( c (1) ) ˆ3 (2) ( c (2) ) ˆ3 b (2) while the right hand side is c (1)(1)(1) S B ( c (1)(2) )( c (2) ) ˆ1 ( c (3) ) ˆ1 b (1)(1)(1) ⊗ ( c (1)(1)(2) ) ˆ1 ( c (2) ) ˆ2 (1) ( c (3) ) ˆ2 b (1)(1)(2) ⊗ ( c (1)(1)(2) ) ˆ2 ( c (2) ) ˆ2 (2) ( c (3) ) ˆ3 b (1)(2) ⊗ ( c (1)(1)(2) ) ˆ3 ( c (2) ) ˆ3 b (2) . Notice that c (1) ⊗ c (2) ⊗ c (3) can be replaced by c (1)(1) ⊗ c (1)(2) ⊗ c (2) or c (1) ⊗ c (2)(1) ⊗ c (2)(2) ,since [ c ] ∈ C . Now we have( c (1) ) ˆ1 ( c (2) ) ˆ1 (1) ⊗ ( c (1) ) ˆ2 ( c (2) ) ˆ1 (2) ⊗ ( c (1) ) ˆ3 (1) ( c (2) ) ˆ2 ⊗ ( c (1) ) ˆ3 (2) ( c (2) ) ˆ3 =( c (1)(1)(1) ) ˆ1 ( c (1)(1)(2) ) ˆ1 (1) c (1)(2)(1)(1)(1) S B ( c (2) ) (1)(1)(1) ⊗ ( c (1)(1)(1) ) ˆ2 ( c (1)(1)(2) ) ˆ1 (2) c (1)(2)(1)(1)(2) S B ( c (2) ) (1)(1)(2) ⊗ ( c (1)(1)(1) ) ˆ3 (1) ( c (1)(1)(2) ) ˆ2 c (1)(2)(1)(2) S B ( c (2) ) (1)(2) ⊗ ( c (1)(1)(1) ) ˆ3 (2) ( c (1)(1)(2) ) ˆ3 c (1)(2)(2) S B ( c (2) ) (2) and c (1)(1)(1) S B ( c (1)(2) )( c (2) ) ˆ1 ( c (3) ) ˆ1 ⊗ ( c (1)(1)(2) ) ˆ1 ( c (2) ) ˆ2 (1) ( c (3) ) ˆ2 ⊗ ( c (1)(1)(2) ) ˆ2 ( c (2) ) ˆ2 (2) ( c (3) ) ˆ3 ⊗ ( c (1)(1)(2) ) ˆ3 ( c (2) ) ˆ3 = c (1)(1)(1)(1)(1) S B ( c (1)(1)(1)(2) )( c (1)(1)(2) ) ˆ1 ( c (1)(1)(3) ) ˆ1 c (1)(2)(1)(1)(1) S B ( c (2) ) (1)(1)(1) ⊗ ( c (1)(1)(1)(1)(2) ) ˆ1 ( c (1)(1)(2) ) ˆ2 (1) ( c (1)(1)(3) ) ˆ2 c (1)(2)(1)(1)(2) S B ( c (2) ) (1)(1)(2) ⊗ ( c (1)(1)(1)(1)(2) ) ˆ2 ( c (1)(1)(2) ) ˆ2 (2) ( c (1)(1)(3) ) ˆ3 c (1)(2)(1)(2) S B ( c (2) ) (1)(2) ⊗ ( c (1)(1)(1)(1)(2) ) ˆ3 ( c (1)(1)(2) ) ˆ3 c (1)(2)(2) S B ( c (2) ) (2) Thus to show this lemma it is sufficient to show( c (1)(1) ) ˆ1 ( c (1)(2) ) ˆ1 (1) c (2)(1)(1)(1) ⊗ ( c (1)(1) ) ˆ2 ( c (1)(2) ) ˆ1 (2) c (2)(1)(1)(2) ⊗ ( c (1)(1) ) ˆ3 (1) ( c (1)(2) ) ˆ2 c (2)(1)(2) ⊗ ( c (1)(1) ) ˆ3 (2) ( c (1)(2) ) ˆ3 c (2)(2) = c (1)(1)(1)(1) S B ( c (1)(1)(2) )( c (1)(2) ) ˆ1 ( c (1)(3) ) ˆ1 c (2)(1)(1)(1) ⊗ ( c (1)(1)(1)(2) ) ˆ1 ( c (1)(2) ) ˆ2 (1) ( c (1)(3) ) ˆ2 c (2)(1)(1)(2) ⊗ ( c (1)(1)(1)(2) ) ˆ2 ( c (1)(2) ) ˆ2 (2) ( c (1)(3) ) ˆ3 c (2)(1)(2) ⊗ ( c (1)(1)(1)(2) ) ˆ3 ( c (1)(2) ) ˆ3 c (2)(2) Using (2.4) the left hand side of the above equation is( c (1)(1) ) ˆ1 ( c (1)(2) ) ˆ1 (1) c (2)(1)(1)(1) ⊗ ( c (1)(1) ) ˆ2 ( c (1)(2) ) ˆ1 (2) c (2)(1)(1)(2) ⊗ ( c (1)(1) ) ˆ3 (1) ( c (1)(2) ) ˆ2 c (2)(1)(2) ⊗ ( c (1)(1) ) ˆ3 (2) ( c (1)(2) ) ˆ3 c (2)(2) =( c (1) ) ˆ1 ( c (2)(1) ) ˆ1 (1) c (2)(2)(1)(1)(1) ⊗ ( c (1) ) ˆ2 ( c (2)(1) ) ˆ1 (2) c (2)(2)(1)(1)(2) ⊗ ( c (1) ) ˆ3 (1) ( c (2)(1) ) ˆ2 c (2)(2)(1)(2) ⊗ ( c (1) ) ˆ3 (2) ( c (2)(1) ) ˆ3 c (2)(2)(2) =( c (1) ) ˆ1 c (2)(1)(1) ⊗ ( c (1) ) ˆ2 c (2)(1)(2) ⊗ ( c (1) ) ˆ3 (1) c (2)(2)(1) ⊗ ( c (1) ) ˆ3 (2) c (2)(2)(2) = c (1) ⊗ c (2)(1) ⊗ c (2)(2)(1) ⊗ c (2)(2)(2) , sing the Proposition 2.4 the right hand side is c (1)(1)(1) S B ( c (1)(2) )( c (2)(1) ) ˆ1 ( c (2)(2)(1) ) ˆ1 c (2)(2)(2)(1)(1)(1) ⊗ ( c (1)(1)(2) ) ˆ1 ( c (2)(1) ) ˆ2 (1) ( c (2)(2)(1) ) ˆ2 c (2)(2)(2)(1)(1)(2) ⊗ ( c (1)(1)(2) ) ˆ2 ( c (2)(1) ) ˆ2 (2) ( c (2)(2)(1) ) ˆ3 c (2)(2)(2)(1)(2) ⊗ ( c (1)(1)(2) ) ˆ3 ( c (2)(1) ) ˆ3 c (2)(2)(2)(2) = c (1)(1)(1) S B ( c (1)(2) )( c (2)(1) ) ˆ1 ( c (2)(2)(1)(1) ) ˆ1 c (2)(2)(1)(2)(1)(1) ⊗ ( c (1)(1)(2) ) ˆ1 ( c (2)(1) ) ˆ2 (1) ( c (2)(2)(1)(1) ) ˆ2 c (2)(2)(1)(2)(1)(2) ⊗ ( c (1)(1)(2) ) ˆ2 ( c (2)(1) ) ˆ2 (2) ( c (2)(2)(1)(1) ) ˆ3 c (2)(2)(1)(2)(2) ⊗ ( c (1)(1)(2) ) ˆ3 ( c (2)(1) ) ˆ3 c (2)(2)(2) = c (1)(1)(1) S B ( c (1)(2) )( c (2)(1) ) ˆ1 c (2)(2)(1)(1) ⊗ ( c (1)(1)(2) ) ˆ1 ( c (2)(1) ) ˆ2 (1) c (2)(2)(1)(2)(1) ⊗ ( c (1)(1)(2) ) ˆ2 ( c (2)(1) ) ˆ2 (2) c (2)(2)(1)(2)(2) ⊗ ( c (1)(1)(2) ) ˆ3 ( c (2)(1) ) ˆ3 c (2)(2)(2) = c (1)(1)(1) S B ( c (1)(2) ) c (2)(1) ⊗ ( c (1)(1)(2) ) ˆ1 c (2)(2)(1)(1) ⊗ ( c (1)(1)(2) ) ˆ2 c (2)(2)(1)(2) ⊗ ( c (1)(1)(2) ) ˆ3 c (2)(2)(2) = c (1)(1)(1)(1) S B ( c (1)(1)(2) ) c (1)(2) ⊗ ( c (1)(1)(1)(2) ) ˆ1 c (2)(1)(1) ⊗ ( c (1)(1)(1)(2) ) ˆ2 c (2)(1)(2) ⊗ ( c (1)(1)(1)(2) ) ˆ3 c (2)(2) = c (1)(1) ⊗ ( c (1)(2) ) ˆ1 c (2)(1)(1) ⊗ ( c (1)(2) ) ˆ2 c (2)(1)(2) ⊗ ( c (1)(2) ) ˆ3 c (2)(2) = c (1) ⊗ ( c (2)(1) ) ˆ1 c (2)(2)(1)(1) ⊗ ( c (2)(1) ) ˆ2 c (2)(2)(1)(2) ⊗ ( c (2)(1) ) ˆ3 c (2)(2)(2) = c (1) ⊗ c (2)(1) ⊗ c (2)(2)(1) ⊗ c (2)(2)(2) , where the 1st, 4th and 6th step use Proposition 2.4, the 2nd, 3rd and last step use (2.4),the 5th step use (6.2). (cid:3) As a result of Lemma 6.2, 5.2, 5.3, 6.3 and 6.4 we have
Theorem 6.5.
Let B be a quasi coassociative Hopf coquasigroup corresponding to acoassociative pair ( C, B, φ ) . If B is commutative, then H = C ⊗ B is a coherent Hopf2-algebra. Finite dimensional coherent Hopf 2-algebra and example
In [6] there is a dual pairing between bialgebra, we can see that there is also a dualpairing between Hopf coquasigroup and Hopf quasigroup. In this section we will makeclear of why quasi coassociative Hopf coquasigroup is the quantization of quasiassociativequasi group.
Definition 7.1.
Given a Hopf quasigroup ( A, ∆ A , ǫ A , m A , A , S A ) and a Hopf coquasi-group ( B, ∆ B , ǫ B , m B , B , S B ). A dual pairing between A and B is a bilinear map h• , •i : B × A → k such that: • h ∆ B ( b ) , a ⊗ a ′ i = h b, aa ′ i and h b ⊗ b ′ , ∆ A ( a ) i = h bb ′ , a i . • ǫ B ( b ) = h b, A i and ǫ A ( a ) = h B , a i .for any a, a ′ ∈ A and b, b ′ ∈ B . A dual pairing between B and A is called nondegenerate if h b, a i = 0 for all b ∈ B implies a = 0 and if h b, a i = 0 for all a ∈ A implies b = 0. Remark . Given two dual pairings h• , •i : B × A → k and h• , •i : B × A → k for two Hopf quasigroups A , A and two Hopf coquasigroup B , B , we can construct anew dual pairing h• , •i : B ⊗ B × A ⊗ A → k , which is given by h b ⊗ b , a ⊗ a i := h a , b i h a , b i for any a ∈ A , a ∈ A and b ∈ B , b ∈ B . Notice that A ⊗ A isalso a Hopf quasigroup with the factorwise (co)product, (co)unit and antipode. Similarly, B ⊗ B is also a Hopf coquasigroup. Moreover, if the both of the dual pairings arenondegenerate, then the new pairing is also nondegenate. f B is a finite dimensional Hopf coquasigroup, there is a nondegenerate dual pairingbetween B and its dual algebra A := Hom( B, k ). More precisely, the dual pairing is givenby h b, a i := a ( b ) for b ∈ B and a ∈ A . In this case A is a Hopf quasigroup, with structuregiven by aa ′ ( b ) := a ( b (1) ) a ′ ( b (2) ), 1 A ( b ) := ǫ ( b ), ∆ A ( a )( b ⊗ b ′ ) := a ( bb ′ ), ǫ A ( a ) := a (1 B ), S A ( a )( b ) := a ( S B ( b )), for any a ∈ A and b ∈ B . We can see that it satisfies the axiomsof Hopf quasigroup, for example, ( S ( a (1) )( a (2) a ′ ))( b ) = a (1) ( S B ( b (1) )) a (2) ( b (2)(1) ) a ′ ( b (2)(2) ) = a ( S B ( b (1) ) b (2)(1) ) a ′ ( b (2)(2) ) = ǫ A ( a ) a ′ ( b ), since the pairing of A and B are nondegenerate,we get the axioms.Given a finite dimensional coquasigroup B with its dual A := Hom( B, k ), recall theassociative elements of a quasigroup, we have similarly a subset of A : N A := { a ∈ A | a ( uv ) = ( au ) v, u ( av ) = ( ua ) v, u ( va ) = ( uv ) a, for ∀ u, v ∈ A } , (7.1)Clearly, N A is an associative algebra. The elements in N A can pass though braketsfor the multiplication. For example, let n ∈ N A and ∀ a , a , a , a ∈ A , then we have n ((( a a ) a ) a ) = ((( na ) a ) a ) a .If ∆ A ( N A ) ⊆ N A ⊗ N A , N A is a Hopf algebra with the structure inherited from A . Inthis case there is also a dual pairing between N A and B by the restriction of dual pairingbetween A and B , which is not nondegenerate. From now on we will assume N A to be aHopf algebra.Define I B := { b ∈ B | h b, a i = 0 for ∀ a ∈ N A } , (7.2)we can see that I B is an ideal of B , since for any b ∈ B , a ∈ N A and i ∈ I B , a ( bi ) = a (1) ( b ) a (2) ( i ) = 0. And I B is also a coideal (i.e. ∆ B ( i ) ∈ I B ⊗ B + B ⊗ I B for any i ∈ I B ), since N A is an algebra. As a result, the quotient algebra C := B/I B isa Hopf coquasigroup. We can see that there is also a dual pairing between N A and C given by h [ b ] , a i := h b, a i , where b ∈ B and [ b ] is the image of the quotient map in C , and a ∈ N A . If h [ b ] , a i = 0 for any b ∈ B , we get a = 0. And if h [ b ] , a i = 0 for any a ∈ N A , weget b ∈ I B , so [ b ] = 0. Thus the dual pairing between C and N A is nondegenerate. Since N A is associative and the dual pairing is nondegenerate, we get C is coassociative. As aresult, C is a Hopf algebra. Remark . Recall the linear map β : B → B ⊗ B ⊗ Bβ ( b ) = b (1)(1) S B ( b (2) ) (1)(1) ⊗ b (1)(2)(1) S B ( b (2) ) (1)(2) ⊗ b (1)(2)(2) S B ( b (2) ) (2) for any b ∈ B . We can see that β is corresponding to the associator β ∗ : A ⊗ A ⊗ A → A ,which is given by β ∗ ( u ⊗ v ⊗ w ) := ( u (1) ( v (1) w (1) ))( S ( w (2) )( S ( v (2) ) S ( u (2) ))) (7.3)for any u, v, w ∈ A . And it is easy to check h β ( b ) , u ⊗ v ⊗ w i = h b, β ∗ ( u ⊗ v ⊗ w ) i . for any b ∈ B . We call A is quasiassociative , if N A is adjoint invariant (i.e. a (1) nS ( a (2) )) ∈ N A for any n ∈ N A and a ∈ A ) and the image of β belong to N A . Thus B is quasicoassicoative if and only if A is quasiassociative. Example . In [8] the unital basis of Cayley algebras G n := {± e a | a ∈ Z n } is aquasigroup, with the product controlled by a 2-cochain F : Z n × Z n → k ∗ , more pre-cisely, e a e b := F ( a, b ) e a + b . From now on we also denote e a := e a and e a := − e a , i.e. G n = { e ia | a ∈ Z n , i ∈ Z } . We define k G n as the linear extension of G n , which is a Hopf uasigroup, thus e ia e jb = F ( a, b ) e i + ja + b , and the coalgebra structure is given by ∆( u ) = u ⊗ u , ǫ ( u ) = 1, and S ( u ) := u − on the basis elements.The dual of G n is a Hopf coquasigroup B := k [ G n ], which are functions on G n . Let f ia ∈ k [ G n ] be the delta function on each element of G n , i.e. f ia ( e jb ) = δ a,b δ i,j . We can see k [ G n ] is an algebra with generators { f ia | a ∈ Z n , i ∈ Z } subject to the relation f ia f i ′ a ′ = ( f ia if a = a ′ and i = i ′ k [ G n ] is P a ∈ Z n ,i ∈ Z n f ia . The coproduct, counit and antipode is given by∆ B ( f ia ) := X b + c = aj + k = i F ( b, c ) f jb ⊗ f kc . (7.4) ǫ B ( f ia ) := δ a, δ i, . (7.5) S B ( f ij ) := F ( a, a ) f ia . (7.6)There is a canonical dual pairing between B = k [ G n ] and A = k G n , which is given by h b, u i := b ( u ). Clearly, this dual pairing is nondegenerate. As we already know in [8] that k G n ≃ C if n = 1 H if n = 2 O if n = 3 . Since k G n is a subalgebra of k G m , for n ≤ m , therefore N k G m ⊆ N k G n . We have N k G n ≃ C if n = 1 H if n = 2 R if n ≥ , since N k G = R . In [[8], Prop 3.6] we know G n is quasiassociative. Thus for any i ∈ I B and u, v, w, x ∈ A h β ( i (1)(2) ) ⊗ i (1)(1) S ( i (2) ) , u ⊗ v ⊗ w ⊗ x i = h i, ( x (1) β ∗ ( u ⊗ v ⊗ w )) S ( x (2) ) i = 0 . Since the dual pairing is nondegenerate, we can see k [ G n ] is quasi coassociative cor-responding to the coassociative pair ( k [ G ] , k [ G n ] , π ), where π : k [ G n ] → k [ G ] is thecanonical projection map and k [ G ] is just the functions on {− e , e } , thus by Theorem6.5 there is a coherent Hopf 2-algebra structure, with C = k [ G ], and H = k [ G ] ⊗ k [ G n ].To be more precise, we give the structure maps:∆( f i ⊗ f la ) = X j + k = i f j ⊗ ⊗ B f k ⊗ f la ; ǫ ( f i ⊗ f la ) = ǫ B ( f i ) f la ; S ( f i ⊗ f la ) = X i + n = l f i ⊗ f na ; s ( f la ) = X m + n = l f m ⊗ f na ; t ( f la ) = 1 ⊗ f la . ll the above is the structure of Hopf algebroid over B = k [ G n ]. Let c = f ia , then c ∈ I B if and only if a = 0, since N A = R . We can see c (2)(1) ⊗ c (1) S B ( c (2)(2) ) = X b + c + d = aj + k + l = i F ( b, c + d ) F ( c, d ) F ( d, d ) f kc ⊗ f jb f ld , (7.7)the right hand of the equality is not zero only if b = d , therefore a = c . So if c ∈ I B , then c (2)(1) ⊗ c (1) S B ( c (2)(2) ) ∈ I B ⊗ B , and similarly c (1)(2) ⊗ c (1)(1) S B ( c (2) ) ∈ I B ⊗ B . If c = f i [ c (2)(1) ] ⊗ c (1) S B ( c (2)(2) ) =[ c (1)(2) ] ⊗ c (1)(1) S B ( c (2) ) = X b + d =0 j + k + l = i F ( b, d ) F ( d, d ) f k ⊗ f jb f ld = X b + d =0 j + k + l = i f k ⊗ f jb f ld = f i ⊗ . For the Hopf coquasigroup structure we have: N ( f i ⊗ f la ) = X m + n = lb + c = aj + k = i F ( b, c ) f j ⊗ f mb ⊗ f k ⊗ f nc ; ǫ H ( f i ⊗ f la ) = ǫ B ( f i f la ) = δ i, δ l, δ a, ; S H ( f i ⊗ f la ) = F ( a, a ) f i ⊗ f la . For β : B → B ⊗ B ⊗ B , we have β ( f ia ) = X j + k + l + m + n + p = ib + c + d + e + f + g = a F ( b + c + d, e + f + g ) F ( b, c + d ) F ( c, d ) F ( e, f + g ) F ( f, g ) F ( e, e ) F ( f, f ) F ( g, g ) f jb f pg ⊗ f kc f nf ⊗ f ld f me We can see if i ∈ I B , β ( i ) = 0. For α : H → B ⊗ B ⊗ B we have α ( f i ⊗ f la ) = X k + m + n = lb + c + d = a β ( f i )( F ( b + c, d ) F ( b, c ) f kb ⊗ f mc ⊗ f nd ) . From the formular of β , we can see that α is controlled by a 3-cocycle correspondingto the 2-cochain F . In fact, β ( f i ) = X b,c,d F ( b + c + d, b + c + d ) F ( b, c + d ) F ( c, d ) F ( d, c + b ) F ( c, b ) F ( d, d ) F ( c, c ) F ( b, b ) f jb ⊗ f kc ⊗ f ld = X b,c,d F ( b + c + d, b + c + d ) ψ ( b, c, d ) F ( d, d ) F ( c, c ) F ( b, b ) f jb ⊗ f kc ⊗ f ld , where ψ is the 3-cocycle given by the 2-cochain F , ψ ( b, c, d ) = F ( b, c + d ) F ( c, d ) F ( d, c + b ) F ( c, b ) = F ( b, c + d ) F ( c, d ) F ( d, c + b ) F ( c, b ) , since F has value in {± } [8]. Acknowledgment:
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Xiao Han
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