Galois linear maps and their construction

The condition of an algebra to be a Hopf algebra or a Hopf(co)quasigroup can be determined by the properties of Galois linear maps.For a bialgebra H,if it is unital and associative as an algebra and counital coassociative as a coalgebra,then the Galois linear maps T1 and T2 can be defined.For such a bialgebra H,it is a Hopf algebra if and only if T1 is bijective.Moreover,T1^-1 is a right H-module map and a left H-comodule map(similar to T2).On the other hand,for a unital algebra(no need to be associative),and a counital coassociative coalgebra A,if the coproduct and counit are both algebra morphisms,then the sufficient and necessary condition of A to be a Hopf quasigroup is that T1 is bijective,and T1^-1 is left compatible with ΔT1-11^r and right compatible with mT1-1^l at the same time(The properties are similar to T2).Furthermore,as a corollary,the quasigroups case is also considered.

Crossed products for Hopf group-algebras

First,the group crossed product over the Hopf group-algebras is defined,and the necessary and sufficient conditions for the group crossed product to be a group algebra are given.The cleft extension theory of the Hopf group algebra is introduced,and it is proved that the crossed product of the Hopf group algebra is equivalent to the cleft extension.The necessary and sufficient conditions for the crossed product equivalence of two Hopf groups are then given.Finally,combined with the equivalence theory of the Hopf group crossed product and cleft extension,the group crossed product constructed by the general 2-cocycle as algebra is determined to be isomorphic to the group crossed product of the 2-cocycle with a convolutional invertible map of the 2-cocycle.The unit property of a general 2-cocycle is equivalent to the convolutional invertible map of the 2-cocycle,and the combination condition of the weak action is equivalent to the convolutional invertible map of the 2-cocycle and the combination condition of the weak action.Similarly,crossed product algebra constructed by the general 2-cocycle is isomorphic to the Hopfπ-crossed product algebra constructed by the 2-cocycle with a convolutional invertible map.

Fundamentals of quasigroup Hopf group-coalgebras

A large class of algebras(possibly nonassociative)with group-coalgebraic structures,called quasigroup Hopf group-coalgebras,is introduced and studied.Quasigroup Hopf group-coalgebras provide a unifying framework for the classical Hopf algebras and Hopf group-coalgebras as well as Hopf quasigroups.Then,basic results similar to those in Hopf algebras H are proved,such as anti-(co)multiplicativity of the antipode S:H→H,and S^(2)=id if H is commutative or cocommutative.

Sweedler's dual of Hopf algebras in HHYDQCM

Firstly,the notion of the left-left Yetter-Drinfeld quasicomodule M=(M,·,ρ)over a Hopf coquasigroup H is given,which generalizes the left-left Yetter-Drinfeld module over Hopf algebras.Secondly,the braided monoidal category HHYDQCM is introduced and the specific structure maps are given.Thirdly,Sweedler's dual of infinite-dimensional Hopf algebras in HHYDQCM is discussed.It proves that if(B,mB,μB,ΔB,εB)is a Hopf algebra in HHYDQCM with antipode SB,then(B^0,(mB0)^op,εB^*,(ΔB0)^op,μB^*)is a Hopf algebra in HHYDQCM with antipode SB^*,which generalizes the corresponding results over Hopf algebras.

BiHom-H-pseudoalgebras and their constructions

The definition and an example of BiHom-associative H-pseudoalgebra are given.A BiHom-H-pseudoalgebra is an H-pseudoalgebra(A,μ)with two mapsα,β∈HomH(A,A)satisfying the BiHom-associative law which generalizes BiHom-associative algebras and associative H-pseudoalgebras.Secondly,a method which is called the Yau twist of constructing BiHom-associative H-pseudoaglebra(A,(IH■H■Hα)μ,α,β)from an associative H-pseudoalgebra(A,μ)and two maps of H-pseudoalgebrasα,β,is introduced.Thirdly,a generalized form of the Yau twist is discussed.It concerns constructing a BiHom-associative H-pseudoalgebra(A,μ(α■β),α^~α,β^~β)from a BiHom-associative H-pseudoalgebra(A,μ,α^~,β^~)and two mapsα,β∈Hom H(A,A).Finally,a method of constructing BiHom-associative H-pseudoalgebra on tensor product space A■B of two BiHom-associative H-pseudoalgebras is given.

Construction of a class of H-pseudoalgebras

Let H be a cocommutative Hopf algebra.First,anew class//-pseudoalgebras o f H-pseudoalgebras are definedby changing the regular action(i.e.left multiplication)of Hon itself into an adjoint action.Secondly,a class o f{H,R)-pseudoalgebras are studied by generalizing the aboveconstruction when(H,R)is a quasitrianglar Hopf algebra.A tthe same time,the(H,R)-pseudoalgebra is constructed byboth the usual algebra and the tensor product o f(H,R)-pseudoalgebras.Finally,some examples of the(H,R)-pseudoalgebra are given explicitly,and the conditions for aHopf algebra to be an(H,R)-pseudoalgebra(resp.Hpseudoalgebra)are discussed.

Diagonal crossed product of multiplier Hopf algebras

Let A and B be two regular multiplier Hopf algebras.First,the notion of diagonal crossed product B#A of multiplier Hopf algebras is constructed using the bimodule algebra,which is a generalization of the diagonal crossed product in the sense of Hopf algebras.The result that the product in B#A is non-degenerate is given.Next,the definition of the comultiplicationΔ#on B#A is introduced,which is composed of the multiplierΔB(b)on B⊗B and the multiplierΔA(a)on A⊗A,and the elementΔ#(b⊗a)is a two-side multiplier of B#A⊗B#A,for any b∈B and a∈A.Then,a sufficient condition for B#A to be a regular multiplier Hopf algebra is described.In particular,Delvaux's main theorem in the case of smash products is generalized.Finally,these integrals on a diagonal crossed product of multiplier Hopf algebras are considered.