| Radford's biproduct is an important research topic in the theory of Hopf algebra,which plays a fundamental role in the classification of finite-dimensional pointed Hopf algebras.Infinitesimal bialgebras were introduced by Joni and Rota,and later studied by Aguiar.In this paper,we mainly study Radford's biproduct and infinitesimal bialgebras.The main contents are as follows:(1)From the definition of the left Brzezi(?)ski cross product algebra and the definition of the right Brzezi(?)ski cross product algebra,this paper constructs the Brzezi(?)ski two-sided cross product algebra,which generalizes the two-sided smash product.Furthermore,we get sufficient and necessary conditions for Brzezi(?)ski two-sided crossed product algebra and the two-sided tensor product coalgebra to be a Brzezi(?)ski double twisted tensor biproducts bialgebra.(2)In this part,the notion of covariant(monoidal)BiHom-bialgebra is introduced generalizing infinitesimal(monoidal)BiHom-bialgebra.In particular,we also give the equivalent characterization of covariant(monoidal)BiHom-bialgebras,and then obtain the notions of associative(monoidal)BiHom-Yang-Baxter pair and associative(monoidal)BiHom-Yang-Baxter equation.Then,the concept of quasitriangular covariant(monoidal)BiHom-bialgebra is introduced,and the equivalent characterization of quasitriangular covariant(monoidal)BiHom-bialgebra is given.(3)As a generalization of Rota-B axter monoidal BiHom-algebra,the notion of Rota-B axter monoidal BiHom-system is introduced,and the relations between it and monoidal BiHom algebra,monoidal BiHom-dendriform algebra,left BiHom-pre-Lie algebra and associative monoidal BiHom-Yang-Baxter pair are investigated.In addition,as another generalization of Rota-B axter monoidal BiHom-algebras,we also consider Rota-Baxter monoidal BiHom-paired modules and some of its properties.(4)Firstly,we introduce the notion of double BiHom-algebra,and then we present the construction of the Drinfeld double for infinitesimal BiHom-bialgebra from the double BiHomalgebra.(5)Firstly,we introduce the notions of an(m,n)-smash product monoidal BiHom-algebra(B#nmH,?B(?)?H,?B(?)?H)and a(p,q)-smash coproduct monoidal BiHom-coalgebra(B×qpH,?B(?)?H,?B(?)?H).Furthermore,we get sufficient and necessary conditions for(B#nmH,?B(?)?H,?B(?)?H)and(B×qp H,?B(?)?H,?B×?H)to be a Radford(m,n,p,q)-biproduct monoidal BiHom-bialgebras(B×n,q#m,pH,?B(?)?H,?B(?)?H).At last,we provide a class of monoidal category which is related to(B×n,q#m,pH,?B(?)?H,?B(?)?H). |
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