The Analysis And Research To BiFrobenius Algebra

As a natural generalization of finite-dimensional Hopf algebras,the concept of a biFrobenius algebra was introduced by Doi and Takeuchi in 2000.These are both Frobenius algebras and Frobenius coalgebras and satisfy some compatibility conditions.They are almost equivalent to Koppinen's double Frobenius algebras under some correspondence. In this paper,our work is mainly around biFrobenius algebras.The following is an outline of the whole paper.In Section 1 we mainly introduce relative definition,proposition about biFrobenius algebras and propose and prove the result that the semilattice graded weak Hopf algebras are biFrobenius algebras.Doi and Takeuchi give the Radford's S~4-formula for biFrobenius algebras and discuss substructures,quotient structures of biFrobenius algebras. Also,they introduce a notion of a group-like algebra,which is an interesting example of biFrobenius algebras.In Section 2 we establish the braided version of the biFrobenius algebras theory. There are some choices about how to define braided biFrobenius algebras.In this paper, we give two definitions way.One choice is to generalize braided Hopf algebras and give relative definition.We prove the relations between the braided biFrobenius algebra and a solution of the Yang-Baxter equation on its module.At the same time we induce a dual concept:cobraided biFrobenius algebras and applications.The other choice is to define the notion of a braided biFrobenius algebras in a braided tensor category by Doi and Takeuchi.In section 3 we discuss some counterexamples to the relative properties of Hopf algebras and biFrobenius algebras problem.