Constructing Quasitriangular Hopf Algebras

Group unified coprodcuts and related quasitriangular structures

“…In this section, we shall introduce the π-unified coproduct as the natural generalization of unified coproduct introduced by the second author in [4]. First, a coextending π-datum is introduced as follows.…”

“…In what follows, we always let A be a Hopf algebra and Ω(A) = (H, ρ, , ω) a crossed semi-Hopf πcoalgebra coextending structure of A with the crossing ϕ. First, we introduce some new definitions which are both the group version of the concepts of [4] and also the dual concepts of Definition 2.1, 2.2 and 2.4 in [2]. Definition 5.1.…”

“…In this paper, we shall introduce a new coproduct named π-unified coproduct as the generalization of unified coproduct introduced by the second author in [4] which is the dual of unified product introduced in [1] as an answer to the restricted extending structures problem for Hopf algebras. Then we discuss the its classification and quasitriangular stuctures.…”

On $4n$-dimensional neither pointed nor semisimple Hopf algebras and the associated weak Hopf algebras

“…Quasi-triangular Hopf algebras play an important in the theory of Hopf algebras and quantum groups, since they provide solutions to quantum Yang-Baxter equations. People try to construct quasi-triangular Hopf algebras and get a lot of results(see [29,20,6,30,16]). In this section, we shall show that H 4n is quasitriangular and give all universal R-matrices explicitly.…”

“…(5) and (6). Let k ∈ {1, 2}, s ∈ {0, n}, j ∈ {0, 1} and v s be the basis of M [1, s], then X • v s = 0 and G • v s = (−1) s n v s , so we have…”