Braided Hopf Algebras

Preface The term " quantum group " was popularized by Drinfeld in his address to the International Congress of Mathematicians in Berkeley. However, the concepts of quantum groups and quasitriangular Hopf algebras are the same. Therefore, Hopf algebras have close connections with various areas of mathematics and physics. The development of Hopf algebras can be divided into five stages. The first stage is that integral and Maschke's theorem are found. Maschke's theorem gives a nice criterion of semisimplicity, which is due to M.E. Sweedler. The second stage is that the Lagrange's theorem is proved, which is due to W.D. Nichols and M.B. Zoeller. The third stage is the research of the actions of Hopf algebras, which unifies actions of groups and Lie algebras. S. Montgomery's " Hopf algebras and their actions on rings " contains the main results in this area. S. Montgomery and R.J. Blattner show the duality theorem. and the author study the relation between algebra R and the crossed product R# σ H of the semisimplicity, semiprimeness, Morita equivalence and radical. S. Montgomery and M. Cohen ask whether R# σ H is semiprime when R is semiprime for a finite-dimensional semiprime Hopf algebra H. This is a famous semiprime problem. The answer is sure when the action of H is inner or H is commutative or cocommutative, which is due to S. Montgomery, H.J. Schneider and the author. This is the best result in the study of the semiprime problem up to now. However, this problem is still open. Y. Doi and M. Takeachi show that the crossed product is a cleft extension. The research of quantum groups is the fourth stage. The concepts of quantum groups and (co)quasitriangular Hopf algebras are the same. The Yang-Baxter equation first came up in a paper by Yang as factorization condition of the scattering S-matrix in the many-body problem in one dimension and in work of Baxter on exactly solvable models in statistical mechanics. It has been playing an important role in mathematics and physics (see [7], [52]). Attempts to find solutions of the Yang-Baxter equation (YBE) in a systematic way have led to the theory of quantum groups. In other words, we can obtain a solution of the Yang-Baxter equation by a (co)quasitriangular Hopf algebra. It is well-known that the universal enveloping algebras of Lie algebras are Hopf algebras. But they are not quasitriangular in general. Drin-feld obtained a …