| It has been shown that quasitriangular Hopf algebras (QTHAs) have been increasingly playing important roles in many areas of mathematics and physics. Some people believe that the theory of quantum groups will be the group theory of next century. The main goal of this thesis is to develop methods to determine the quasitriangular structures (R-matrices) of a finite dimensional Hopf algebra over a field. The primary research that I have done in this thesis touched quantum groups from several directions. We prove the main results in this thesis that are stated as follows. Let H be a finite dimensional Hopf algebra over a field k. If H is unimodular, then the R-matrices of H can be embedded in the center of the quantum double D(H), a QTHA associated to H that was discovered by Drinfel'd. If H is cosemisimple (equivalently, if the dual algebra of H is semisimple), then the R-matrices of H correspond to central idempotents in D(H). Hence, for a finite dimensional cosemisimple Hopf algebra H (such as the group algebra of a finite group), one can possibly locate all the R-matrices among the set of central idempotents of D(H), which is a finite set in many general contexts. We will see that there are many non-trivial R-matrices arising from finite non-abelian groups. Non-trivial R-matrices of non-abelian group algebras allow us to use groups to construct quantum groups. |
