Braided Hopf algebras obtained from coquasitriangular Hopf algebras

Abstract

Let $(H,\sigma )$ be a coquasitriangular Hopf algebra, not necessarily finite dimensional. Following methods of Doi and Takeuchi, which parallel the constructions of Radford in the case of finite dimensional quasitriangular Hopf algebras, we define $H_{\sigma }$, a sub-Hopf algebra of $H^0$, the finite dual of $H$. Using the generalized quantum double construction and the theory of Hopf algebras with a projection, we associate to $H$ a braided Hopf algebra structure in the category of Yetter-Drinfeld modules over $H_{\sigma }^{\mathrm{cop}}$. Specializing to $H={\mathrm{SL}}_q(N)$, we obtain explicit formulas which endow ${\mathrm{SL}}_q(N)$ with a braided Hopf algebra structure within the category of left Yetter-Drinfeld modules over $U_q^{\mathrm{ext}}({\mathrm{sl}}_N{)}^{\mathrm{cop}}$.

43 pages, 1 figure