The Quantum Double as a Hopf Algebra

In the last lecture we have learned that the category of modules over a braided Hopf algebra H is a braided monoidal category. A braided Hopf algebra is a rather sophisticated algebraic object, it is not easy to give interesting nontrivial examples. In this text we develop a theory that will lead to a concrete recipe which produces a nontrivial braided Hopf algebra D(A) (called Drinfeld’s quantum double) for any finite dimensional Hopf algebra A with invertible antipode! We will furthermore use this technique to produce an important example of a quantum group, namely the quantized universal enveloping algebra of the Lie algebra sl2 of traceless 2× 2-matrices. Conventions: In this text Hopf algebras will be assumed to have invertible antipodes. Although we will be working with several Hopf algebras at the same time, we use the same notations for the associated (co)algebra maps, units and antipodes: it will always be clear from context which maps we are dealing with.