## U_q(sl(2)), SL_q(2), and duality for Hopf algebras

### Steve Lack – 30 October 1996

This was the third talk in the series on Kassel's book Quantum Groups. I recalled the Hopf algebras SL(2), SLq(2), and U(sl(2)) defined in previous talks, and showed how the algebra Uq(sl(2)) of a previous talk can be given the structure of a Hopf algebra; trying always to stress the connections between these various algebras. I then looked at the notion of duality for Hopf algebras given in Kassel: A (perfect) duality between Hopf algebras H and K is a (non-degenerate) bilinear form on H x K such that the induced maps H --> K* and K --> H* are both morphisms of algebras. Finally I gave a duality between SL(2) and U(sl(2)) and a duality between SLq(2) and Uq(sl(2)).

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