Dual Pairs of Hopf *‐Algebras

Abstract

If A is a Hopf *‐algebra, the dual space A′ is again a *‐algebra. There is a natural subalgebra A° of A′ that is again a Hopf *‐algebra. In many interesting examples, A° will be large enough (to separate points of A). More generally, one can consider a pair (A, B) of Hopf *‐algebras and a bilinear form on A × B with conditions such that, if the pairing is non‐degenerate, one algebra can be considered as a subalgebra of the dual of the other.

In these notes, we study such pairs of Hopf *‐algebras. We start from the notion of a Hopf *‐algebra A and its reduced dual A°. We give examples of pairs of Hopf *‐algebras, and discuss the problem of non‐degeneracy. The first example is an algebra paired with itself. The second example is the pairing of a Hopf *‐algebra (due to Jimbo) and the twisted SU(n) of Woronowicz. We also discuss the notion of the quantum double of Drinfeld in this framework of dual pairs.