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Dual pairs of Hopf *-algebras. (English) Zbl 0796.16034

If \(A\) is a Hopf *-algebra, the dual space \(A'\) is again a *-algebra. There is a natural subalgebra \(A^ 0\) of \(A'\) that is again a Hopf *- algebra. In many interesting examples \(A^ 0\) 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 \(A\times B\) 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 \(A^ 0\). We give examples of pairs of Hopf *-algebras and we 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 \(\text{SU}(n)\) of Woronowicz. We also discuss the notion of the quantum double of Drinfeld in this framework of dual pairs.

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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