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.


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