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A characterization of coboundary Poisson Lie groups and Hopf algebras. (English) Zbl 0893.22011

Budzyński, Robert (ed.) et al., Quantum groups and quantum spaces. Lectures delivered during the minisemester, Warsaw, Poland, December 1, 1995. Warszawa: Polish Academy of Sciences, Inst. of Mathematics, Banach Cent. Publ. 40, 273-278 (1997).
It is shown that a Poisson Lie group \((G,\pi)\) is a coboundary, i.e. \(\pi(g)= rg-gr\) for some \(r\in {\mathbf g}\wedge {\mathbf g}\) where \({\mathbf g} =\text{Lie} (G)\), if and only if the natural left-right action of \(G\times G\) on \(G\) is a Poisson action with respect to some Poisson structure on \(G\). A similar condition is analyzed in the context of Hopf algebras with a possible application in the theory of quantum groups.
For the entire collection see [Zbl 0865.00041].

MSC:

22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
22E43 Structure and representation of the Lorentz group
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81S05 Commutation relations and statistics as related to quantum mechanics (general)
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