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Poisson Lie groups, quantum duality principle, and the quantum double. (English) Zbl 0841.17008
Sally, Paul J. jun. (ed.) et al., Mathematical aspects of conformal and topological field theories and quantum groups. AMS-IMS-SIAM summer research conference, June 13-19, 1992, Mount Holyoke College, South Hadley, MA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 175, 219-248 (1994).
Quantum groups still form a young and fast growing theory, and perhaps the major argument in support of this is an undisputable fact that there is no universally accepted definition of a quantum group as such. Apparently, one should no longer expect mathematicians ever to come up with such an all-embracing single notion. Instead, the likeliest scenario would be for the five or so existing approaches to quantum groups to crystallise and a comprehensive scheme of precise relationships between them to emerge; a global pattern would be of the same variety as that presently formed by discrete groups, locally compact groups, Lie groups, and algebraic groups, along with duality theories added on top of that. From this viewpoint, the papers like the one under review are of great interest in that they are examining links between different passages to the concept of a quantum group.
The present paper deals with the so-called quantum duality principle, which is a way to relate to each other the Drinfeld-Jimbo approach (quantum groups as deformations of the universal enveloping Hopf algebras of Lie algebras) and the Faddeev-Reshetikhin-Takhtajan approach (quantum groups as deformations of coordinate rings of functions on Lie groups).
The article contains a brief yet self-contained introduction to Poisson-Lie groups, including a survey of Poisson structures on simply connected Lie groups, the actions of Poisson-Lie groups on Poisson manifolds, and the theory of Lie bialgebras. Quasitriangular Hopf algebras are discussed, and the function algebras on a Poisson Lie group and its dual are quantized. The author considers the Heisenberg double and the Drinfeld double of Hopf algebras from a certain class (the so-called quasi-classical factorizable Hopf algebras), along with their operator realization.
Then the author proceeds to the quantum duality principle, stating that the quantized function algebras, $$\text{Fun}_q (G)$$ (on $$G$$) and $$\text{Fun}_q (G^*)$$ (on the dual of $$G$$) are dual to each other as Hopf algebras in the usual sense through a non-degenerate pairing $$\text{Fun}_q (G)\otimes \text{Fun}_q (G^*)\to \mathbb{C}[[h]]$$; as a corollary, the appropriate completion of $$\text{Fun}_q (G)$$ and of $$U_q ({\mathfrak g})$$ are isomorphic, where $${\mathfrak g}$$ stands for the Lie algebra of $$G$$. It is observed that the quantum duality principle is meaningful already for the trivial Lie bialgebras, in which case it includes the concept of Fourier transform.
In the quantum case, the quantum duality principle is related to the suggested version of the quantum Fourier transform.
The last section is devoted to the construction of the twisted double of a Poisson-Lie group, and its relationship with the quantum structures considered earlier.
For the entire collection see [Zbl 0801.00049].

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 53D50 Geometric quantization 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 22D35 Duality theorems for locally compact groups
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