Lie-poisson Structure on Some Poisson Lie Groups

Poisson Lie groups appeared in the work of Drinfel'd (see, e.g., [Drl, Dr2]) as classical objects corresponding to quantum groups. Going in the other direction, we may say that a Poisson Lie group is a group of symmetries of a phase space that are allowed to "twist," in a certain sense, the symplectic or Poisson structure. The Poisson structure on the group controls this twisting in a precise way. Quantizing both the phase space and the symmetries, one may obtain a quantum group acting on a quantum phase space. In recent work, Lu and Ratiu [LR] used so-called standard Poisson structures on a compact semisimple Lie group K and on its Poisson dual K* in order to give a new proof of the nonlinear convexity theorem of Kostant [Ko]. Their method is analogous to the famous symplectic proof of the linear convexity theorem given by Atiyah [A] and Guillemin and Sternberg [GS]. The nonlinear convexity theorem, like the linear one, follows from a very general result on convexity of the image of the momentum map [A, GS]; however, in [LR] it is applied not to a coadjoint orbit in e* , but to a symplectic leaf in K* . The main result of this paper is that the standard Poisson structure on the Poisson dual K* to a compact semisimple Poisson Lie group K is actually isomorphic to the linear one on e* . This theorem seems to be related to some facts in the theory of quantum groups. Namely, (for generic q) the universal enveloping algebra Vee) and its quantum deformation Vq(e) are isomorphic as algebras (though not as coalgebras, of course, since the quantum version is not cocommutative). In particular, there is a bijective correspondence between their representations. A direct connection between our work and its quantum analogues, though, is still to be found. The present work supplies a positive answer to Question 5.1 in [LR] and strongly depends on that paper. To simplify reading, we keep the notations of [LR] wherever possible, on one hand, but give all necessary definitions, on the other. Our work is also related to [Du], in which the nonlinear convexity theorem is reduced to the linear one by a deformation argument not unlike the one that we use in §5. The paper is organized as follows. In §2 we define Poisson Lie groups, discuss