Poisson Lie groups, dressing transformations, and Bruhat decompositions. (English) Zbl 0673.58018

Lu, Jiang-Hua; Weinstein, Alan

J. Differ. Geom. 31, No. 2, 501-526 (1990).

A Poisson Lie group is a Lie group G with a Poisson structure for which the multiplication \(G\times G\to G\) is a Poisson map. Poisson actions of these groups are defined by an analogous condition. According to V. G. Drinfel’d [Sov. Math., Dokl. 27, 68-71, translation from Dokl. Akad. Nauk SSSR 268, 285-287 (1983; Zbl 0526.58017)] and M. A. Semenov- Tian-Shansky [Publ. Res. Inst. Math. Sci. 21, 1237-1260 (1985; Zbl 0673.58019)], every Poisson Lie group has a dual Poisson group \(G^*\) with a (locally defined) action on G by so-called dressing transformations. These structures were originally motivated by considering the classical limit of completely integrable quantum systems.
Using the Iwasawa decomposition, it is shown that every compact semisimple Lie group G admits a compatible Poisson structure which passes in a natural way to each coadjoint orbit of G. (These structures were also found by S. Majid via the Drinfel’d-Jimbo solution of the Yang- Baxter equation.) The symplectic leaves on the coadjoint orbits, which are also orbits of an action of the dual group, turn out to be precisely the cells of the Bruhat decomposition.
The paper ends with some remarks about possible connections of the results with recent work on quantum groups and quantum spheres.

Reviewer: J.-H.Lu

MSC:

37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
35A30	Geometric theory, characteristics, transformations in context of PDEs
53D50	Geometric quantization
37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Keywords:

Poisson Lie group; dressing transformations; completely integrable quantum systems; Iwasawa decomposition; Poisson structure; Yang-Baxter equation; Bruhat decomposition

Citations:

Zbl 0526.58017; Zbl 0673.58019