## Lie-Poisson structure on some Poisson Lie groups.(English)Zbl 0766.58018

Let $$K$$ be a compact semisimple group endowed with the standard Poisson structure and let $$K^*$$ be its Poisson dual (also endowed with its standard Poisson structure). Lu and Ratiu have recently used this framework to obtain a new proof of the nonlinear convexity theorem of Kostant. Let $${\mathcal G}$$ be a complex semisimple Lie algebra and $$K$$ its compact real form. Let $${\mathcal L}$$ be the Lie algebra of $$K$$ viewed as the compact part in the Iwasawa decomposition of $$G$$ considered as a real group.
The main result of this paper is that the standard Poisson structure on the Poisson dual $$K^*$$ is actually globally diffeomorphic to the linear one on the dual of $${\mathcal L}$$.
Reviewer: J.Lacroix (Paris)

### MSC:

 53D17 Poisson manifolds; Poisson groupoids and algebroids 53D10 Contact manifolds (general theory) 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 17B56 Cohomology of Lie (super)algebras 58H05 Pseudogroups and differentiable groupoids

### Keywords:

Poisson structure; compact Lie algebras; cohomology
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### References:

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