## A geometric approach to Conn’s linearization theorem.(English)Zbl 1229.53085

This paper gives another proof of Conn’s linearization theorem. It is well-known that with any zero $$x_0$$ of a Poisson bracket is naturally associated a Lie algebra called the isotropy algebra at $$x_0$$. Moreover, the corresponding tangent space carries a canonical linear Poisson bracket called the linear approximation at $$x_0$$. Conn’s linearization theorem stipulated that when the isotropy Lie algebra at $$x_0$$ is semisimple of compact type, then the Poisson structure is linearizable around $$x_0$$. Conn’s proof is analytic and difficult. In this work, the authors obtain a soft geometric proof of the result using Moser’s path method. This geometrical approach can be used in other situations. For instance, the case of Lie algebroids is discussed.

### MSC:

 53D17 Poisson manifolds; Poisson groupoids and algebroids 53C05 Connections (general theory)
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### References:

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