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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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