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Levi decomposition for smooth Poisson structures. (English) Zbl 1085.53074
Let $$\Pi$$ be a $$C^p$$ Poisson structure ($$p$$ may be infinite) in a neighborhood of 0 in $$\mathbb R^n$$, which vanishes at the origin. Denote by $$\mathfrak l$$ the $$n$$-dimensional Lie algebra of linear functions in $$\mathbb R^n$$ under the Lie-Poisson bracket $$\Pi_1$$, which is the linear part of $$\Pi$$ at 0, and by $$\mathfrak g$$ a compact semi-simple subalgebra of $$\mathfrak l$$. Denote by $$(x_1,\dots,x_m,y_1,\dots,y_{n-m})$$ a linear basis of $$\mathfrak l$$, such that $$x_1,\dots,x_m$$ $$\text{span\,}{\mathfrak g}$$ and $$y_1,\dots,y_{n-m}$$ span the linear complement $$\mathfrak r$$ of $$\mathfrak g$$ in $$\mathfrak l$$ which is invariant under the adjoint action of $$\mathfrak g$$. Denote by $$c_{ij}^k$$ and $$a^k_{ij}$$ the structural constants of $$\mathfrak g$$ and $$\mathfrak r$$ such that $$[x_i,x_j]=\sum c^k_{ij}x_k$$, and $$[x_i,y_j]=\sum a^k_{ij}y_k$$. Now we say that $$\Pi$$ admits a local $$C^q$$-smooth Levi-decomposition with respect to $$\mathfrak g$$ if there exists a local $$C^q$$-smooth system of coordinates $$(x^\infty_1,\dots,x_m^\infty,y^\infty_1,\dots,y^\infty_{n-m})$$, with $$x^\infty_i=x_i+$$ higher order terms and $$y^\infty_i = y_i+$$ higher order terms, such that in this coordinates system, the Poisson structure has the form $\Pi = \frac 12 \left[\sum c^k_{ij} x^\infty_k\partial/\partial x^\infty_i\wedge \partial/\partial x^\infty_j+\sum a^k_{ij}y^\infty_j\partial/\partial x^\infty_i\wedge \partial/\partial y^\infty_j+\sum F_{ij}\partial/\partial y^\infty_i\wedge\partial/\partial y^\infty_j\right]$ where $$F_{ij}$$ are some functions in a neighbourhood of 0 in $$\mathbb R^n$$.
The main theorem in this paper is the following: There exists a positive integer $$\ell$$ such that any $$C^{2q-1}$$-smooth Poisson structure $$\Pi$$ in a neighborhood of 0 in $$\mathbb R^n$$ which vanishes at 0, where $$q\geq \ell$$, admits a local $$C^q$$-smooth Levi-decomposition with respect to $$\mathfrak g$$ as above. This result generalizes the theorem due to J. F. Conn [Ann. Math. (2) 121, 565–593 (1985; Zbl 0592.58025)].

##### MSC:
 53D17 Poisson manifolds; Poisson groupoids and algebroids
##### Keywords:
Poisson structures; Levi decomposition
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