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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
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