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Subalgebras of finite codimension in symplectic Lie algebra. (English) Zbl 1064.17508

This short paper proves that each subalgebra of finite codimension in the algebra of symplectic vector fields on \(\mathbb R^{2n}\) vanishing at the origin contains the whole subalgebra of all fields infinitely flat at origin. This result clearly should be expected and it plays the role of Borel’s theorem on formal power series in the theory of natural bundles on symplectic manifolds, but the present proof relies on a simple observation from linear algebra and thus it is quite nice.

MSC:

17B66 Lie algebras of vector fields and related (super) algebras
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