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By Luther Pfahler Eisenhart

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Each element of a semisimple Lie algebra of rank r is contained in an r-dimensional commutative subalgebra. Proof l 0 • One of the possible dennitions of a semisimple Lie algebra consists in the nondegeneracy of its Killing form (x, y) = tr(adxo ad,). This form is invariant with respect to the adjoint action; therefore the adjoint representation of a semisimple Lie algebra is isomorphic to the coadjoint one. r. e. the co rank of its Poisson structure at a generic point). It follows from the annihilator theorem that the rank of the annihilator g~ of an arbitrary E g* is not less than the rank of g.

A sufficiently small neighbourhood of a Lagrangian submanifold is symplectomorphic to a neighbourhood of the zero section in its cotangent bundle. The corank of the restriction of the symplectic form to a (co)isotropic submanifold is equal to its (co)dimension and is therefore constant. Consequently (see sect. 1, chap. 2), a germ of a (co)isotropic submanifold reduces in suitable Darboux coordinates to the linear normal form of sect. 2, chap. 1. The symplectic type of an isotropic submanifold Nk c M 2" in its tubular neighbourhood is determined in a one-one fashion by the equivalence class ofthe following 2(n- k)-dimensional symplectic vector bundle with base space Nk: the fibre of this bundle at a point x is the quotient space of the skew-orthogonal complement to the tangent space T"N by the space T"N itself.

Poisson Manifolds. A Poisson structure on a manifold is a bilinear form { , } on the space of smooth functions on it satisfying the requirement of anticommutativity, the Jacobi identity and the Leibniz rule (see corollary 2 of the preceding item). This form we shall still call a Poisson bracket. The first two properties of the Poisson bracket mean that it gives a Lie algebra structure on the space of smooth functions on the manifold. From the Leibniz rule it follows that the Poisson bracket of an arbitrary function with a function having a secondorder zero at a given point vanishes at this point; therefore a Poisson structure defines an exterior 2-form on each cotangent space to the manifold, depending smoothly on the point of application.

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