a linear Poisson bracket, known in the literature under the names of Lie-Poisson, Kirillov-Poisson or KKS (Kostant-Kirillov-Souriau) structure:where and the derivatives are interpreted as elements of the bidual . Equivalently, the Poisson bivector can be locally expressed aswhere are coordinates on and are the associated structure constants of , Conversely, any linear Poisson structure on must be of this form, i.e. there exists a natural Lie algebra structure induced on whose Lie-Poisson bracket recovers . The symplectic leaves of the Lie-Poisson structure on are the orbits of the coadjoint action of on . Fibrewise linear Poisson structures The previous example can be generalised as follows. A Poisson structure on the total space of a vector bundle is called fibrewise linear when the bracket of two smooth functions , whose restrictions to the fibres are linear, is still linear when restricted to the fibres. Equivalently, the Poisson bivector field is asked to satisfy for any , where is the scalar multiplication . The class of vector bundles with linear Poisson structures coincides actually with that of (dual of) Lie algebroids. Indeed, the dual of any Lie algebroid carries a fibrewise linear Poisson bracket, uniquely defined bywhere is the evaluation by . Equivalently, the Poisson bivector can be locally expressed aswhere are coordinates around a point , are fibre coordinates on , dual to a local frame of , and and are the structure function of , i.e. the unique smooth functions satisfyingConversely, any fibrewise linear Poisson structure on must be of this form, i.e. there exists a natural Lie algebroid structure induced on whose Lie-Poisson backet recovers . The symplectic leaves of are the cotangent bundles of the algebroid orbits ; equivalently, if is integrable to a Lie groupoid , they are the connected components of the orbits of the cotangent groupoid . For one recovers linear Poisson structures, while for the fibrewise linear Poisson structure is the nondegenerate one given by the canonical symplectic structure of the cotangent bundle . Other examples and constructions Any constant bivector field on a vector space is automatically a Poisson structure; indeed, all three terms in the Jacobiator are zero, being the bracket with a constant function. Any bivector field on a 2-dimensional manifold is automatically a Poisson structure; indeed, is a 3-vector field, which is always zero in dimension 2. Given any Poisson bivector field on a 3-dimensional manifold , the bivector field