Extensions of Lie-Rinehart algebras and cotangent bundle reduction (Q2864129)

If \(Q\) is a smooth manifold, the cotangent bundle \(T^*Q\) carries a canonical symplectic structure. When a Lie group \(G\) acts smoothly on \(Q\), one can lift the action to \(T^*Q\); hence \(G\) acts on the Poisson algebra \(C^\infty(T^*Q)\). The Poisson algebra of \(G\)-invariant smooth functions on \(T^*Q\) induces a Poisson structure on the space of \(G\)-orbits \(T^*Q/G\). This space is not smooth in general. It will be, for example, if the \(G\)-action on \(Q\) is principal.NEWLINENEWLINEA Lie-Rinehart algebra is a pair \((A,L)\), where \(A\) is a commutative algebra over a commutative ring \(R\) and \(L\) is a Lie algebra over \(R\). It is furthermore required that \(L\) has a left \(A\)-module structure and that there exists an \(A\)-linear Lie algebra homomorphism \(L\to\mathrm{Der}_R(A)\), satisfying the Leibnitz rule. A Lie-Rinehart algebra can be seen as an algebraic incarnation of a Lie algebroid.NEWLINENEWLINEIn the paper under review, the Poisson structure on smooth functions on \(T^*Q/G\) is described in terms of (extensions of) Lie-Rinehart algebras. The key ingredient, introduced by the authors, is the tautological Poisson structure of a Lie-Rinehart algebra. This is a substantial generalization of the well-known correspondence between a Lie algebroid structure on a vector bundle \(E\to M\) and a Poisson structure on its dual \(E^*\to M\), \(M\) being a smooth manifold.NEWLINENEWLINEWhen the \(G\)-action is principal there is an explicit description of the Poisson algebra structure of \(C^{\infty}(T^*Q/G)\), see [\textit{R. Montgomery} et al., Contemp. Math. 28, 101--114 (1984; Zbl 0546.58026)]. The authors explain how this description fits in the general framework of Lie-Rinehart algebras.NEWLINENEWLINEThey also show with various examples that when the \(G\)-action is not free, the description of the Poisson structure on functions on \(T^*Q/G\) in terms of Lie-Rinehart algebras and their extensions is only partially possible: indeed, the Lie-Rinehart algebra description recovers the \(G\)-invariant functions on \(T^*Q\) which are polynomial on the fibres; these functions clearly inject into \(C^{\infty}(T^*Q)^G\), but the embedding is not surjective in the Fréchet topology.