On symplectomorphisms of generalized Lie-Rinehart algebras

Lie-Rinehart algebras provide an algebraic abstraction of Lie algebroids, encoding simultaneously a Lie algebra structure and a module structure over a commutative algebra, together with a compatible anchor map acting by derivations. Since their introduction by Herz, these structures have played a central role in the algebraic formulation of differential geometry, Poisson geometry, and deformation theory, serving as the natural algebraic setting for derivations, differential forms, and cohomology on commutative algebras. An analogous construction for a general Lie algebroid has been introduced by Iglesias and Marrero under the name of a generalized Lie algebroid. In parallel, Jacobi algebras generalize Poisson algebras by allowing the Leibniz rule to hold up to a derivation, capturing algebraically the geometry of contact and locally conformal symplectic manifolds. While Poisson algebras correspond to Lie algebroids via the cotangent bundle construction, Jacobi algebras require a more flexible framework that incorporates both a Lie bracket and a distinguished derivation.

In our talk, we describe generalized Lie-Rinehart algebras as an extension of Lie-Rinehart algebras and give some examples. These structures extend classical Lie-Rinehart algebras by allowing the anchor to take values in first-order differential operators. We characterize generalized symplectic manifolds and contact manifolds in terms of generalized symplectic Lie-Rinehart algebras and we show the existence of Jacobi structure induced by a generalized symplectic Lie-Rinehart algebra. We introduce the notion of symplectomorphisms of generalized Lie-Rinehart algebras and describe them particularly in the case of contact manifolds, generalized symplectic manifolds and in the case of non-degenerate Jacobi structures