Connections, Poisson brackets, and affine algebras (Q1911783)

The paper is the third in a series aimed at examining algebraic themes in symplectic and Poisson geometry [see also Commun. Algebra 23, 4669-4886 (1995; Zbl 0841.17012)]. Namely, the goal of this paper is to uncover the algebraic essence of connections and use it to analyse Poisson algebras. First, the paper finds a global algebraic description of Poisson regularity. Then, the relation between the existence of parallel connection and Poisson regularity is shown. Eventually, a Poisson connection for a Poisson regular algebra is explicitly exhibited.