From Lie--Rinehart Algebras to $F$-Manifold Algebras

Abstract

For every Lie--Rinehart algebra, we construct an $F$-manifold algebra on the direct sum of its base algebra and module. Contrary to the assertion in \cite[Proposition 13.3.26]{LodayVallette}, the resulting structure is generally not Poisson. We determine when powers of the positive-degree ideal in the associated symmetric Poisson algebra are Poisson ideals, and relate the Leibnizator to the Lie--Rinehart differential. For a finite projective module of constant rank, the trace of the Leibnizator recovers the anchor and yields a rigidity result for injective anchors. We conclude with algebraic and geometric examples.