Differential Batalin-Vilkovisky algebras arising from twilled Lie-Rinehart algebras

TL;DR

This paper explores the algebraic structures called twilled Lie-Rinehart algebras, their relation to complex structures on manifolds, and their connection to Batalin-Vilkovisky algebras, providing new insights into mirror symmetry and homological algebra.

Contribution

It characterizes twilled Lie-Rinehart structures via differential graded Lie and G-algebras, linking them to Batalin-Vilkovisky algebras and extending Koszul's results.

Findings

Characterization of twilled LR structures in terms of differential graded Lie and G-algebras

Identification of Batalin-Vilkovisky algebra structures arising from these algebras

A conceptual proof of the Tian-Todorov lemma using these algebraic frameworks

Abstract

Twilled L(ie-)R(inehart)-algebras generalize, in the Lie-Rinehart context, complex structures on smooth manifolds. An almost complex manifold determines an "almost twilled pre-LR algebra", which is a true twilled LR-algebra iff the almost complex structure is integrable. We characterize twilled LR structures in terms of certain associated differential (bi)graded Lie and G(erstenhaber)-algebras; in particular the G-algebra arising from an almost complex structure is a d(ifferential) G-algebra iff the almost complex structure is integrable. Such G-algebras, endowed with a generator turning them into a B(atalin-)V(ilkovisky)-algebra, occur on the B-side of the mirror conjecture. We generalize a result of Koszul to those dG-algebras which arise from twilled LR-algebras. A special case thereof explains the relationship between holomorphic volume forms and exact generators for the corresponding dG-algebra and thus yields in particular a conceptual proof of the Tian-Todorov lemma. We give a differential homological algebra interpretation for twilled LR-algebras and by means of it we elucidate the notion of generator in terms of homological duality for differential graded LR-algebras.