Representations of Homotopy Lie-Rinehart Algebras

TL;DR

This paper introduces a new framework for defining connections on strong homotopy Lie-Rinehart algebras, unifying and extending concepts of representations and actions in homotopical algebra.

Contribution

It provides a novel definition of connections on strong homotopy Lie-Rinehart algebras, generalizing representations of Lie algebroids and actions of homotopy Lie algebras.

Findings

Defined left/right connections on strong homotopy Lie-Rinehart algebras

Unified representations up to homotopy and actions on graded manifolds

Discussed Schouten-Nijenhuis calculus in this context

Abstract

I propose a definition of left/right connection along a strong homotopy Lie-Rinehart algebra. This allows me to generalize simultaneously representations up to homotopy of Lie algebroids and actions of strong homotopy Lie algebras on graded manifolds. I also discuss the Schouten-Nijenhuis calculus associated to strong homotopy Lie-Rinehart connections.