Shifted double Lie-Rinehart algebras

TL;DR

This paper extends the concepts of shifted double Poisson and Lie-Rinehart structures to monoids in symmetric monoidal categories, establishing an equivalence between shifted and non-shifted structures via a shift operation.

Contribution

It introduces a generalized framework for shifted double Lie-Rinehart algebras in a categorical setting and proves their equivalence to non-shifted counterparts with a shift in the module.

Findings

Generalization of shifted double Poisson and Lie-Rinehart structures to monoids in symmetric monoidal categories.

Establishment of an equivalence between shifted and non-shifted structures via a shift operation.

Provides a categorical foundation for studying shifted algebraic structures.

Abstract

We generalize the notions of shifted double Poisson and shifted double Lie-Rinehart structures, defined by Van den Bergh in [VdB08a, VdB08b], to monoids in a symmetric monoidal abelian category. The main result is that an n-shifted double Lie-Rinehart structure on a pair (A, M) is equivalent to a non-shifted double Lie-Rinehart structure on the pair (A, M [--n]).