Lie Algebroids in the Loday-Pirashvili Category

TL;DR

This paper explores the structure of Lie-Rinehart algebras within the Loday-Pirashvili tensor category and establishes a functorial relationship to Leibniz algebroids, advancing the understanding of algebraic structures in this context.

Contribution

It introduces a novel perspective on Lie-Rinehart algebras in the Loday-Pirashvili category and constructs a functor to Leibniz algebroids, linking these algebraic frameworks.

Findings

Defined Lie-Rinehart algebras in the Loday-Pirashvili category

Constructed a functor from these algebras to Leibniz algebroids

Provided new insights into algebraic structures in tensor categories

Abstract

We describe Lie-Rinehart algebras in the tensor category $\mathcal{LM}$ of linear maps in the sense of Loday and Pirashvili and construct a functor from Lie-Rinehart algebras in $\mathcal{LM}$ to Leibniz algebroids.