$\mathcal{L}$-Lie algebroids over topological ringed spaces

TL;DR

This paper introduces generalized Lie algebroids and Gerstenhaber algebras over topological ringed spaces, unifying geometric and algebraic structures and extending classical correspondences in this broader context.

Contribution

It defines $\\mathcal{L}$-Lie algebroids and $\\mathcal{A}$-Gerstenhaber algebras over topological ringed spaces, expanding the theoretical framework of geometric and algebraic structures.

Findings

Established the concept of $\mathcal{L}$-Lie algebroids.

Introduced $\mathcal{A}$-Gerstenhaber algebras in this setting.

Extended classical correspondences to the generalized framework.

Abstract

The notion of Lie algebroids over a topological ringed space provides a unified framework to study various geometric structures. This geometric concept is intimately connected with well-known algebraic structures, including Gerstenhaber algebras and Batalin--Vilkovisky algebras. We introduce more general concepts such as $\mathcal{L}$-Lie algebroids and $\mathcal{A}$-Gerstenhaber algebras, associated with a given Lie algebroid $\mathcal{L}$ and Gerstenhaber algebra $\mathcal{A}$ over a topological ringed space, respectively. Following this, we explore how several standard correspondences extend within this broader framework.