Homological sections of Lie algebroids

TL;DR

This paper explores Lie superalgebroids with a special homological section, generalizing Q-manifolds and Lie superalgebras, and introduces an odd Loday-Leibniz bracket structure on their sections.

Contribution

It introduces the concept of inner Q-algebroids with homological sections and demonstrates their connection to odd Loday-Leibniz brackets via derived brackets.

Findings

Inner Q-algebroids generalize Q-manifolds and Lie superalgebras.

The space of sections admits an odd Loday-Leibniz bracket.

Derived bracket formalism links homological sections to algebraic structures.

Abstract

We examine Lie (super)algebroids equipped with a homological section, i.e., an odd section that `self-commutes', we refer to such Lie algebroids as inner Q-algebroids: these provide natural examples of suitably "superised" Q-algebroids in the sense of Mehta. Such Lie algebroids are a natural generalisation of Q-manifolds and Lie superalgebras equipped with a homological element. Amongst other results, we show that, via the derived bracket formalism, the space of sections of an inner Q-algebroid comes equipped with an odd Loday-Leibniz bracket.