The Supergeometry of Loday Algebroids

TL;DR

This paper introduces the concept of Loday algebroids, explores their algebraic and geometric structures, and relates them to existing structures like Courant algebroids, providing new insights into their properties and applications.

Contribution

It proposes a new definition of Loday algebroids, studies their relation to Lie pseudoalgebras, and interprets them as homological vector fields on supercommutative manifolds.

Findings

Loday pseudoalgebras reduce naturally to Lie pseudoalgebras.

Loday algebroids can be viewed as homological vector fields on supermanifolds.

Several examples including Courant algebroids and Nambu-Poisson structures are provided.

Abstract

A new concept of Loday algebroid (and its pure algebraic version - Loday pseudoalgebra) is proposed and discussed in comparison with other similar structures present in the literature. The structure of a Loday pseudoalgebra and its natural reduction to a Lie pseudoalgebra is studied. Further, Loday algebroids are interpreted as homological vector fields on a `supercommutative manifold' associated with a shuffle product and the corresponding Cartan calculus is introduced. Several examples, including Courant algebroids, Grassmann-Dorfman and twisted Courant-Dorfman brackets, as well as algebroids induced by Nambu-Poisson structures, are given.