Modular Classes of Loday Algebroids

TL;DR

This paper introduces Loday algebroids as a generalization of Courant algebroids, defining their cohomology and modular class, and explores properties of their doubles and relations between different cohomologies.

Contribution

It defines Loday algebroids, introduces their cohomology and modular class, and investigates their properties and relations to Courant algebroids and Lie bialgebroids.

Findings

Modular class of the double of a Lie bialgebroid vanishes.

Relations between naive and standard cohomologies for Courant algebroids are described.

Conjecture that naive and standard cohomologies are isomorphic for transitive Courant algebroids.

Abstract

We introduce the concept of Loday algebroids, a generalization of Courant algebroids. We define the naive cohomology and modular class of a Loday algebroid, and we show that the modular class of the double of a Lie bialgebroid vanishes. For Courant algebroids, we describe the relation between the naive and standard cohomologies and we conjecture that they are isomorphic when the Courant algebroid is transitive.