AV-Courant algebroids and generalized CR structures

TL;DR

This paper introduces a broad class of algebraic structures called AV-Courant algebroids, classifies them via third cohomology, and defines generalized CR structures that encompass classical CR and contact structures.

Contribution

It generalizes Courant algebroids through a cohomological classification and introduces generalized CR structures extending classical geometric structures.

Findings

AV-Courant algebroids include Courant algebroids and $ ext{E}^1(M)$ structures.

Generalized CR structures encompass classical CR and contact structures.

Classification of AV-Courant algebroids via third cohomology group $H^3(A,V)$.

Abstract

We construct a generalization of Courant algebroids which are classified by the third cohomology group $H^3(A,V)$, where $A$ is a Lie Algebroid, and $V$ is an $A$-module. We see that both Courant algebroids and $\mathcal{E}^1(M)$ structures are examples of them. Finally we introduce generalized CR structures on a manifold, which are a generalization of generalized complex structures, and show that every CR structure and contact structure is an example of a generalized CR structure.