# VB-Courant algebroids, E-Courant algebroids and generalized geometry

## TL;DR

This paper explores the relationships between VB-Courant and E-Courant algebroids, introduces generalized complex structures on E-Courant algebroids, and connects these structures to complex Lie algebras, unifying various geometric frameworks.

## Contribution

It establishes the connection between VB-Courant and E-Courant algebroids, introduces generalized complex structures on E-Courant algebroids, and relates these to complex Lie algebra structures.

## Key findings

- Constructed examples of E-Courant algebroids.
- Unified generalized complex structures on different manifolds.
- Showed correspondence between structures on omni-Lie algebras and complex Lie algebras.

## Abstract

In this paper, we first discuss the relation between VB-Courant algebroids and E-Courant algebroids and construct some examples of E-Courant algebroids. Then we introduce the notion of a generalized complex structure on an E-Courant algebroid, unifying the usual generalized complex structures on even-dimensional manifolds and generalized contact structures on odd-dimensional manifolds. Moreover, we study generalized complex structures on an omni-Lie algebroid in detail. In particular, we show that generalized complex structures on an omni-Lie algebra $\gl(V)\oplus V$ correspond to complex Lie algebra structures on V.