The Free Courant Algebroid

TL;DR

This paper introduces the category of generalized Courant algebroids and constructs a free Courant algebroid using symmetric Leibniz algebroids, expanding the theoretical framework of algebroid structures.

Contribution

It defines symmetric Leibniz algebroids and constructs the free Courant algebroid from these, providing new insights into their structure and relationships.

Findings

Construction of the free Courant algebroid from symmetric Leibniz algebroids

Comparison between symmetric Leibniz algebroids and Loday algebroids

Introduction of the category of generalized Courant algebroids

Abstract

We introduce the category of generalized Courant algebroids and show that it admits a free object on any anchored vector bundle. The free Courant algebroid is built from two components: the generalized Courant algebroid associated to a symmetric Leibniz algebroid and the free symmetric Leibniz algebroid on an anchored vector bundle. Our construction is thus based on the new concept of symmetric Leibniz algebroid. We compare this subclass of Leibniz algebroids with the subclass of Loday algebroids that was introduced and studied in a previous paper of Grabowski, Khudaverdyan and Poncin.