A Courant algebroid is defined as a real vector bundle equipped with a product and a symmetric bilinear on its space of sections satisfying a particular set of conditions. What would be the definition of a Courant algebroid on the associated principal bundle? In other words, what is the structure that a principal bundle must have in order for the associated vector bundle to be a Courant algebroid?


Thanks.



Principal bundle associated to a Courant algebroid
Since the bracket (you call it product) is a bidifferential operator on the space of sections of the vector bundle, the translation to the principal bundles is not straightforward. You have to use the first jet bundle of a principal bundle and to associate to it the jet bundle of the associated vector bundle. Bilinearity has to be linearized by using the tensor-product (of the space of sections), etc.
This works like a bilinear extension of lifting a connection to a principal bundle. 


