Categorification of VB-Lie algebroids and VB-Courant algebroids
Abstract
In this paper, first we introduce the notion of a -Lie 2-algebroid, which can be viewed as the categorification of a -Lie algebroid. The tangent prolongation of a Lie 2-algebroid is a -Lie 2-algebroid naturally. We show that after choosing a splitting, there is a one-to-one correspondence between -Lie 2-algebroids and flat superconnections of a Lie 2-algebroid on a 3-term complex of vector bundles. Then we introduce the notion of a - 2-algebroid, which can be viewed as the categorification of a -Courant algebroid. We show that there is a one-to-one correspondence between split Lie 3-algebroids and split - 2-algebroids. The notion of a -Lie 2-bialgebroid is introduced and the double of a -Lie 2-bialgebroid is a - 2-algebroid. Finally, we introduce the notion of an E- 2-algebroid and show that associated to a - 2-algebroid, there is an E- 2-algebroid structure on the graded fat bundle naturally. By this result, we give a construction of a Lie 3-algebra from a given Lie 3-algebra, which provides interesting examples of Lie 3-algebras including the higher analogue of the string Lie 2-algebra.