Categorification of VB-Lie algebroids and VB-Courant algebroids
Yunhe Sheng

TL;DR
This paper develops the categorification of VB-Lie and VB-Courant algebroids, establishing correspondences with Lie 2- and 3-algebroids, and introduces new structures like VB-LWX 2-algebroids and Lie 2-bialgebroids.
Contribution
It introduces the notions of VB-Lie 2-algebroids and VB-LWX 2-algebroids, establishing their relations with Lie 2- and 3-algebroids, and constructs new examples of Lie 3-algebras.
Findings
One-to-one correspondence between VB-Lie 2-algebroids and flat superconnections.
Equivalence between split Lie 3-algebroids and split VB-LWX 2-algebroids.
Construction of Lie 3-algebras including higher string Lie 2-algebra.
Abstract
In this paper, first we introduce the notion of a -Lie -algebroid, which can be viewed as the categorification of a -Lie algebroid. The tangent prolongation of a Lie -algebroid is a -Lie -algebroid naturally. We show that after choosing a splitting, there is a one-to-one correspondence between -Lie -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 -bialgebroid is introduced and the double of a -Lie -bialgebroid is a - 2-algebroid. Finally, we introduce the notion of an - 2-algebroid and show that…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Advanced Topics in Algebra · Sphingolipid Metabolism and Signaling
