Lie 2-algebras of vector fields
Abstract
We show that the category of vector fields on a geometric stack has the structure of a Lie 2-algebra. This proves a conjecture of R.~Hepworth. The construction uses a Lie groupoid that presents the geometric stack. We show that the category of vector fields on the Lie groupoid is equivalent to the category of vector fields on the stack. The category of vector fields on the Lie groupoid has a Lie 2-algebra structure built from known (ordinary) Lie brackets on multiplicative vector fields of Mackenzie and Xu and the global sections of the Lie algebroid of the Lie groupoid. After giving a precise formulation of Morita invariance of the construction, we verify that the Lie 2-algebra structure defined in this way is well-defined on the underlying stack.
Cite
@article{arxiv.1609.03944,
title = {Lie 2-algebras of vector fields},
author = {Daniel Berwick-Evans and Eugene Lerman},
journal= {arXiv preprint arXiv:1609.03944},
year = {2020}
}
Comments
32 pages. (v2) Changes in exposition. In particular we exercise more care in handling maps of Lie 2-algebras. (v3) major reorganization and additional content (v4) more new content