Vol. 309, No. 1, 2020

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Lie 2-algebras of vector fields

Daniel Berwick-Evans and Eugene Lerman

Vol. 309 (2020), No. 1, 1–34
DOI: 10.2140/pjm.2020.309.1
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.

Keywords
Lie 2-algebra, stack, vector field, Lie groupoid
Mathematical Subject Classification
Primary: 17B66
Secondary: 18D05
Milestones
Received: 29 November 2019
Revised: 9 June 2020
Accepted: 7 August 2020
Published: 26 December 2020
Authors
Daniel Berwick-Evans
Department of Mathematics
University of Illinois at Urbana-Champaign
Urbana, IL
United States
Eugene Lerman
Department of Mathematics
University of Illinois at Urbana-Champaign
Urbana, IL
United States