Affine structures on Lie groupoids

Honglei Lang, Zhangju Liu and Yunhe Sheng

Vol. 307 (2020), No. 2, 353–382

DOI: 10.2140/pjm.2020.307.353

Abstract

We study affine structures on a Lie groupoid, including affine $k$-vector fields, $k$-forms and $\left(p,q\right)$-tensors. We show that the space of affine structures is a $2$-vector space over the space of multiplicative structures. Moreover, the space of affine multivector fields with the Schouten bracket and the space of affine vector-valued forms with the Frölicher–Nijenhuis bracket are graded strict Lie $2$-algebras, and affine $\left(1,1\right)$-tensors constitute a strict monoidal category. Such higher structures can be seen as the categorification of multiplicative structures on a Lie groupoid.

Keywords

affine structure, multiplicative structure, $2$-vector space, strict monoidal category, Lie 2-algebra

Mathematical Subject Classification 2010

Primary: 53D17, 53D18

Secondary: 22A22, 70G45

Milestones

Received: 18 March 2019

Revised: 21 February 2020

Accepted: 1 April 2020

Published: 4 September 2020