Vol. 307, No. 2, 2020

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Affine structures on Lie groupoids

Honglei Lang, Zhangju Liu and Yunhe Sheng

Vol. 307 (2020), No. 2, 353–382
Abstract

We study affine structures on a Lie groupoid, including affine k-vector fields, k-forms and (p,q)-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 (1,1)-tensors constitute a strict monoidal category. Such higher structures can be seen as the categorification of multiplicative structures on a Lie groupoid.

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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
Authors
Honglei Lang
Department of Applied Mathematics
China Agricultural University
Beijing
China
Zhangju Liu
School of Mathematical Sciences
Peking University
Beijing
China
Yunhe Sheng
Department of Mathematics
Jilin University
Changchun
China