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

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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affine structure, multiplicative structure, $2$-vector space, strict monoidal category, Lie 2-algebra
Mathematical Subject Classification 2010
Primary: 53D17, 53D18
Secondary: 22A22, 70G45
Received: 18 March 2019
Revised: 21 February 2020
Accepted: 1 April 2020
Published: 4 September 2020
Honglei Lang
Department of Applied Mathematics
China Agricultural University
Zhangju Liu
School of Mathematical Sciences
Peking University
Yunhe Sheng
Department of Mathematics
Jilin University