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 $\left(p,q\right)$-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.

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