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Journal of Lie Theory 28 (2018), No. 3, 711--733
Copyright Heldermann Verlag 2018

Representations up to Homotopy from Weighted Lie Algebroids

Andrew James Bruce
Mathematics Research Unit, University of Luxembourg, Maison du Nombre, 6 Avenue de la Fonte, 4364 Esch-sur-Alzette, Luxembourg

Janusz Grabowski
Institute of Mathematics, Polish Academy of Sciences, ul. Sniadeckich 8, 00-656 Warszawa, Poland

Luca Vitagliano
Dept. of Mathematics, UniversitÓ degli Studi di Salerno, Via Giovanni Paolo II n. 123, 84084 Fisciano, Italy

Weighted Lie algebroids were recently introduced as Lie algebroids equipped with an additional compatible non-negative grading, and represent a wide generalisation of the notion of a VB-algebroid. There is a close relation between two term representations up to homotopy of Lie algebroids and VB-algebroids. In this paper we show how this relation generalises to weighted Lie algebroids and in doing so we uncover new and natural examples of higher term representations up to homotopy of Lie algebroids. Moreover, we show how the van Est theorem generalises to weighted objects.

Keywords: Graded manifolds, Lie algebroids, Lie groupoids, representations up to homotopy.

MSC: 16W50, 22A22, 53D17

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