Modular class of Lie $ \infty $-algebroids and adjoint representations

Raquel Caseiro; Camille Laurent-Gengoux

doi:10.3934/jgm.2022008

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2022, Volume 14, Issue 2: 273-305. Doi: 10.3934/jgm.2022008

This issue Previous Article Constrained systems, generalized Hamilton-Jacobi actions, and quantization Next Article The evolution equation: An application of groupoids to material evolution

Modular class of Lie $ \infty $-algebroids and adjoint representations

Raquel Caseiro^1, and
Camille Laurent-Gengoux^2,

1.
University of Coimbra, CMUC, Department of Mathematics, Apartado 3008, EC Santa Cruz, 3001-501 Coimbra, Portugal
2.
Institut Élie Cartan de Lorraine, UMR 7502 du CNRS, Université de Lorraine, Metz

Received: August 2021

Early access: April 21, 2022

Published online: April 21, 2022

Raquel Caseiro is partially supported by the Centre for Mathematics of the University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES. Camille Laurent-Gengoux thanks the "Granum" project MITI 80 primes CNRS for its financial support while this project was completed.

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Abstract

We study the modular class of $ Q $-manifolds, and in particular of negatively graded Lie $ \infty $-algebroid. We show the equivalence of several descriptions of those classes, that it matches the classes introduced by various authors and that the notion is homotopy invariant. In the process, the adjoint and coadjoint actions up to homotopy of a Lie $ \infty $-algebroid are spelled out. We also wrote down explicitly some dualities, e.g. between representations up to homotopies of Lie $ \infty $-algebroids and their $ Q $-manifold equivalent, which we hope to be of use for future reference.

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Mathematics Subject Classification: Primary: 57R30, 53D17.

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