Modules and representations up to homotopy of Lie n-algebroids Article Swipe
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· 2023
· Open Access
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· DOI: https://doi.org/10.1007/s40062-022-00322-x
· OA: W2997198201
This paper studies differential graded modules and representations up to homotopy of Lie n -algebroids, for general $$n\in {\mathbb {N}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> . The adjoint and coadjoint modules are described, and the corresponding split versions of the adjoint and coadjoint representations up to homotopy are explained. In particular, the case of Lie 2-algebroids is analysed in detail. The compatibility of a Poisson bracket with the homological vector field of a Lie n -algebroid is shown to be equivalent to a morphism from the coadjoint module to the adjoint module, leading to an alternative characterisation of non-degeneracy of higher Poisson structures. Moreover, the Weil algebra of a Lie n -algebroid is computed explicitly in terms of splittings, and representations up to homotopy of Lie n -algebroids are used to encode decomposed VB-Lie n -algebroid structures on double vector bundles.
