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Multi-Derivations and Kähler Forms

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Poisson Structures

Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 347))

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Abstract

The algebraic structure of the space of differential forms on a manifold is well-known; the algebraic structure on the space of multi-vector fields may be less well-known. We present both here, together with the algebraic counterpart of these spaces, which are respectively the spaces of Kähler forms and of skew-symmetric multi-derivations of an algebra. It leads to the Gerstenhaber structure on the space of multi-vector fields of a given manifold (or multi-derivations of a given algebra), a space in which the Poisson structure lives and which acts on the space of differential forms on the manifold (Kähler forms of the algebra), via the generalized Lie derivative.

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Laurent-Gengoux, C., Pichereau, A., Vanhaecke, P. (2013). Multi-Derivations and Kähler Forms. In: Poisson Structures. Grundlehren der mathematischen Wissenschaften, vol 347. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31090-4_3

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