The supergeometry of Loday algebroids

Janusz Grabowski; David Khudaverdyan; Norbert Poncin

doi:10.3934/jgm.2013.5.185

Article Contents

2013, Volume 5, Issue 2: 185-213. Doi: 10.3934/jgm.2013.5.185

This issue Previous Article Leibniz-Dirac structures and nonconservative systems with constraints Next Article Semi-global symplectic invariants of the Euler top

The supergeometry of Loday algebroids

1.
Polish Academy of Sciences, Institute of Mathematics, Śniadeckich 8, P.O. Box 21, 00-956 Warsaw
2.
University of Luxembourg, Campus Kirchberg, Mathematics Research Unit, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg City

Received: June 2012

Revised: April 2013

Published: June 2013

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Abstract

A new concept of Loday algebroid (and its pure algebraic version -- Loday pseudoalgebra) is proposed and discussed in comparison with other similar structures present in the literature. The structure of a Loday pseudoalgebra and its natural reduction to a Lie pseudoalgebra is studied. Further, Loday algebroids are interpreted as homological vector fields on a `supercommutative manifold' associated with a shuffle product and the corresponding Cartan calculus is introduced. Several examples, including Courant algebroids, Grassmann-Dorfman and twisted Courant-Dorfman brackets, as well as algebroids induced by Nambu-Poisson structures, are given.

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Mathematics Subject Classification: 53D17, 58A50, 17A32, 17B66.

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