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By Michael K. (-insb-cs) Weinstein and Alan D. (-ca


Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces. (English summary) Amer. J. Math. 123 (2001), no. 3, 525–550. Skew-symmetric bilinear operations which satisfy weakened versions of the Jacobi identity arise from a number of constructions in algebra and differential geometry. The purpose of this paper is to show how certain of these operations can be realized in a natural way on the tangent spaces of reductive homogeneous spaces. The authors also use such a construction to take steps toward finding group-like objects which “integrate ” these not-quite-Lie algebras. The main examples include the Courant algebroids and the omni-Lie algebras. The central result in this paper (Theorems 2.6 and 2.9) is that the skew-symmetrization of every Leibniz algebra structure can be extended in a natural way to a Lie-Yamaguti structure. Hence it can be realized as the projection of a Lie algebra bracket onto a reductive complement of a subalgebra. The (left) Leibniz algebra structure, introduced by Loday, is a bilinear operation on a vector space satisfying the derivation identity x · (y · z) = (x · y) · z + y · (x · z)

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