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Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces. (English) Zbl 0986.17001
Recall that in K. Yamaguti [J. Sci. Hiroshima Univ., Ser. A 21, 155-160 (1958; Zbl 0084.18404)] an algebraic description of Nomizu’s affine connections with parallel torsion and curvature was given in terms of a bilinear and a trilinear operation on a vector space, called a Lie-Yamaguti algebra in the paper under review. Furthermore, J.-L. Loday has defined a left Leibniz algebra as vector space $${\mathbf g}$$ together with a bilinear operation $$[\;,\;]$$ which satisfies the (left) derivation property $[x,[y,z]]= [[x,y],z]+ [y,[x,z]],$ although the bracket $$[\;,\;]$$ is not necessarily skew-symmetric [J.-L. Loday and T. Pirashvili, Math. Ann. 296, 139-158 (1993; Zbl 0821.17022)]. The main result of the paper under review is that the skew-symmetrization of every Leibniz algebra $[[x,y]]= \tfrac 12 ([x,y]- [y,x])$ can be extended to a Lie-Yamaguti structure, which can then be realized as the projection of a Lie bracket onto a reductive complement of a subalgebra.
A key construction of the paper is the hemisemidirect product of a Lie algebra $${\mathbf h}$$ and an $${\mathbf h}$$-module $$V$$. Let $$E={\mathbf h}\times V$$ and define a binary operation $$\cdot$$ on $$E$$ via $(\xi,x)\cdot (\eta,y)= ([\xi,\eta], x y).$ Then $$(E,\cdot)$$ is a left Leibniz algebra with $${\mathbf h}$$-equivariant projection $$\pi:E\to{\mathbf h}$$. From the above construction, the authors define an enveloping Lie algebra of any Leibniz algebra $$(E,\cdot)$$ as a triple $$({\mathbf g},{\mathbf h},f)$$, where $${\mathbf h}$$ is a Lie algebra which acts on $$E$$ via (left) derivations, and $$f:E\to{\mathbf h}$$ is an $${\mathbf h}$$-equivariant projection that factors through the left multiplication map $$\lambda:E\to \text{Der}(E)$$. Above $${\mathbf g}={\mathbf h}\times E$$ is the semidirect product Lie algebra, where $$E$$ is given the zero bracket. The notion of an enveloping Lie algebra of a Leibniz algebra is not the same as Loday’s enveloping algebra of a Leibniz algebra. It is proved that every Leibniz algebra has enveloping Lie algebras, and that every Leibniz algebra can be embedded as a subalgebra in a hemisemidirect product. The space of smooth sections of a Courant algebroid is known to carry the structure of a Leibniz algebra [see, for example, D. Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds (Ph.D. thesis, University of California, Berkeley) (1999)], and an enveloping Lie algebra for this space of sections is constructed.
Finally as an open question, it is asked whether the Leibniz algebra arising from a Courant algebroid can be realized as the infinitesimal construction associated to a group-like structure, a question having its roots in Loday’s work.

##### MSC:
 17A32 Leibniz algebras 53C30 Differential geometry of homogeneous manifolds
##### Citations:
Zbl 0084.18404; Zbl 0821.17022
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