Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces.

*(English)*Zbl 0986.17001Recall 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.

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.

Reviewer: J.M.Lodder (Las Cruces)