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

Abstract

Abstract. We show that the skew-symmetrized product on every Leibniz algebra E can be realized on a reductive complement to a subalgebra in a Lie algebra. As a consequence, we construct a nonassociative multiplication on E which, when E is a Lie algebra, is derived from the integrated adjoint representation. We apply this construction to realize the bracket operations on the sections of Courant algebroids and on the "omni-Lie algebras" recently introduced by the second author.

Omni-Lie superalgebras and Lie 2-superalgebras

“…An omni-Lie algebra is actually a Lie 2-algebra since Roytenberg and Weinstein proved that every Courant algebroid gives rise to a Lie 2-algebra ( [13]). Rencently, omni-Lie algebras are studied from several aspects ( [4], [9], [16]) and are generalized to omni-Lie algebroids and omni-Lie 2-algebras in [5,6,15]. The corresponding Dirac structures are also studied therein.…”

On Killing Forms and Invariant Forms of Lie-Yamaguti Superalgebras

“…Later on, these algebraic objects were called "Lie triple algebras" [3] and the terminology of "Lie-Yamaguti algebras" is introduced in [4] for these algebras. For further development of the theory of Lie-Yamaguti algebras one may refer, for example, to [5][6][7][8].…”

Cohomology and Deformations of Courant Pairs

“…It is also important to note that a Courant algebras can be viewed as a model to get Leibniz algebras in the form of hemisemidirect products (as introduced in ref. [7]) by starting with a Lie algebra g and a g-modules. One of the Lie algebras mostly encountered in geometry or in mathematical physics is the Lie algebra χ(M) of smooth vector fields on a smooth manifold M. This Lie algebra (which is also a Leibniz algebra) can also be described as the space of linear derivations on the associative algebra C ∞ (M) of smooth functions on the given manifold.…”