Omni-Lie superalgebras and Lie 2-superalgebras

Abstract

We introduce the notion of omni-Lie superalgebra as a super version of the omni-Lie algebra introduced by Weinstein. This algebraic structure gives a nontrivial example of Leibniz superalgebra and Lie 2-superalgebra. We prove that there is a one-to-one correspondence between Dirac structures of the omni-Lie superalgebra and Lie superalgebra structures on subspaces of a super vector space.

Conformal Lie 2-algebras and conformal omni-Lie algebras

“…The notion of omni-Lie algebras was generalized to omni-Lie superalgebras in [22]. In thus subsection, we introduce the concept of conformal omni-Lie algebras and construct 2-term conformal L ∞ -algebras from them.…”

“…Omni-Lie algebras are studied from several aspects and are generalized to omni-Lie algebroids and omni-Lie 2-algebras in [4,5,9,19]. In a recent paper [22], we study dirac structures of omni-Lie superalgebras. Now a natural question arise, does there exist a categorification of a Lie conformal algebra or a vertex Lie algebra?…”

Non-abelian tensor product and homology of Lie superalgebras

“…Now we are ready to introduce the following notion of crossed modules of Lie superalgebras (see also [23,Definition 5]). Definition 2.5.…”

Generalized omni-Lie algebras

“…Actually, an omni-Lie algebra is a Lie 2-algebra since Roytenberg and Weinstein proved that every Courant algeboid gives rise to a Lie 2-algebra [5]. Recently, omni-Lie algebras were generalized to omni-Lie superalgebras, omni-Lie color algebras and omni-Lie algebroids [1,8]. In [2], they generalized omni-Lie algebras from a linear space to a linear bundle E in order to characterize all possible Lie algebroid structures on E. Dirac structures were also studied from several aspects [2,3,6].…”