Omni-Lie superalgebras and Lie 2-superalgebras (Q2258096)

In this paper, one introduces the notion of omni-Lie superalgebras as a super analogue of an omni-Lie algebra introduced by A. Weinstein. This algebraic structure gives a nontrivial example of Leibniz superalgebras and Lie 2-superalgebras. We also study Dirac structures of omni-Lie superalgebra in order to characterize all possible Lie superalgebra structures on a super vector space. We prove that there is a one-to-one correspondence between Dirac structures of the omni-Lie superalgebra and Lie superalgebra structures on a subspace of a super vector space. More precisely, we prove that an omni-Lie superalgebra is a Leibniz superalgebra as well as a Lie 2-superalgebra, which is a super version of a Lie 2-algebra or a 2-term \(L_\infty\)-algebra.