Omni-Lie 2-algebras and their Dirac structures

Abstract

We introduce the notion of omni-Lie 2-algebra, which is a categorification of Weinstein's omni-Lie algebras. We prove that there is a one-to-one correspondence between strict Lie 2-algebra structures on 2-sub-vector spaces of a 2-vector space $\V$ and Dirac structures on the omni-Lie 2-algebra $ \gl(\V)\oplus \V $. In particular, strict Lie 2-algebra structures on $\V$ itself one-to-one correspond to Dirac structures of the form of graphs. Finally, we introduce the notion of twisted omni-Lie 2-algebra to describe (non-strict) Lie 2-algebra structures. Dirac structures of a twisted omni-Lie 2-algebra correspond to certain (non-strict) Lie 2-algebra structures, which include string Lie 2-algebra structures.