Omni-Lie 2-algebras and their Dirac structures

TL;DR

This paper introduces omni-Lie 2-algebras as a categorification of omni-Lie algebras, establishing a correspondence between Lie 2-algebra structures and Dirac structures, including non-strict cases like string Lie 2-algebras.

Contribution

It defines omni-Lie 2-algebras, explores their Dirac structures, and extends the framework to twisted omni-Lie 2-algebras for non-strict Lie 2-algebras.

Findings

One-to-one correspondence between strict Lie 2-algebras and Dirac structures.

Dirac structures of graphs correspond to Lie 2-algebra structures on V.

Introduction of twisted omni-Lie 2-algebras capturing non-strict Lie 2-algebras.

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.