Omni-Lie 2-algebras and their Dirac structures

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.

💡 Research Summary

The paper introduces the omni‑Lie 2‑algebra, a categorified analogue of Weinstein’s omni‑Lie algebra, and shows that Dirac structures on this object encode both strict and non‑strict Lie‑2‑algebra structures on a given 2‑vector space 𝓥.

Construction of the omni‑Lie 2‑algebra.
A 2‑vector space 𝓥 is a linear category (objects and morphisms are vector spaces) and its endomorphism 2‑category 𝔤𝔩(𝓥) plays the role of a “general linear” 2‑algebra. The authors define
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