Left-invariant vector fields on a Lie 2-group

Abstract

A Lie 2-group $G$ is a category internal to the category of Lie groups. Consequently it is a monoidal category and a Lie groupoid. The Lie groupoid structure on $G$ gives rise to the Lie 2-algebra $\mathbb{X}(G)$ of multiplicative vector fields, see (Berwick-Evans -- Lerman). The monoidal structure on $G$ gives rise to a left action of the 2-group $G$ on the Lie groupoid $G$, hence to an action of $G$ on the Lie 2-algebra $\mathbb{X}(G)$. As a result we get the Lie 2-algebra $\mathbb{X}(G)^G$ of left-invariant multiplicative vector fields. On the other hand there is a well-known construction that associates a Lie 2-algebra $\mathfrak{g}$ to a Lie 2-group $G$: apply the functor $\mathsf{Lie}: \mathsf{Lie Groups} \to \mathsf{Lie Algebras}$ to the structure maps of the category $G$. We show that the Lie 2-algebra $\mathfrak{g}$ is isomorphic to the Lie 2-algebra $\mathbb{X}(G)^G$ of left invariant multiplicative vector fields.