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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 X(G) of multiplicative vector fields. 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 X(G). As a result we get the Lie 2-algebra 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 g to a Lie 2-group G: apply the functor Lie : LieGp -> LieAlg to the structure maps of the category G. We show that the Lie 2-algebra g is isomorphic to the Lie 2-algebra X(G)^G of left invariant multiplicative vector fields.
@article{TAC_2019_34_a20,
author = {Eugene Lerman},
title = {Left-invariant vector fields on a {Lie} 2-group},
journal = {Theory and applications of categories},
pages = {604--634},
publisher = {mathdoc},
volume = {34},
year = {2019},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2019_34_a20/}
}
Eugene Lerman. Left-invariant vector fields on a Lie 2-group. Theory and applications of categories, Tome 34 (2019), pp. 604-634. http://geodesic.mathdoc.fr/item/TAC_2019_34_a20/