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Left-invariant vector fields on a Lie 2-group

##
Eugene Lerman

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.

Keywords: Lie 2-group, Lie 2-algebra, invariant vector fields, 2 limit

2010 MSC: 18D05, 22A22, 22E, 17B

*Theory and Applications of Categories,*
Vol. 34, 2019,
No. 21, pp 604-634.

Published 2019-08-02.

http://www.tac.mta.ca/tac/volumes/34/21/34-21.pdf

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