Left-invariant vector fields on a lie 2-group

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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, see [2]. 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.

Original languageEnglish (US)
Article number21
Pages (from-to)604-634
Number of pages31
JournalTheory and Applications of Categories
StatePublished - 2019


  • 2 limit
  • Invariant vector fields
  • Lie 2-algebra
  • Lie 2-group

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

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