### 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 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 language | English (US) |
---|---|

Article number | 21 |

Pages (from-to) | 604-634 |

Number of pages | 31 |

Journal | Theory and Applications of Categories |

Volume | 34 |

State | Published - Jan 1 2019 |

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### Keywords

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

### ASJC Scopus subject areas

- Mathematics (miscellaneous)

### Cite this

*Theory and Applications of Categories*,

*34*, 604-634. [21].

**Left-invariant vector fields on a lie 2-group.** / Lerman, Eugene M.

Research output: Contribution to journal › Article

*Theory and Applications of Categories*, vol. 34, 21, pp. 604-634.

}

TY - JOUR

T1 - Left-invariant vector fields on a lie 2-group

AU - Lerman, Eugene M

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - 2 limit

KW - Invariant vector fields

KW - Lie 2-algebra

KW - Lie 2-group

UR - http://www.scopus.com/inward/record.url?scp=85071145581&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85071145581&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85071145581

VL - 34

SP - 604

EP - 634

JO - Theory and Applications of Categories

JF - Theory and Applications of Categories

SN - 1201-561X

M1 - 21

ER -