## Homology, Homotopy and Applications

- Homology Homotopy Appl.
- Volume 9, Number 2 (2007), 101-135.

### From loop groups to 2-groups

John C. Baez, Danny Stevenson, Alissa S. Crans, and Urs Schreiber

#### Abstract

We describe an interesting relation between Lie 2-algebras,
the Kac-Moody central extensions of loop groups, and the
group String($n$). A Lie 2-algebra is a categorified version of
a Lie algebra where the Jacobi identity holds up to a natural
isomorphism called the 'Jacobiator.' Similarly, a Lie 2-group is
a categorified version of a Lie group. If $G$
is a simply-connected
compact simple Lie group, there is a 1-parameter family of Lie
2-algebras $\mathfrak{g}_k$ each having $\mathfrak{g}$ as its Lie algebra of objects, but
with a Jacobiator built from the canonical 3-form on $G$. There
appears to be no Lie 2-group having $\mathfrak{g}_\mathcal{k}$ as its Lie 2-algebra,
except when $k = 0$. Here, however, we construct for integral $k$
an infinite-dimensional Lie 2-group $\mathcal{P}_k G$ whose Lie 2-algebra
is *equivalent* to $\mathfrak{g}_\mathcal{k}$. The objects of $\mathfrak{g}_\mathcal{k}$ are based paths in
$G$-, while the automorphisms of any object form the level-$k$
Kac-Moody central extension of the loop group $\Omega{G}$. This 2-
group is closely related to the $k$th power of the canonical gerbe
over $G$. Its nerve gives a topological group $|\mathcal{P}_k G|$ that is an
extension of $G$ by $K(\mathbb{Z}, 2)$. When $k = \pm 1, |\mathcal{P}_k G|$ can also be
obtained by killing the third homotopy group of $G$. Thus, when
$G = \rm{Spin}(n), |PkG|$ is none other than String($n$).

#### Article information

**Source**

Homology Homotopy Appl., Volume 9, Number 2 (2007), 101-135.

**Dates**

First available in Project Euclid: 23 January 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.hha/1201127333

**Mathematical Reviews number (MathSciNet)**

MR2366945

**Zentralblatt MATH identifier**

1122.22003

**Subjects**

Primary: 22E67: Loop groups and related constructions, group-theoretic treatment [See also 58D05]

**Keywords**

gerbe Kac–Moody extension Lie 2-algebra loop group string group 2-group

#### Citation

Baez, John C.; Stevenson, Danny; Crans, Alissa S.; Schreiber, Urs. From loop groups to 2-groups. Homology Homotopy Appl. 9 (2007), no. 2, 101--135. https://projecteuclid.org/euclid.hha/1201127333