nLab
loop group

Idea

The loop space of a topological group inherits itself the structure of a group under pointwise group multiplication of loops. This is called a loop group.

Properties

Loop groups of compact Lie groups have canonical central extensions, often called Kac-Moody central extensions . A detailed discussion is in (PressleySegal). A review is in (BCSS)

Related concepts

• string group

References

The standard textbook on loop groups is

• Andrew Pressley, Graeme Segal, Loop groups Oxford University Press (1988)

A review of some aspects is in

• BCSS, From loop groups to 2-groups (web)

(with an eye towards the string 2-group).

The relation between representations of loop groups and twisted K-theory over the group is the topic of

• Dan Freed, Mike Hopkins, Constantin Teleman, Loop Groups and Twisted K-Theory