A group corresponding to a Kac-Moody algebra is a Kac-Moody group.
This subsumes the case of affine Lie algebras in which case the corresponding Kac-Moody group is a centrally extended loop group (see there for more) and often the words “Kac-Moody group” are used synonomously with “centrally extended loop group”.
For $G$ a topological group, write $B G$ for its delooping, the classifying space for topological $G$-principal bundles.
The classifying space $B G$ of every Kac-Moody group is a homotopy colimit over classifying spaces $B G_i$, $i \in I$ of compact connected Lie groups $G_i$:
Moreover, the diagram $I$ may be taken to be a sieve in the poset of subobjects of the $n$-element set, for some $n \in \mathbb{B}$.
This is due to Nitu Kitchloo, 1998, see for instance Kitchloo’s survey, p. 9.
This means that the classifying space of every Kac-Moody group has a smooth refinement to a smooth moduli stack given by forming
in Smooth∞Grpd.
Kay moody groups appear as U-duality groups in 11-dimensional supergravity compactified to low dimensions.
A standard textbook is
A survey is
Original articles include
Nitu Kitchloo, On the Topology of Kac-Moody groups (arXiv), (Phd thesis)
Carles Broto, Nitu Kitchloo, Classifying spaces of Kac-Moody groups, Math. Z. 240,621–649 (2002) (pdf)
Udo Baumgartner, Jacqueline Ramagge, Bertrand Remy, Contraction groups in complete Kac-Moody groups (2008) (pdf)
Andreas Mars, On the topology and geometry of Kac-Moody groups, PhD thesis (2011) (web)
Christof Geiss, Bernard Leclerc, Jan Schröer, Kac-Moody groups and cluster algebras (arXiv:1001.3545)