## Groups and group functors attached to Kac-Moody data.(English)Zbl 0572.17010

Arbeitstag. Bonn 1984, Proc. Meet. Max-Planck-Inst. Math., Bonn 1984, Lect. Notes Math. 1111, 193-223 (1985).
[For the entire collection see Zbl 0547.00007.]
According to the classification due to Killing and Cartan of complex semi-simple Lie algebras, isomorphism classes of these algebras are in one-to-one correspondence with the systems (*) $${\mathfrak H}$$, $$(\alpha_ i)_{1\leq i\leq \ell}$$, $$(h_ i)_{1\leq i\leq \ell}$$. Here, for some representative $${\mathfrak G}$$ of the class, $${\mathfrak H}$$ is a Cartan subalgebra, $$(\alpha_ i)$$ is a basis for $${\mathfrak H}^*$$, and $$(h_ i)$$ is a dual basis for $${\mathfrak H}$$ such that the matrix $${\mathcal A}=(\alpha_ j(h_ i))$$ is a Cartan matrix. Given such a system, one can write down a presentation of the corresponding Lie algebra. If the assumptions about the system (*) are changed in a certain way then the same presentation yields an infinite dimensional Lie algebra called a Kac-Moody algebra. Just as a finite dimensional Lie algebra can be associated with a Lie group, one would like to associate a Kac-Moody algebra with a ”Kac-Moody group”. This is the topic of Tit’s lecture.
The lecture begins with a brief but detailed introduction to Kac-Moody algebras. This is followed by a discussion of several different methods for attaching a group to a Kac-Moody algebra and a specific example is presented: groups of type $${}^ 2\tilde E_ 6.$$
While much of this article surveys work already published by the author and by others, there are two appendices which seem to present new results. One is a description of the ”Kac-Moody group” (in this case it is actually a group functor on the category of rings) in the case where the matrix $${\mathcal A}$$ is of irreducible, affine type.
Reviewer: Th.Farmer

### MSC:

 17B65 Infinite-dimensional Lie (super)algebras 22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties 17B20 Simple, semisimple, reductive (super)algebras

Zbl 0547.00007