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**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.

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 |