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Uniqueness and presentation of Kac-Moody groups over fields. (English) Zbl 0626.22013

In his famous paper, Chevalley has defined analogues over arbitrary fields of the complex simple Lie groups. Further, Chevalley and Demazure have constructed a certain group scheme G over \({\mathbb{Z}}\) associated to every reductive group over \({\mathbb{C}}\) which is characterized by a few simple properties, making G(R) for any commutative ring R the natural analogue of the group over R. The group schemes are classified by data \(D=(I,\Lambda,(\alpha_ i)_{i\in I}\), \((h_ i)_{i\in I})\) consisting of a finite set I, a finitely generated free abelian group \(\Lambda\), and two maps \(i\mapsto \alpha_ i\) and \(i\mapsto h_ i\) of I in \(\Lambda\) and in its dual \(\Lambda^{\vee}\), respectively, these data being subject only to the condition that the matrix \(A=(A_{ij})=(<\alpha_ j\), \(h_ i>)\) be a Cartan matrix. If we assume for A the only conditions \(A_{ii}=2\), \(A_{ij}\leq 0\) if \(i\neq j\) and \(A_{ij}=0 \Leftrightarrow A_{ij}=0\), then A is called a generalized Cartan matrix.
In this paper, to any data D with generalized Cartan matrix A, the author associates a group functor \(G_ D\) on the category of commutative rings with 1. Further, the author states a few axioms which should hold for any reasonable extension of the Chevalley-Demazure group schemes to the Kac- Moody situation and shows that any functor satisfying these axioms coincides with \(G_ D\) over fields.
The author has treated already these problems [cf. Annuaire College de France 82, 91-105 (1981/82)], but the present method is simpler and more direct and it is based on a presentation “à la Steinberg” of the Kac- Moody groups. An important feature of the situation is that it leads to groups endowed with two distinct BN-pairs having the same group N (double BN-pairs), and the new structure turns out to be much richer than that consisting of a single BN-pair.
These axiomatic characterization of Kac-Moody groups provides an easy answer to certain recognition problems. For instance, Chevalley groups of classical types are isomorphic with corresponding classical groups, that has been proved in various ways, and it also can be applied to Kac-Moody groups of affine types which also have “classical interpretation”. Further, the axiomatic set up should also apply to suitably defined twisted Kac-Moody groups.
Reviewer: E.Abe

MSC:

22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
14L15 Group schemes
20F05 Generators, relations, and presentations of groups
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
20G25 Linear algebraic groups over local fields and their integers
20G15 Linear algebraic groups over arbitrary fields
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