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On the Steinberg presentation for Lie-type groups of type C2

✍ Scribed by C. Müller

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
104 KB
Volume
252
Category
Article
ISSN
0021-8693

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✦ Synopsis

Let Φ be an irreducible root-system of rank 2 satisfying the crystallographic condition. (That is, Φ is one of types A , B , C , l 2, D , 4, E , 6 8, F 4 or G 2 .) Inspired by the Steinberg presentation of Chevalley groups, recently Timmesfeld considered the following situation (cf. [1-3]):

Let G be an abstract group generated by subgroups A α , α ∈ Φ, satisfying the following hypothesis denoted by (H):

is a rank-one group with unipotent subgroups A α and A -α . (For definition of a rank-one group see Section 2.) Clearly, all Chevalley groups satisfy (H). Hence the question arises which possibilities exist in general for the structure of a group satisfying (H). For the case that in condition (i) equality always holds, Timmesfeld solved this problem in [1]:

1.1. Theorem. Suppose G satisfies (H) with equality holding in condition (i). Then there exists a surjective homomorphism ϕ : G → G, where G is a group of Lie-type B, B an irreducible, spherical Moufang building, which maps the A α ,

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