TY - JOUR

T1 - Net subgroups of Chevalley groups. II. Gauss decomposition

AU - Vavilov, N. A.

AU - Plotkin, E. B.

N1 - Copyright: Copyright 2007 Elsevier B.V., All rights reserved.

PY - 1984/11

Y1 - 1984/11

N2 - This paper is a continuation of RZhMat 1980, 5A439, where there was introduced the subgroup γ(δ) of the Chevalley group G(Φ,R) of type Φ over a commutative ring R that corresponds to a net δ, i.e., to a set b{cyrillic}=(b{cyrillic}∝),∝∈Φ, of ideals b{cyrillic}∝ of R such that b{cyrillic}∝b{cyrillic}β{square image of or equal to}b{cyrillic}∝+β whenever α,Β,α+Β ∃Φ. It is proved that if the ring R is semilocal, then Γ(b{cyrillic}) coincides with the group γ0δ considered earlier in RZhMat 1976, 10A151; 1977, 10A301; 1978, 6A476. For this purpose there is constructed a decomposition of γ(δ) into a product of unipotent subgroups and a torus. Analogous results are obtained for sub-radical nets over an arbitrary commutative ring.

AB - This paper is a continuation of RZhMat 1980, 5A439, where there was introduced the subgroup γ(δ) of the Chevalley group G(Φ,R) of type Φ over a commutative ring R that corresponds to a net δ, i.e., to a set b{cyrillic}=(b{cyrillic}∝),∝∈Φ, of ideals b{cyrillic}∝ of R such that b{cyrillic}∝b{cyrillic}β{square image of or equal to}b{cyrillic}∝+β whenever α,Β,α+Β ∃Φ. It is proved that if the ring R is semilocal, then Γ(b{cyrillic}) coincides with the group γ0δ considered earlier in RZhMat 1976, 10A151; 1977, 10A301; 1978, 6A476. For this purpose there is constructed a decomposition of γ(δ) into a product of unipotent subgroups and a torus. Analogous results are obtained for sub-radical nets over an arbitrary commutative ring.

UR - http://www.scopus.com/inward/record.url?scp=0039106638&partnerID=8YFLogxK

U2 - 10.1007/BF01410741

DO - 10.1007/BF01410741

M3 - Article

AN - SCOPUS:0039106638

VL - 27

SP - 2874

EP - 2885

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 4

ER -