A Bruhat decomposition for subgroups containing the group of diagonal matrices.~II
Zapiski Nauchnykh Seminarov POMI, Modules and algebraic groups, Tome 114 (1982), pp. 50-61
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This paper is a continuation of RZhMat 1981, 7A438. Suppose $R$ is a commutative ring generated by its group of units $R^*$ and there exist such that. Suppose also that $\mathfrak J$ is the Jacobson radical of $R$, and $B(\mathfrak J)$ is a subgroup of $GL(n,R)$ consisting of the matrices $a=(a_{ij})$ such that $a_{ij}\in\mathfrak J $ for$i>j$. If a matrix $a\in B(\mathfrak J)$ is represented in the form $a=udv$, where $u$ is upper unitriangular, $d$ is diagonal, and $v$ is lower unitriangular, then $u,v\in\langle D,ada^{-1}\rangle$, where $D=D(n,R)$ is the group of diagonal matrices. In particular, $D$ is abnormal in $B(\mathfrak J)$.
@article{ZNSL_1982_114_a5,
author = {N. A. Vavilov},
title = {A {Bruhat} decomposition for subgroups containing the group of diagonal {matrices.~II}},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {50--61},
publisher = {mathdoc},
volume = {114},
year = {1982},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_1982_114_a5/}
}
N. A. Vavilov. A Bruhat decomposition for subgroups containing the group of diagonal matrices.~II. Zapiski Nauchnykh Seminarov POMI, Modules and algebraic groups, Tome 114 (1982), pp. 50-61. http://geodesic.mathdoc.fr/item/ZNSL_1982_114_a5/