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 Zap. Nauchn. Sem. LOMI, 1982, Volume 114, Pages 50–61 (Mi znsl1766)

A Bruhat decomposition for subgroups containing the group of diagonal matrices. II

N. A. Vavilov

Abstract: 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 $B(\mathfrak J)$.

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English version:
Journal of Soviet Mathematics, 1984, 27:4, 2865–2874

Bibliographic databases:

UDC: 519.46

Citation: N. A. Vavilov, “A Bruhat decomposition for subgroups containing the group of diagonal matrices. II”, Modules and algebraic groups, Zap. Nauchn. Sem. LOMI, 114, "Nauka", Leningrad. Otdel., Leningrad, 1982, 50–61; J. Soviet Math., 27:4 (1984), 2865–2874

Citation in format AMSBIB
\Bibitem{Vav82} \by N.~A.~Vavilov \paper A Bruhat decomposition for subgroups containing the group of diagonal matrices.~II \inbook Modules and algebraic groups \serial Zap. Nauchn. Sem. LOMI \yr 1982 \vol 114 \pages 50--61 \publ "Nauka", Leningrad. Otdel. \publaddr Leningrad \mathnet{http://mi.mathnet.ru/znsl1766} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=669559} \zmath{https://zbmath.org/?q=an:0548.20033|0521.20030} \transl \jour J. Soviet Math. \yr 1984 \vol 27 \issue 4 \pages 2865--2874 \crossref{https://doi.org/10.1007/BF01410740} 

• http://mi.mathnet.ru/eng/znsl1766
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This publication is cited in the following articles:
1. E. A. Sopkina, “Classitification of group subschemes in $\operatorname{GL}_n$, that contain a split maximal torus”, J. Math. Sci. (N. Y.), 136:3 (2006), 3988–3995
2. N. A. Vavilov, “Subgroups of $\operatorname{SL}_n$ over a semilocal ring”, J. Math. Sci. (N. Y.), 147:5 (2007), 6995–7004
3. N. Vavilov, “Geometry of 1-tori in $\mathrm{GL}(n,T)$”, St. Petersburg Math. J., 19:3 (2008), 407–429
4. N. Vavilov, “Weight elements of Chevalley groups”, St. Petersburg Math. J., 20:1 (2009), 23–57