Elementary calculus in chevalley groups over rings

ALEXEI STEPANOV

Author(s)

ALEXEI STEPANOV Department of Mathematics and Mechanics, Sankt-Petersburg State University, Russia; Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan.

Abstract

The article studies structure theory of Chevalley groups over commutative rings. Main results of the article are relative dilation and local-global principles. and an economic set of generators of relative elementary subgroup. These statements proved by computations with elementary unipotents (hence the title) are very important in further development of the subject. No restrictions on the ground ring or the root system are imposed except that the rank of is not less than 2. The results improve previous results in the area. The article contains a brief survey of the subject, some gaps in proofs or incorrect references are discussed. Proofs of some known related results are substantially simplified.

Publication Details

Page(s) 79-95

DOI DOI not available

Published Journal: Journal of Prime Research in Mathematics, Volume: 9, Issue: 1, Year: 2013

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Keywords

Chevalley groups, AMS SUBJECT, principal congruence subgroup, local global principle, dilation principle, 20G35

References

Abe E. .1983 .Whitehead groups of Chevalley groups over polynomial rings. Comm. Algebra, 12(1271) : 1307.

.1989 .[2] , Normal subgroups of Chevalley groups over commutative rings. , 1 : 17.

Abe E.,Suzuki K. .1976 .On normal subgroups of Chevalley groups over commutative rings. T^ohoku Math. J, 28(185) : 198.

Apte H.,Chattopadhyay P.,Rao R. .2012 .A local global theorem for extended ideals. J. Ramanujan Mathematical Society, 27(17) : 30.

Apte H.,Stepanov A. .2013 .A local global theorem for extended ideals. Algebra Colloquium, : .

Bak A. .1991 .Nonabelian K-theory: The nilpotent class of K1 and general stability, K-theory 4 (. , 363 : 397.

Bak A.,Vavilov N. A. .1995 .Normality for elementary subgroup functors. Math. Proc. Cambridge Philos. Soc, 118(35) : 47.

.2000 .[8] , Structure of hyperbolic unitary groups I: Elementary subgroups. Algebra Colloquium, 7(159) : 196.

Bass H.,Benjamin H. .1986 .Algebraic K-theory. , : .

Bourbaki N. .2008 .. Lie groups and Lie algebras. Chapters 4-6, : .

Chevalley C. . .Sur certains groupes simples. Tohoku Math. J. 7, 14(1955) : 66.

Grunewald F.,Mennicke J.,Vaserstein L. .1991 .On symplectic groups over polynomial rings. Math. Zeitschrift, 35 : 56.

Hazrat R. .2002 .Dimension theory and nonstable K1 of quadratic modules. K-Theory, 27(293) : 328.

Hazrat R.,Stepanov A.,Vavilov N.,Z. Zhang, N. .2011 .The yoga of commutators. , 179(662) : 678.

Hazrat R.,Vavilov N. .2003 .K1 of Chevalley groups are nilpotent. J. Pure and Appl. Algebra, 99 : 116.

Hazrat R.,Vavilov N.,Z. Zhang, N. .2013 .Relative commutator calculus in Chevalley groups. J.Algebra, 262 : 293.

Hazrat R.,Zhang Z. .2013 .. , 195(481) : 505.

W. van der Kallen .1983 .A group structure on certain orbit sets of unimodular rows. J. Algebra, 363 : 397.

Khlebutin S. G. .1984 .Sufficient conditions for the normality of the subgroup of elementary matrices. Russian Mathematical Surveys, 39(203) : 204.

V. I. Kopejko .1978 .The stabilization of symplectic groups over a polynomial ring. Mathematics of the USSR { Sbornik, 34(655) : 669.

Luzgarev A.,Stavrova A. .2012 .Elementary subgroup of an isotropic reductive group is perfect, St. , 23(881) : 890.

Petrov V.,Stavrova A. .2009 .Elementary subgroups in isotropic reductive groups. St. Petersburg Mathematical Journal, 20(625) : 644.

Quillen D. .1976 .Projective modules over polynomial rings. Invent. Math, 167 : 171.

Stavrova A.,J. A. .2014 .Homotopy invariance of non-stable K1-functors,. , 10 : .

Stein M. R. .1971 .coverings of Chevalley groups over commutative rings. Amer. J, 965 : 1004.

Stepanov A.,Vavilov N. .2011 .Length of commutators in Chevalley groups. , 253 : 276.

Stepanov A. V. .2014 .Structure of chevalley groups over rings via universal localization. , : .

Suslin A. A. .1977 .On the structure of the special linear group over polynomial rings. Math. USSR. Izvestija, 221 : 238.

Suslin A. A.,V. I. Kopejko A. A. .2031 .Quadratic modules over polynomial rings. J. Sov. Math, 17(4) : .

.1982 .Quadratic modules and orthogonal group over polynomial rings. J. Sov. Math, 20(2665) : 2691.

Taddei G. .1986 .Normalit´e des groupes ´el´ementaire dans les groupes de Chevalley sur un anneau, Contemp. , 693 : 710.

Tits J.,Acad C. R. .1976 .. , 693 : 695.

Tulenbaev M. S. .2067 .Schur multiplier of a group of elementary matrices of finite order. , 17(4) : .

Vaserstein L. N. .1981 .On the normal subgroups of GLn over a ring. , 456 : 465.

.1986 .On normal subgroups of Chevalley groups over commutative rings. T^ohoku Math. J, 219 : 230.

Vaserstein L. N.,Suslin A. A. .1976 .Serre's problem on projective modules over polynomial rings, and algebraic K-theory. Math. USSR. Izvestija, 10(937) : 1001.

Vavilov N. A. .1982 .Parabolic subgroups of the general linear group over Dedekind domain of arithmetic type. J. Sov. Math, 20(2546) : 2555.

Vavilov N. A.,Stepanov A. V. .2008 .Standard commutator formulae, Vestnik St. , (9) : 14.

Wendt M. .2010 .-homotopy of Chevalley groups. J. K-Theory, 245 : 287.