Net subgroups of Chevalley groups

We introduce and study net subgroups of Chevalley groups of normal and certain twisted types. Another subgroup of Chevalley groups related to a net was discussed in Ref. Zh. Mat. 1976; 10A151; 1977, 10A301; 1978, 6A476.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 40–49, 1979.In conclusion, the authors would like to thank Z. I. Borevich for his.interest in this paper.     

Weakly closed unipotent subgroups in Chevalley groups

The goal of this note is to classify the weakly closed unipotent subgroups in the split Chevalley groups. In an application we show under some mild assumptions on the characteristic that the Lie algebra of a connected simple algebraic group fails to be a so-called 2F-module.     

Parabolic subgroups of Chevalley groups over a semilocal ring

Let G be the Chevalley group over a commutative semilocal ring R which is associated with a root system . The parabolic subgroups of G are described in the work. A system =() of ideals in R ( runs through all roots of the system ) is called a net of ideals in the commutative ring R if ₊ for all those roots and for which + is also a root. A net is called parabolic if =R for >0. The main theorem: under minor additional assumptions all parabolic subgroups of G are in bijective correspondence with all parabolic nets . The paper is related to two works of K. Suzuki in which the parabolic subgroups of G are described under more stringent conditions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 43–58, 1978.     

Parabolic subgroups of twisted Chevalley groups over a semilocal ring

It is shown that, under minor additional assumptions, the standard parabolic subgroups of a Chevalley group G (, R) of twisted type =A_l,l odd, D_l, E₆ over a commutative semilocal ring R with involution are in one-to-one correspondence with the -invariant parabolic nets of ideals of R of type , i.e., with the sets, of ideals of R such that: (l) whenever; (2) = for all ; (3) =R for > 0. For Chevalley groups of normal types, analogous results were obtained in Ref. Zh. Mat. 1976, 10A151; 1977, 10A 301; 1978, 6A476.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 21–36, 1979.     

The decomposition of permutation module for infinite Chevalley groups

Let G be a connected reductive group defined over F_q, the finite field with q elements. Let B be a Borel subgroup defined over F_q. In this paper, we completely determine the composition factors of the induced module M(tr) = kG ■tr(where tr is the trivial B-module) for any field k.     

Bruhat decomposition for long root tori in Chevalley groups

It is proved that all elements of a given long root torus (i.e., a one-parameter subgroup of long root semisimple elements) in a Chevalley group over a field, except for at most three, belong to the same Bruhat decomposition class.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 175, pp. 12–23, 1989.     

Automorphisms and normal structure of unipotent subgroups of finitary Chevalley groups

We propose a method of constructing vector fields with certain vortex properties by means of transformations that change the value of the field vector at every point, the form of the field lines, and their mutual position. We discuss and give concrete examples of the prospects of using the method in applications involving solution of partial differential equations, including nonlinear ones.     

Gauss decomposition for quantum groups and supergroups

The Gauss decompositions of quantum groups related to classical Lie groups and supergroups are considered by the elementary algebraic and R-matrix methods. The commutation relations between the generators of the new basis introduced by the decomposition are described in detail. It is shown that it is possible to reduce a number of independent generators to the dimension of the related classical group. The symplectic quantum group Sp_q(2) and supergroups GL_q(1, 1) and GL_q(2, 1) are considered as examples. Bibliography: 73 titles. In memory of Victor Nikolaevich Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 155–177. Translated by E. V. Damanskinskii.     

Products of conjugacy classes in Chevalley groups II. Covering and generation

In this paper we establish a relationship between generating numbers and covering numbers of conjugacy classes in Chevalley groups over algebraically closed fields. The authors gratefully acknowledge EPSRC grant GR58542. The first author acknowledges further a grant SFB 343 “Diskrete Strukturen in der Mathematik”. The second author thanks the Institute for Advanced Studies at The Hebrew University for its hospitality.     

Co-local subgroups of abelian groups II

In [J. Buckner, M. Dugas, Co-local subgroups of abelian groups, in: Abelian Groups, Rings, Modules, and Homological Algebra, in: Lect. Notes Pure and Applied Math., vol. 249, Taylor and Francis/CRC Press, pp. 25-33] the notion of a co-local subgroup of an abelian group was introduced. A subgroup K of A is called co-local if the natural map is an isomorphism. At the center of attention in [J. Buckner, M. Dugas, Co-local subgroups of abelian groups, in: Abelian Groups, Rings, Modules, and Homological Algebra, in: Lect. Notes Pure and Applied Math., vol. 249, Taylor and Francis/CRC Press, pp. 25-33] were co-local subgroups of torsion-free abelian groups. In the present paper we shift our attention to co-local subgroups K of mixed, non-splitting abelian groups A with torsion subgroup t(A). We will show that any co-local subgroup K is a pure, cotorsion-free subgroup and if D/t(A) is the divisible part of A/t(A)=D/t(A)⊕H/t(A), then K∩D=0, and one may assume that K⊆H. We will construct examples to show that K need not be a co-local subgroup of H. Moreover, we will investigate connections between co-local subgroups of A and A/t(A).     

Total positivity for loop groups II: Chevalley generators

This is the second in a series of papers developing a theory of total positivity for loop groups. In this paper, we study infinite products of Chevalley generators. We show that the combinatorics of infinite reduced words underlies the theory, and develop the formalism of infinite sequences of braid moves, called a braid limit. We relate this to a partial order, called the limit weak order, on infinite reduced words. The limit semigroup generated by Chevalley generators has a transfinite structure. We prove a form of unique factorization for its elements, in effect reducing their study to infinite products which have the order structure of ?. For the latter infinite products, we show that one always has a factorization which matches an infinite Coxeter element. One of the technical tools we employ is a totally positive exchange lemma which appears to be of independent interest. This result states that the exchange lemma (in the context of Coxeter groups) is compatible with total positivity in the form of certain inequalities.     

Extensions of Chevalley groups

LetG ₀ be a split simple Chevalley group of any type over the fieldK andG its universal group; and let? ₀ be the group of automorphisms of the corresponding Chevalley algebra,L _K, generated byG ₀ and all the diagonal automorphisms. A group? (and appropriate homorphisms) is constructed which generalizes the groupGL _n (K) whenG ₀ is specialized to typeA _n?1.     

Ultraproducts and Chevalley groups

Given a simple non-trivial finite-dimensional Lie algebra L, fields and Chevalley groups , we first prove that is isomorphic to . Then we consider the case of Chevalley groups of twisted type . We obtain a result analogous to the previous one. Given perfect fields having the property that any element is either a square or the opposite of a square and Chevalley groups , then is isomorphic to . We apply our results to prove the decidability of the set of sentences true in almost all finite groups of the form L(K) where K is a finite field and L a fixed untwisted Chevalley type. Received: 19 November 1993 / Revised version: 15 November 1995