Algebraically and verbally closed subgroups and retracts of finitely generated nilpotent groups

We study algebraically and verbally closed subgroups and retracts of finitely generated nilpotent groups. A special attention is paid to free nilpotent groups and the groups UT _n (Z) of unitriangular (n×n)-matrices over the ring Z of integers for arbitrary n. We observe that the sets of retracts of finitely generated nilpotent groups coincides with the sets of their algebraically closed subgroups. We give an example showing that a verbally closed subgroup in a finitely generated nilpotent group may fail to be a retract (in the case under consideration, equivalently, fail to be an algebraically closed subgroup). Another example shows that the intersection of retracts (algebraically closed subgroups) in a free nilpotent group may fail to be a retract (an algebraically closed subgroup) in this group. We establish necessary conditions fulfilled on retracts of arbitrary finitely generated nilpotent groups. We obtain sufficient conditions for the property of being a retract in a finitely generated nilpotent group. An algorithm is presented determining the property of being a retract for a subgroup in free nilpotent group of finite rank (a solution of a problem of Myasnikov). We also obtain a general result on existentially closed subgroups in finitely generated torsion-free nilpotent with cyclic center, which in particular implies that for each n the group UT _n (Z) has no proper existentially closed subgroups.     

Constructible Matrix Groups

We prove that the additive group of a ring K is constructible if the group GL ₂ (K) is constructible. It is stated that under one extra condition on K, the constructibility of GL ₂ (K) implies that K is constructible as a module over its subring L generated by all invertible elements of the ring L; this is true, in particular, if K coincides with L, for instance, if K is a field or a group ring of an Abelian group with the specified property. We construct an example of a commutative associative ring K with 1 such that its multiplicative group K* is constructible but its additive group is not. It is shown that for a constructible group G represented by matrices over a field, the factors w.r.t. members of the upper central series are also constructible. It is proved that a free product of constructible groups is again constructible, and conditions are specified under which relevant statements hold of free products with amalgamated subgroup; this is true, in particular, for the case where an amalgamated subgroup is finite. Also we give an example of a constructible group GL ₂ (K) with a non-constructible ring GL. Similar results are valid for the case where the group SL ₂ (K) is treated in place of GL ₂ (K) .     

Ideals of Some Matrix Rings

For a commutative ring K the conception of a strongly maximal ideal J was introduced by Kuzucuoglu and Levchuk in 2000. Denote by R_n(K,J) the ring of all n×n-matrices over K with elements from J on and above the main diagonal. Recent results on ideals of the ring R_n(K,J) for this case, ideals of the associated Lie ring and normal subgroups of the adjoint group are considered in this paper. Also ideals of R_n(K,J) for the case of an arbitrary associative ring K with the identity are investigated.     

Subgroups of unitriangular groups of infinite matrices

We show that for any associative ring R, the subgroup UT _r(∞, R) of row-finite matrices in UT(∞, R), the group of all infinite-dimensional (indexed by ℕ) upper unitriangular matrices over R, is generated by the so-called strings (block-diagonal matrices with finite blocks along the main diagonal). This allows us to define a large family of subgroups of UT _r(∞, R) associated with some growth functions. The smallest subgroup in this family, called the group of banded matrices, is generated by 1-banded simultaneous elementary transvections (a slight generalization of the usual notion of elementary transvection). We introduce the notion of net subgroup and characterize the normal net subgroups of UT(∞, R). Bibliography: 26 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 338, 2006, pp. 137–154.     

Linear Groups Over Integral Extensions of Semilocal Commutative Rings

Let K be an associative and commutative ring with 1, k a subring of K such that 1 ∈ k, K is an integral finitely generated extension of k, the element 2 invertible in k, and k is semilocal. The paper studies subgroups of the general linear group GL _n (K) with n ≥ 2 containing the special linear group SL _n (k).     

Non-closed Lie subgroups of Lie groups

We obtain several homotopy obstructions to the existence of non-closed connected Lie subgroupsH in a connected Lie groupG.First we show that the foliationF(G, H) onG determined byH is transversely complete [4]; moreover, forK the closure ofH inG, F(K, H) is an abelian Lie foliation [2].Then we prove that ₁(K) and ₁(H) have the same torsion subgroup, _n (K)= _n (H) for alln 2, and rank₁(K) — rank₁(H) > codimF(K, H). This implies, for instance, that a contractible (e.g. simply connected solvable) Lie subgroup of a compact Lie group must be abelian. Also, if rank₁(G) 1 then any connected invariant Lie subgroup ofG is closed; this generalizes a well-known theorem of Mal'cev [3] for simply connected Lie groups.Finally, we show that the results of Van Est on (CA) Lie groups [6], [7] provide many interesting examples of such foliations. Actually, any Lie group with non-compact centre is the (dense) leaf of a foliation defined by a closed 1-form. Conversely, when the centre is compact, the latter is true only for (CA) Lie groups (e.g. nilpotent or semisimple).     

Quasipolar Property of Generalized Matrix Rings

The article concerns the question of when a generalized matrix ring K _s (R) over a local ring R is quasipolar. For a commutative local ring R, it is proved that K _s (R) is quasipolar if and only if it is strongly clean. For a general local ring R, some partial answers to the question are obtained. There exist noncommutative local rings R such that K _s (R) is strongly clean, but not quasipolar. Necessary and sufficient conditions for a single matrix of K _s (R) (where R is a commutative local ring) to be quasipolar is obtained. The known results on this subject in [5 Cui , J. , Chen , J. ( 2011 ). When is a 2 × 2 matrix ring over a commutative local ring quasipolar? Comm. Alg. 39 : 3212 – 3221 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] are improved or extended.     

The Derived Picard Group is a Locally Algebraic Group

Let A be a finite-dimensional algebra over an algebraically closed field K. The derived Picard group DPic _K (A) is the group of two-sided tilting complexes over A modulo isomorphism. We prove that DPic _K (A) is a locally algebraic group, and its identity component is Out⁰ _K (A). If B is a derived Morita equivalent algebra then DPic _K (A)DPic _K (B) as locally algebraic groups. Our results extend, and are based on, work of Huisgen-Zimmermann, Saorín and Rouquier.     

Structure of the automorphism group for partially commutative class two nilpotent groups

Let R be a ring, which is either a ring of integers or a field of zero characteristic. For every finite graph Γ, we construct an R-arithmetic linear group H(Γ). The group H(Γ) is realized as the factor automorphism group of a partially commutative class two nilpotent R-group G _Γ. Also we describe the structure of the entire automorphism group of a partially commutative nilpotent R-group of class two.     

ON SYMMETRIC UNITS IN GROUP ALGEBRAS

Let U(KG) be the group of units of the group ring KG of the group G over a commutative ring K. The anti-automorphism g → g ^?1 of G can be extended linearly to an anti-automorphism a → a ^* of KG. Let S _* (KG) = {x ∈ U(KG) | x ^* = x} be the set of all symmetric units of U(KG). We consider the following question: for which groups G and commutative rings K it is true that S _* (KG) is a subgroup in U(KG). We answer this question when either a) G is torsion and K is a commutative G-favourable integral domain of characteristic p≥ 0 or b) G is non-torsion nilpotent group and KG is semiprime.     

Real elements in T n (K)

We study the real elements in triangular matrix groups. We describe some classes of elements that are real in T _n (K) – the groups of upper triangular matrices over a commutative field K. From the obtained results there follow some applications for finding real elements in general linear groups – GL _n (K).     

Ultrafilters and Ramsey-type shadowing phenomena in topological dynamics

The graded exponent is an important invariant of group graded PI-algebras. In this paper we study a specific elementary grading on the algebra of upper triangular matrices UT _n , compute its codimensions, and use this grading to find the asymptotic behaviour of the codimensions of any elementary grading on UT _n , for any group. Moreover, we extend this to the Lie case, and obtain, for any elementary grading on the Lie algebra UT_n^(?), an upper bound and a lower bound for the asymptotic behaviour of its codimensions. Also, we obtain the graded exponent of any grading on UT_n^(?) and for any grading on the Jordan algebra UJ _n .It turns out that the graded exponent for UT _n , considered as an associative, Jordan or Lie algebra, for any grading, coincides with the exponent of the ordinary case. In the associative case, the asymptotic behaviour of the codimensions of any grading on UT _n coincides with the asymptotic behaviour of the ordinary codimensions. But this is not the case for the graded asymptotics of the codimensions of the Lie algebra UT_n^(?).     

Homogeneity of Certain Invariant Distributions on the Lie Algebra of p-adic GLn

Let F be a non-Archimedean local field with ring of integers R and prime ideal . Suppose T is a GL _n (F)-invariant distribution on =M _n (F), the Lie algebra of GL _n (F). If T has support in the set of topologically nilpotent elements, then the restriction of T to the set of functions which are compactly supported and invariant under M _n ( ) may be expressed as a linear combination of nilpotent orbital integrals restricted to the same set of functions.     

The real spectrum of an ideal and KO-theory exact sequences

A construction in abstract real algebra is used to define invariants S _n(A) of commutative rings, with or without identity. If A=C(X) is the ring of continuous real functions on a compact space, then S _n(A) = k0^–n(X), and, for any A, S _n(A) Z[1/2]-W _n(A) Z[1/2], where the W _n(A) are the Witt groups of A. In addition, a short exact sequence of rings yields a long exact sequence of the groups S _n. The functors S _n(A) thus provide a solution of a problem proposed by Karoubi. This paper primarily deals with the exact sequences involving a ring A and an ideal I A. Work supported in part by NSF Grant DMS85-06816.     

On Lie Nilpotent Rings and Cohen's Theorem

We study certain (two-sided) nil ideals and nilpotent ideals in a Lie nilpotent ring R. Our results lead us to showing that the prime radical rad(R) of R comprises the nilpotent elements of R, and that if L is a left ideal of R, then L + rad(R) is a two-sided ideal of R. This in turn leads to a Lie nilpotent version of Cohen's theorem, namely if R is a Lie nilpotent ring and every prime (two-sided) ideal of R is finitely generated as a left ideal, then every left ideal of R containing the prime radical of R is finitely generated (as a left ideal). For an arbitrary ring R with identity we also consider its so-called n-th Lie center Z _n (R), n ≥ 1, which is a Lie nilpotent ring of index n. We prove that if C is a commutative submonoid of the multiplicative monoid of R, then the subring ?Z _n (R) ∪ C? of R generated by the subset Z _n (R) ∪ C of R is also Lie nilpotent of index n.     

Une variante à la Coxeter du groupe de Steinberg

LetG _n ()be the semi-direct product of the symmetric groupS _n by the Steinberg groupSt _n ()of a ringWe first prove thatG _n ()has a Coxeter-type presentation. The canonical morphism St _n () GL _n ()extends to a group homo G_n() GL _n ()We next determine the kernel of for n = We also give an expression for the generator of the algebraic K group K ₂(Z)of the integers in terms of permutation matrices.