On simple periodic linear groups— Dense subgroups,permutation representations,and induced modules

We determine the Zariski-dense subgroups of Chevalley groups and their twisted analogues over infinite algebraic extensions of finite fields. It turns out that these are essentially forms of the same group (possibly becoming twisted) over smaller infinite fields. It follows from our classification that if is a simple algebraic group over the algebraic closure of a finite field, then a dense subgroup of can never be maximal, and so the maximal subgroups of are necessarily closed. It follows that Seitz’s determination of the closed maximal subgroups of actually gives all the maximal subgroups. It also enables us to prove that ifG is a simple Chevalley group or twisted type over an infinite algebraic extension of a finite field, then in every non-trivial permutation representation ofG, every finite subgroup has a regular orbit. It follows that every non-trivial permutation module forG over a fieldk iskG-faithful. This is relevant to a programme of studying ideals in group rings of simple locally finite groups. To John Thompson in recognition of his many outstanding contributions to group theory     

The spectrum of the averaging operator for a finite homogeneous graph

We give an estimate for the spectrum of the averaging operator T₁(Γ, 1) over the radius 1 for the finite (q+1)-homogeneous quotient graph Γ/X, where X is an infinite (q+1)-homogeneous tree associated with the free group G over a finite set of generators S={x₁ ..., x_p} (2p=q+1), and Γ, a subgroup of finite index in G. T₁(Γ, 1) is defined on the subspace L²(Γ/G, 1) ⊖ E_ex, where E_ex is the subspace of eigenfunctions of T₁(Γ, 1) with eigenvalue λ such that |λ|=q+1. We present a construction of some finite homogeneous graphs such that the spectrum of their adjacency matrices can be calculated explicitly. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 205, 1993, pp. 92–109. Translated by A. M. Nikitin.     

Free group of infinite unitriangular matrices

In the group of infinite unitriangular matrices over the field with two elements, a free subgroup of rank two is constructed which is a group of finite-automata transformations over a two-element alphabet. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 382–386, March, 2000.     

Representation Theory of some Infinite-dimensional Algebras Arising in Continuously Controlled Algebra and Topology

In this paper we determine the representation type of some algebras of infinite matrices continuously controlled at infinity by a compact metrizable space. We explicitly classify their finitely presented modules in the finite and tame cases. The algebra of row-column-finite (or locally finite) matrices over an arbitrary field is one of the algebras considered in this paper, its representation type is shown to be finite.Received October 2003     

Homological identities for a finite homogeneous graph

An application of the Atiyah-Bott trace identity to the study of the spectral characteristics of a finite (q+1)-homogeneous factorgraph Y=Γ/X is given (X is an infinite (q+1)-homogeneous tree, Γ a free group of isometries of X). Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 205, 1993, pp. 110–121.     

Convergence Analysis of the Finite Section Method and Banach Algebras of Matrices

The finite section method is a classical scheme to approximate the solution of an infinite system of linear equations. Based on an axiomatic framework we present a convergence analysis of the finite section method for unstructured matrices on weighted ℓ ^p -spaces. In particular, the stability of the finite section method on ℓ ² implies its stability on weighted ℓ ^p -spaces. Our approach uses recent results from the theory of Banach algebras of matrices with off-diagonal decay. Furthermore, we demonstrate that Banach algebra theory provides a natural framework for deriving a finite section method that is applicable to large classes of unstructured non-hermitian matrices as well as to least squares problems.     

Some infinite groups with strongly embedded subgroup

An involution i of a group G is said to be finite if |ii^g|<∞ for all g ∃ G. Suppose that G contains a finite involution and an infinite elementary Abelian 2-subgroup S and, moreover, the normalizer H=N_G(S)=SλT is strongly embedded in G and is a Frobenius group with locally cyclic complement T. It is proved that G is isomorphic to L₂(Q) over a locally finite field Q of characteristic 2. In particular, part (a) of Question 10.76 raised by Shunkkov in the Kourovka Notebook is answered in the affirmative. Supported by RFFR grant No. 99-01-00542. Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 602–617, September–October, 2000.     

Description of the characters and factor representations of the infinite symmetric inverse semigroup

A complete list of indecomposable characters of the infinite symmetric semigroup is given. In comparison with a similar list for the infinite symmetric group, only one new parameter appears, which has a clear combinatorial meaning. The results rely on the representation theory of finite symmetric semigroups and the representation theory of the infinite symmetric group.     

On direct decompositions in modules over group rings

In the theory of infinite groups, one of the most important useful generalizations of the classical Maschke theorem is the Kovačs-Newman theorem, which establishes sufficient conditions for the existence of G-invariant complements in modules over a periodic group G finite over the center. We genralize the Kovačs-Newman theorem to the case of modules over a group ring KG, where K is a Dedekind domain. Dnepropetrovsk University, Dnepropetrovsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 255–261, February, 1997.     

Fields interpretable in rosy theories

We are working in a monster model ℭ of a rosy theory T. We prove the following theorems, generalizing the appropriate results from the finite Morley rank case and o-minimal structures. If R is a ⋁-definable integral domain of positive, finite U^t-rank, then its field of fractions is interpretable in ℭ. If A and M are infinite, definable, abelian groups such that A acts definably and faithfully on M as a group of automorphisms, M is A-minimal and U^t(M) is finite, then there is an infinite field interpretable in ℭ. If G is an infinite, solvable but non nilpotent-by-finite, definable group of finite U^t-rank and T has NIP, then there is an infinite field interpretable in 〈G, ·〉.     

A generalization of the Razmyslov-Procesi theorem

It is proved that all relations between the invariants of several n x n-matrices over an infinite field of arbitrary characteristic follow from σ_n+1,σ_n+2,... where σ_i is the ith coefficient of a characteristic polynomial extended to matrices of any order ≥i. Similarly, all relations between the concomitants are implied by X_n+1, X_n+2, …, where X_i is a characteristic polynomial in the general n x n-matrix, also extended to matrices of any order. Supported by RFFR grant No. 95-01-00513. Translated fromAlgebra i Logika, Vol. 35, No. 4, pp. 433–457, July–August, 1996.     

A representation of the Belavkin equation via Feynman path integrals

 The Belavkin equation, describing the continuous measurement of the position of a quantum particle, is studied. A rigorous representation of its solution by means of an infinite dimensional oscillatory integral (Feynman path integral) defined on the complex Cameron-Martin space is given. Received: 7 January 2002 / Revised version: 20 June 2002 / Published online: 19 December 2002 Mathematics Subject Classification (2000): 81, 81S40, 60H15 Key words or phrases: Belavkin equation – Continuous measurement – Quantum theory – Oscillatory integrals – Feynman path integrals     

On the composition length of finite primitive linear groups

LetG be a finite primitive linear group over a fieldK, whereK is a finite field or a number field. We bound the composition length ofG in terms of the dimension of the underlying vector space and of the degree ofK over its prime subfield. As a byproduct, we prove a result of number theory which bounds the number of prime factors (counting multiplicities) ofq ⁿ−1, whereq, n>1 are integers, improving a result of Turull and Zame [6].     

Quantum affine Cartan matrices,Poincaré series of binary polyhedral groups,and reflection representations

We first review some invariant theoretic results about the finite subgroups of SU(2) in a quick algebraic way by using the McKay correspondence and quantum affine Cartan matrices. By the way it turns out that some parameters (a, b, h; p, q, r) that one usually associates with such a group and hence with a simply-laced Coxeter–Dynkin diagram have a meaningful definition for the non-simply-laced diagrams, too, and as a byproduct we extend Saito’s formula for the determinant of the Cartan matrix to all cases. Returning to invariant theory we show that for each irreducible representation i of a binary tetrahedral, octahedral, or icosahedral group one can find a homomorphism into a finite complex reflection group whose defining reflection representation restricts to i.     

Subregular characters of the unitriangular group over a finite field

We prove a formula for subregular characters of the unitriangular group over a finite field in terms of coefficients of minors of the characteristic matrix. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 5, pp. 103–125, 2007.     

Vector Spaces of Non-measurable Functions

We show that there exists an infinite dimensional vector space every non-zero element of which is a non-measurable function. Moreover, this vector space can be chosen to be closed and to have dimensionβ for any cardinalityβ. Some techniques involving measure theory and density characters of Banach spaces are used.     

On same-invariant linear groups

A linear group G ≤ GL(V) is called same-invariant if the subspace of linear invariants V^g is one and the same for all g ∈ G, g ≠ 1. In this paper, we consider finite same-invariant linear groups of orders pq, (p, q) = 1, or p² over a field of characteristic p. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 321, 2005, pp. 224–239.     

Free-field resolutions of unitary representations of theN=2 virasoro algebra: I1

We construct a free-field resolution of unitary representations of theN=2 superconformal algebra. The resolution is degenerated by the action of one fermionic screening operator (which generates the “gravitational descendants” in the field theory picture). The irreducible representation is singled out from the free-field space as the cohomology of this screening operator. this paper is written at the request of the Editorial Board. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 1, pp. 89–109, October, 1999.     

A superrank of varieties of Lie algebras

A variety of Lie algebras over a field of characteristic 0 has a finite superrank if it is generated by the Grassmann envelope of a finitely generated Lie superalgebra. We prove that every commutator variety not in N_cA has infinite superrank. Consequently, infinite are superranks of all polynilpotent varieties of Lie algebras except N_c and N_cA. Supported by RFFR grant No. 96-01-00146. Translated fromAlgebra i Logika, Vol. 37, No. 4, pp. 394–412, July–August. 1998.     

Asymptotic growth of averaged Dehn functions for nilpotent groups

It is proved that in any finite representation of any finitely generated nilpotent group of nilpotency class l ⩾ 1, the averaged Dehn function σ(n) is subasymptotic w.r.t. the function n^l+1. As a consequence it is stated that in every finite representation of a free nilpotent group of nilpotency class l of finite rank r ⩾ 2, the Dehn function σ(n) is Gromov subasymptotic. Supported by RFBR grant No. 04-01-00489. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 60–74, January–February, 2007.