Dimension expanders

We show that there exists k \in \bbn and 0 < \e \in\bbr such that for every field $F$ of characteristic zero and for every n \in \bbn, there exists explicitly given linear transformations $T_1,..., T_k: F^n \to F^n$ satisfying the following: For every subspace $W$ of $F^n$ of dimension less or equal $\frac n2$, \dim(W+\suml^k_{i=1} T_iW) \ge (1+\e) \dim W. This answers a question of Avi Wigderson [W]. The case of fields of positive characteristic (and in particular finite fields) is left open

Classification of Lie bialgebras over current algebras

In the present paper we present a classification of Lie bialgebra structures on Lie algebras of type g[[u]] and g[u], where g is a simple finite dimensional Lie algebra.Comment: 26 page

Split structures in general relativity and the Kaluza-Klein theories

We construct a general approach to decomposition of the tangent bundle of pseudo-Riemannian manifolds into direct sums of subbundles, and the associated decomposition of geometric objects. An invariant structure {\cal H}^r defined as a set of r projection operators is used to induce decomposition of the geometric objects into those of the corresponding subbundles. We define the main geometric objects characterizing decomposition. Invariant non-holonomic generalizations of the Gauss-Codazzi-Ricci's relations have been obtained. All the known types of decomposition (used in the theory of frames of reference, in the Hamiltonian formulation for gravity, in the Cauchy problem, in the theory of stationary spaces, and so on) follow from the present work as special cases when fixing a basis and dimensions of subbundles, and parameterization of a basis of decomposition. Various methods of decomposition have been applied here for the Unified Multidimensional Kaluza-Klein Theory and for relativistic configurations of a perfect fluid. Discussing an invariant form of the equations of motion we have found the invariant equilibrium conditions and their 3+1 decomposed form. The formulation of the conservation law for the curl has been obtained in the invariant form.Comment: 30 pages, RevTeX, aps.sty, some additions and corrections, new references adde

Incorporation of Spacetime Symmetries in Einstein's Field Equations

In the search for exact solutions to Einstein's field equations the main simplification tool is the introduction of spacetime symmetries. Motivated by this fact we develop a method to write the field equations for general matter in a form that fully incorporates the character of the symmetry. The method is being expressed in a covariant formalism using the framework of a double congruence. The basic notion on which it is based is that of the geometrisation of a general symmetry. As a special application of our general method we consider the case of a spacelike conformal Killing vector field on the spacetime manifold regarding special types of matter fields. New perspectives in General Relativity are discussed.Comment: 41 pages, LaTe

The ranks of central factor and commutator groups

The Schur Theorem says that if $G$ is a group whose center $Z(G)$ has finite index $n$, then the order of the derived group $G'$ is finite and bounded by a number depending only on $n$. In the present paper we show that if $G$ is a finite group such that $G/Z(G)$ has rank $r$, then the rank of $G'$ is $r$-bounded. We also show that a similar result holds for a large class of infinite groups

Hawking Radiation as Tunneling: the D-dimensional rotating case

The tunneling method for the Hawking radiation is revisited and applied to the $D$ dimensional rotating case. Emphasis is given to covariance of results. Certain ambiguities afflicting the procedure are resolved.Comment: Talk delivered at the Seventh International Workshop Quantum Field Theory under the influence of External Conditions, QFEXT'05, september 05,Barcelona, Spain. To appear in Journal of Phys.

Nonsoluble and non-p-soluble length of finite groups

Every finite group G has a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. We define the nonsoluble length λ(G) as the number of nonsoluble factors in a shortest series of this kind. Upper bounds for λ(G) appear in the study of various problems on finite, residually finite, and profinite groups. We prove that λ(G) is bounded in terms of the maximum 2-length of soluble subgroups of G, and that λ(G) is bounded by the maximum Fitting height of soluble subgroups. For an odd prime p, the non-p-soluble length λ p (G) is introduced, and it is proved that λ p (G) does not exceed the maximum p-length of p-soluble subgroups. We conjecture that for a given prime p and a given proper group variety V the non-p-soluble length λ p (G) of finite groups G whose Sylow p-subgroups belong to V is bounded. In this paper we prove this conjecture for any variety that is a product of several soluble varieties and varieties of finite exponent. As an application of the results obtained, an error is corrected in the proof of the main result of the second author’s paper Multilinear commutators in residually finite groups, Israel Journal of Mathematics 189 (2012), 207–224

On the algebraic structures connected with the linear Poisson brackets of hydrodynamics type

The generalized form of the Kac formula for Verma modules associated with linear brackets of hydrodynamics type is proposed. Second cohomology groups of the generalized Virasoro algebras are calculated. Connection of the central extensions with the problem of quntization of hydrodynamics brackets is demonstrated

Bogolyubov Quasiparticles in Constrained Systems

The paper is devoted to the formulation of quantum field theory for an early universe in General Relativity considered as the Dirac general constrained system. The main idea is the Hamiltonian reduction of the constrained system in terms of measurable quantities of the observational cosmology: the world proper time, cosmic scale factor, and the density of matter. We define " particles" as field variables in the holomorphic representation which diagonalize the measurable density. The Bogoliubov quasiparticles are determined by diagonalization of the equations of motion (but not only of the initial Hamiltonian) to get the set of integrals of motion (or conserved quantum numbers, in quantum theory). This approach is applied to describe particle creation in the models of the early universe where the Hubble parameter goes to infinity.Comment: 13 pages, Late

Automorphisms and isomorphisms of Chevalley groups and algebras

An adjoint Chevalley group of rank at least 2 over a rational algebra (or a similar ring), its elementary subgroup, and the corresponding Lie ring have the same automorphism group. These automorphisms are explicitly described.Comment: 8 pages. A Russian version of this paper is at http://mech.math.msu.su/department/algebra/staff/klyachko/papers.htm . V4: minor changes in Introduction and Reference