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

Groups with identities

This is a survey of a still evolving subject. The purpose is to develop a theory of prounipotent (respectively pro-$p$) groups satisfying a prounipotent (respectively pro-$p$) identity that is parallel to the theory of PI-algebra

Length-type parameters of finite groups with almost unipotent automorphisms

Let $\alpha$ be an automorphism of a finite group $G$. For a positive integer $n$, let $E_{G,n}(\alpha )$ be the subgroup generated by all commutators $[...[[x,\alpha ], \alpha ],\dots ,\alpha ]$ in the semidirect product $G\langle\alpha \rangle$ over $x\in G$, where $\alpha$ is repeated $n$ times. By Baer's theorem, if $E_{G,n}(\alpha )=1$, then the commutator subgroup $[G,\alpha ]$ is nilpotent. We generalize this theorem in terms of certain length parameters of $E_{G,n}(\alpha )$. For soluble $G$ we prove that if, for some $n$, the Fitting height of $E_{G,n}(\alpha )$ is equal to $k$, then the Fitting height of $[G,\alpha ]$ is at most $k+1$. For nonsoluble $G$ the results are in terms of the nonsoluble length and generalized Fitting height. The generalized Fitting height $h^*(H)$ of a finite group $H$ is the least number $h$ such that $F^*_h(H)=H$, where $F^*_0(H)=1$, and $F^*_{i+1}(H)$ is the inverse image of the generalized Fitting subgroup $F^*(H/F^*_{i}(H))$. Let $m$ be the number of prime factors of the order $|\alpha |$ counting multiplicities. It is proved that if, for some $n$, the generalized Fitting height of $E_{G,n}(\alpha )$ is equal to $k$, then the generalized Fitting height of $[G,\alpha ]$ is bounded in terms of $k$ and $m$. The nonsoluble length~$\lambda (H)$ of a finite group~$H$ is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if $\lambda (E_{G,n}(\alpha ))=k$, then the nonsoluble length of $[G,\alpha ]$ is bounded in terms of $k$ and $m$. We also state conjectures of stronger results independent of $m$ and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups

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

On a conjecture of Goodearl: Jacobson radical non-nil algebras of Gelfand-Kirillov dimension 2

For an arbitrary countable field, we construct an associative algebra that is graded, generated by finitely many degree-1 elements, is Jacobson radical, is not nil, is prime, is not PI, and has Gelfand-Kirillov dimension two. This refutes a conjecture attributed to Goodearl

Classical Monopoles: Newton, NUT-space, gravomagnetic lensing and atomic spectra

Stimulated by a scholium in Newton's Principia we find some beautiful results in classical mechanics which can be interpreted in terms of the orbits in the field of a mass endowed with a gravomagnetic monopole. All the orbits lie on cones! When the cones are slit open and flattened the orbits are exactly the ellipses and hyperbolae that one would have obtained without the gravomagnetic monopole. The beauty and simplicity of these results has led us to explore the similar problems in Atomic Physics when the nuclei have an added Dirac magnetic monopole. These problems have been explored by others and we sketch the derivations and give details of the predicted spectrum of monopolar hydrogen. Finally we return to gravomagnetic monopoles in general relativity. We explain why NUT space has a non-spherical metric although NUT space itself is the spherical space-time of a mass with a gravomagnetic monopole. We demonstrate that all geodesics in NUT space lie on cones and use this result to study the gravitational lensing by bodies with gravomagnetic monopoles. We remark that just as electromagnetism would have to be extended beyond Maxwell's equations to allow for magnetic monopoles and their currents so general relativity would have to be extended to allow torsion for general distributions of gravomagnetic monopoles and their currents. Of course if monopoles were never discovered then it would be a triumph for both Maxwellian Electromagnetism and General Relativity as they stand!Comment: 39 pages, 9 figures and 2 tables available on request from the author

Conformal and Affine Hamiltonian Dynamics of General Relativity

The Hamiltonian approach to the General Relativity is formulated as a joint nonlinear realization of conformal and affine symmetries by means of the Dirac scalar dilaton and the Maurer-Cartan forms. The dominance of the Casimir vacuum energy of physical fields provides a good description of the type Ia supernova luminosity distance--redshift relation. Introducing the uncertainty principle at the Planck's epoch within our model, we obtain the hierarchy of the Universe energy scales, which is supported by the observational data. We found that the invariance of the Maurer-Cartan forms with respect to the general coordinate transformation yields a single-component strong gravitational waves. The Hamiltonian dynamics of the model describes the effect of an intensive vacuum creation of gravitons and the minimal coupling scalar (Higgs) bosons in the Early Universe.Comment: 37 pages, version submitted to Gen. Rel. Gra