Continuity conditions for finite-dimensional representations of some locally bounded groups

Necessary and sufficient conditions are obtained for the continuity of finite-dimensional representations of “almost divisible” locally pseudocompact groups in terms of the oscillations of these representations at a point. Applications to the continuity problem for finite-dimensional representations of almost connected locally compact groups and to the theory of quasirepresentations are given. To the blessed memory of Eckehart Hotzel Partially supported by the Russian Foundation for Basic Research under grant no. 02-01-00574, by the Program of Supporting the Leading Scientific Schools under grant no. NSh 619.203.1, and by an INTAS grant.     

Fourier-Stieltjes localization in neighborhoods of finite-dimensional irreducible representations of locally compact groups

An analog of the localization principle for the almost convergence for any element of the Fourier-Stieltjes algebra (Fourier-Stieltjes transforms of bounded measures) is established for any unimodular amenable locally compact groups in neighborhoods of finite-dimensional representations of these groups. To the blessed memory of Yuri Solov’ev Partially supported by the Russian Foundation for Basic Research under grant no. 02-01-00574, by the Program of Supporting the Leading Scientific Schools under grant no. NSh 619.203.1, and by the INTAS grant.     

Norm continuous representations of locally compact groups

The structure of norm continuous representations of locally compact groups in Banach spaces is specified. Partially supported by the RFBR grant no. 08-01-00034 and by the Russian Government Program of Supporting Leading Scientific Schools under grant no. NSh-1562.2008.1. To the Memory of S. A.     

Quantization of triangular Lie bialgebras

We derive a universal twisting element for an arbitrary triangularγ-matrix using a simple analogue of the Fedosov quantization method. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002. The work of SLL is partially supported by RFBR grant 00-02-17-956 and the grant INTAS 00-262. The work of AASh is supported by RFBR grant 02-02-06879 and Russian Ministry of Education under the grant E-00-33-184. The work of VAD is partially supported by the grant INTAS 00-561 and by the Grant for Support of Scientific Schools 00-15-96557. The work of API is partially supported by the RFBR grant 00-01-00299.     

Equations of motion in the null formalism

An outline is presented for an approach to a two-body problem. The method uses null coordinates attached to one body and an expansion away from flat space. Unfortunately, the method fails to exhibit the Newtonian interaction between two particles. Therefore, no detailed calculations are given.Research supported in part by National Science Foundation under grant No. MPS74-15246.Based in part on a dissertation for the Ph.D. at Syracuse University, June 1971.     

Rigorous entropy-energy arguments

We present a method for making rigorous various arguments which predict that certain situations are unstable because of a balance of energy vs. entropy. As applications, we give yet another proof that the two-dimensional plane rotor has no spontaneous magnetization and we make rigorous Thouless' arguments on the one-dimensional Ising model with couplingJ/n ².on leave from Departments of Mathematics and Physics, Princeton University. Research partially supported by U.S.N.S.F. under grant No. MCS-78-01885.Danforth Fellow. Research partially supported by U.S.N.S.F. under grant No. PHY 78-23952.     

Ergosphere instability

We consider stationary asymptotically flat spacetimes having an ergosphere but with no horizon. In the framework of linear perturbation theory such configurations are unstable or marginally unstable to scalar and electromagnetic perturbations.Research supported in part by the National Science Foundation under grant MPS 74-17456 with the University of Chicago and grant MPS 74-7456 at the University of Wisconsin-Milwaukee     

Faithful representations of groups by automorphisms of topologies

It is proved that every (in general, infinite) group is the full automorphism group of some topology. Supported by RFBR under grant no. 07-01-00101-a.     

Six generatorq-deformed Lorentz algebra

The six generator deformation of the Lorentz algebra is presented. The Hopf algebra structure and the reality conditions are found. The chiral decomposition of SL(2, C) is generalized to theq-case. Casimir operators for theq-Lorentz algebra are given.This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY-85-15857.     

The smooth cohomology ofN=2 supersymmetric Landau-Ginzburg field theories

We compute the smooth cohomology (both unrestricted and compactly supported) of the supercharge of an ultraviolet cutoffN=2 supersymmetric Landau-Ginzburg field theory.Supported in part by the National Science Foundation under grant DMS-9206936.Supported in part by the Department of Energy under grant DE-FG02-88ER25065.     

On ergodic properties for harmonic crystals

We consider the dynamics of a harmonic crystal in n dimensions with d components, where d and n are arbitrary, d, n ⩾ 1. The initial data are given by a random function with finite mean energy density which also satisfies a Rosenblatt-or Ibragimov-type mixing condition. The random function is close to diverse space-homogeneous processes as x _n → ±∞, with the distributions μ±. We prove that the phase flow is mixing with respect to the limit measure of statistical solutions. Partially supported by RFBR under grant no. 06-01-00096.     

Van der Waerden continuity theorem for semisimple lie groups

As was proved by van der Waerden in 1933, every finite-dimensional locally bounded representation of a semisimple compact Lie group is continuous. This is the famous “van der Waerden continuity theorem,” and it motivated a vast literature. In particular, relationships between the assertion of the theorem (and of the inverse, in a sense, to this theorem) and some properties of the Bohr compactifications of topological groups were established, which led to the introduction and the study of certain classes of the so-called van der Waerden groups and algebras. Until now, after more than 70 years have passed, the van der Waerden theorem appears in monographs and surveys in diverse forms; new proofs were found and then simplified in important special cases. In this paper, we prove that the statement of the van der Waerden theorem remains valid for all (not necessarily compact) real semisimple Lie groups, i.e., any given finite-dimensional representation of a real semisimple Lie group is continuous if and only if this representation is locally bounded. More subtle results are also obtained. The main theorem contains several conditions equivalent to the continuity condition for a finite-dimensional representation. In particular, it is proved that every finite-dimensional representation of a real semisimple Lie group is continuous if and only if the restriction of this representation to the “compact” part, to the Abelian part, or to the nilpotent part of the Iwasawa decomposition is locally bounded, and the original van der Waerden theorem is also somewhat refined. For instance, the following assertion holds: every finite-dimensional representation of a compact semisimple Lie group is continuous if and only if the restriction of this representation to some maximal torus is locally bounded. Dedicated to the memory of George Mackey (1916–2006) Partially supported by the Russian Foundation for Basic Research under grant no. 02-01-00574, by the Program of Supporting the Leading Scientific Schools under grant no. NSh 619.203.1, and by the INTAS grant.     

Reality in the differential calculus on q-Euclidean spaces

The nonlinear reality structure of the derivatives and the differentials for the Euclidean q-spaces are found. A real Laplacian is constructed and reality properties of the exterior derivative are given.This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY-85-15857.     

Barrier penetration and initial values in Kaluza-Klein theories

Topology change is discussed as a barrier penetration process and illustrated by explicit calculations in Witten's Kaluza-Klein bubble geometry. Initial values are given for other bubble geometries, including ones with negative total mass.Work supported in part by the National Science Foundation, under grant No. PHY82-05659, and by the Humboldt Foundation.     

Binding of3He atoms to liquid helium

Summary We calculate the³He−⁴He chemical potential and change in volume in bulk liquid⁴He employing integral equation and Monte Carlo procedures. Detailed numerical results on these quantities and their dependence on bulk density and on the isotopic-mass difference are presented and discussed. This work has been supported by the Deutsche Forschungsgemeinschaft under grant Ri 267 and by the U.S. Department of energy under contract DE-AC 02-76 ER03077 and by the NSF under grant DMR-77-18329.     

Estimates of homogenization for a parabolic equation with periodic coefficients

The asymptotic behavior of the operator exponent related to the Cauchy problem for a parabolic equation with periodic coefficients is studied either under the reduction of the periodicity cell or for large times. Estimates for the closeness of the operator exponentials (the original and the limit) with respect to the L ²-operator norm and the related H ¹-estimates are obtained under minimal assumptions concerning the smoothness of the heat matrix and of the initial data. Financially supported by RFBR under grant no. 05-01-00621.