Infinitely divisible distribution functions of class L and the Lee-Yang theorem

It is shown that the free-energy density of a large class of ferromagnets satisfying the Lee-Yang property is to be connected with the limit characteristic function of a suitably renormalized sum of independent and non-identically distributed random variables. Using the canonical representation formulae of such characteristic functions, various chains of inequalities are derived for the Ursell functions.     

On the spherical and zonal spherical functions on a compact quantum group

Invariance properties of the functions satisfying an integral spherical equation on a compact quantum group are discussed. It is shown that spherical and zonal spherical functions are conncected with the spherical representation of a compact quantum group.Supported by Polish Scientific Grant RPI10.     

Unitary positive energy representations of the gauge group

We investigate the positive energy representations (also called highest weight representations) of the gauge groupC (T ^v,G ₀),G ₀ being a compact simple Lie group, and discuss their unitarity, using the technique of Verma modules constructed from generalized loop algebras (a simple generalization of Kac-Moody affine Lie algebras). We show that the unitarity of the representation imposes severa restrictions in it. In particular, we show, as a part of a more general result, that the gauge group does not admit faithful unitary positive energy representations.Allocataire du MRT.     

Phase transitions and positive definite couplings

The method of infrared bounds is extended to a large class of nearest neighbour interactions in classical spin systems. Temperature controlled bounds on fluctuations follow whenever the coupling function is a positive definite kernel. Existence of phase transitions is demonstrated for the RP ^Nmodel for d3.     

Notes on quaternionic group representations

We study quaternionic group representations of finite groups systematically and obtain some basic tools of the theory, such as orthogonality relations and the Clebsch-Gordan series for reducible representations. We also derive all irreducible inequivalentQ-representations of a groupG, classifying them according to a suitable generalization of the Wigner-Frobenius-Schur classification.     

A counterexample on idempotent states on a compact quantum group

A simple example is given to illustrate that an idempotent state may not be the Haar state of any subgroup in the case of compact quantum groups.     

Faithful compact quantum group actions on connected compact metrizable spaces

We construct faithful actions of quantum permutation groups on connected compact metrizable spaces. This disproves a conjecture of Goswami.     

On the geometry of some unitary Riemann surface braid group representations and Laughlin-type wave functions

In this note we construct the simplest unitary Riemann surface braid group representations geometrically by means of stable holomorphic vector bundles over complex tori and the prime form on Riemann surfaces. Generalised Laughlin wave functions are then introduced. The genus one case is discussed in some detail also with the help of noncommutative geometric tools, and an application of Fourier–Mukai–Nahm techniques is also given, explaining the emergence of an intriguing Riemann surface braid group duality.     

Fourier analysis on multiplier representations of locally compact Abelian groups

Generalising known results [2] for vector groups, it is shown that, for an arbitrary multiplier ω for an arbitrary locally compact Abelian group G, there is a faithful normal semifinite trace on the von Neuman algebra generated by the regular ω-representation of G which is translation invariant in a certain sense. Analogues of the Fourier transformation, the Plancherel identity and the Fourier inversion formula are obtained in which this trace replaces the Haar measure on the dual group.     

All unitary ray representations of the conformal group SU(2,2) with positive energy

We find all those unitary irreducible representations of the -sheeted covering group of the conformal group SU(2,2)/₄ which have positive energyP ⁰0. They are all finite component field representations and are labelled by dimensiond and a finite dimensional irreducible representation (j ₁,j ₂) of the Lorentz group SL(2). They all decompose into a finite number of unitary irreducible representations of the Poincaré subgroup with dilations.     

An effective method of investigation of positive maps on the set of positive definite operators

Let $\text{A}$₁ be the algebra of linear operators on the n-dimensional Hilbert space $\text{H}$₁, and let $\text{A}$₂ be the algebra of linear operators of the m-dimensional Hilbert space $\text{H}$₂. Let $\text{L}$($\text{A}$₁, $\text{A}$₂) denote the complex space of linear maps from $\text{A}$₁ to $\text{A}$₂. By a positive map we mean an element of the space $\text{L}$($\text{A}$₁, $\text{A}$₂) (superoperator with respect to $\text{H}$₁) which maps positive definite operators in $\text{A}$₁ into positive definite operators in $\text{A}$₂. The aim of this paper is to present an effective method which allows to verify whether a given superoperator Λ∈$\text{L}$($\text{A}$₁, $\text{A}$₂) is a positive map. Besides that necessary and sufficient conditions for the positive definiteness of even-degree forms in many variables are given.     

Nonlinear group representations and evolution equations

The theory of nonlinear evolution equations developed by M. Flato, J. Simon and a few others is reviewed. The method of construction of global solutions is described and the cohomological and analytic properties of linearizability of these equations are described.Invited talk at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.     

Maps from a surface into a compact Lie group and curvature

Letters in Mathematical Physics - This is a brief note on curvature properties of Sobolev Lie groups of maps from a Riemann surface into a compact Lie group K. Freed showed that, in a necessarily...     

The representations of the Poincaré group as functions of the eigenvalues of casimir operators

In this paper explicit basis functions are defined for the Poincaré group. Both these functions and the representation matrix elements are continuous functions of the momentum variables for the whole real p² spectrum, including the p² = 0 point. The essence of our method is to enlarge previously obtained SL(2, C) basis functions and representations of a similar nature.     

Invariant star products and representations of compact semisimple Lie groups

Starting from the work by F. A. Berezin, and earlier paper by the author defined an invariant star product on every nonexceptional Kähler symmetric space. In this Letter a recursion formula is obtained to calculate the corresponding invariant Hochschild 2-cochains for spaces of types II and III. An invariant star product is defined on every integral symplectic (Kähler) homogeneous space of simply-connected compact Lie groups (on every integral orbit of the coadjoint representation). The invariant 2-cochains are obtained from the Bochner-Calabi function of the space. The leading term of the lth-2-cochain is determined by the l-power of the Laplace operator.     

The Thirring-model current group and its representations

The Gel'fand-Araki method is used to construct universally covariant representations of the Thirring-model current group. In this representation the currents are recovered as infinitesimal generators of the corresponding one-parametric subgroups. The determination of the generators of the two-dimensional Poincaré group is discussed and the existence of a selfadjoint Hamiltonian is shown. The possibility of determining the charges and their connection with the quantities defining the representation of the Thirring-model current group (algebra) is considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 57–62, November, 1975.     

Almost periodic functions on connected locally compact groups

Some automatic continuity conditions for almost periodic functions on connected Lie groups and connected locally compact groups are discussed.     

Non-unitary representations and fourier transform on the poincaré group

Two families of representations of the Poincaré group $\text{P}$ associated to complex orbits (with complex linear momenta) are introduced. The conditions for equivalence and irreducibility are stated. Fourier transforms related to them are defined and inversion formulas obtained for functions infinitely differentiable and of compact support on $\text{P}$.     

A criterion for a topological group to admit a continuous embedding in a locally compact group

It is proved that a topological group admits a continuous embedding in a locally compact group if and only if the group in question admits a separating family of unitary representations defining a symmetric Hopf-von Neumann algebra whose intrinsic group coincides with the set of nonzero characters of the predual algebra. To the blessed memory of Leonid Romanovich Volevich The author is indebted to Victor Kac for a pdf-file of the paper [13]. The research was partially supported by the Russian Foundation for Basic Research under grant no. 08-01-00034.     

Deformation and contraction of Poincaré group representations

The irreducible representations of the Poincaré group with p ≠ 0 and discrete spin are analytically deformed to representations of the Galilei group. Via decomposition, reducible representations are also deformed. Consequences for physical interpretation of taking the limit c → ∞ are discussed.