Centrosymmetric lie algebras and boost-dilations

We define the Lie algebrac(n) of centrosymmetric matrices. It generates a noncompact and nonsemisimple local Lie group with the unusual property that expc(n) c(n). The group contains an invariant subgroup of Lorentz boost/ dilation transformations. Forn even, these form a subgroup of the conformal group of the Lorentzian metric with signature (– + – + – +).     

Covariant wave functions with invariant helicity for massless particles

Considering a wave function for a massless particle, transforming according to an arbitrary irreducible representation (IR) of the homogeneous Lorentz group, we determine the basic conditions for to be an eigenfunction with a specified value of the helicity inall Lorentz frames. The method used is direct and elementary, requiring no knowledge of the IR's of the Poincaré group. It is shown that there existsno invariant helicity state in unitary representations of the Lorentz group, and one such state in any non-unitary representation (with one extra in special cases).     

The topological structure of the unitary and automorphism groups of a factor

It is proved that a large class ofII ₁ factors have unitary group which is contractible in the strong operator topology, but whose fundamental group in the norm topology is isomorphic to the additive real numbers as proven by Araki-Smith-Smith [1]. The class includes the approximately finite dimensional factor of typeII ₁ and the group factor associated with the free group on infinitely many generators. This contractibility is used to prove the contractibility of the automorphism group of the approximately finite dimensional factor of typeII ₁ and typeII . It is further shown that the fundamental group of the automorphism group of the approximately finite dimensional factor of typeIII , 0<<1, is isomorphic to the integer group .Dedicated to Huzihiro ArakiThis research is supported in part by NSF Grant DMS-9206984     

Some remarks on braided group reconstruction and braided doubles

The cross coproduct braided group AutC)B is obtained by Tannaka-Krein reconstruction from C ^B C for a braided group B in braided category C. We apply this construction to obtain partial solutions to two problems in braided group theory, namely the tensor problem and the braided double. We obtain AutC) Aut(C) Aut(C) Aut(C) and higher braided group spin chains. The example B(R) B(R) ... B(R) is described explicitly by R-matrix relations. We also obtain Aut(C) Aut(C)* as a dual quasitriangular codouble braided group by reconstruction from the dual category C° C. General braided double crossproducts B C are also considered.     

Symmetries and the Einstein-Maxwell field equations the null field case

This paper deals with space-times that satisfy the Einstein-Maxwell field equations in the presence of a perfect fluid, which may be charged. The electromagnetic field is assumed to be null. It is proved that if the space-time admits a group of isometrics then the fluid velocityu ⁱ, energy density, pressurep, and charge density are invariant under the group. In addition, if the charge density is nonzero, the electromagnetic field tensorf _ij is also invariant. On the other hand, examples of exact solutions are given which establish that if = 0, thenF _ij is not necessarily invariant under the group. In the case of spherically symmetric space-times, however, in which the group of isometries acting isSO (3),f _ij is invariant, independently of whether or not is nonzero. This result leads to the conclusion that in a spherically symmetric space-time the field equations in question admit no solutions with non-trivial null electromagnetic field.     

p-Adic string compactified on a torus

The open p-adic string world sheet is a coset space F=T/, where T is the Bruhat-Tits tree for the p-adic linear group GL(2, _p ) and PGL(2, _p ) is some Schottky group. The string dynamics is governed by the local action on F, with the fields taking values in a compact group G. We find the correlation functions and partition functions for the p-adic string surfaces of arbitrary genus and G=U(1)^xD (D-dimensional torus).     

Remarks on the Schur—Howe—Sergeev Duality

We establish a new Howe duality between a pair of two queer Lie superalgebras (q(m),q(n)). This gives a representation theoretic interpretation of a well-known combinatorial identity for Schur Q-functions. We further establish the equivalence between this new Howe duality and the Schur–Sergeev duality between q(n) and a central extension of the hyperoctahedral group H _k. We show that the zero-weight space of a q(n)-module with highest weight given by a strict partition of n is an irreducible module over the finite group parameterized by . We also discuss some consequences of this Howe duality.     

The Automorphism Group of a Simple Tracially AI Algebra

The structure of the automorphism group of a simple TAI algebra is studied. In particular, we show that is isomorphic (as a topological group) to an inverse limit of discrete abelian groups for a unital, simple, AH algebra with bounded dimension growth. Consequently, is totally disconnected. Another consequence of our results is the following: Suppose A is the transformation group C*-algebra of a minimal Furstenberg transformation with a unique invariant probability measure. Then the automorphism group of A is an extension of a simple topological group by the discrete 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.     

Poisson-Lie group of pseudodifferential symbols

We introduce a Lie bialgebra structure on the central extension of the Lie algebra of differential operators (with scalar or matrix coefficients) on the line and on the circle. This defines a Poisson-Lie structure on the dual group of pseudodifferential symbols of an arbitrary real (or complex) order. We show that the usual (second) Benney, GL _n -KdV (or GL _n -Adler-Gelfand-Dickey) and KP Poisson structures are naturally realized as restrictions of this Poisson structure to submanifolds of this universal Poisson-Lie group. Moreover, the reduced (=SL _n ) versions of these manifolds (orW _n -algebras in physical terminology) can be viewed as certain subspaces of the quotient of this Poisson-Lie group by the dressing action of the group of functions on the circle (or as a result of a Poisson reduction). Finally we define an infinite set of commuting functions on the Poisson-Lie group that give the standard families of Hamiltonians when restricted to the submanifolds mentioned above. The Poisson structure and Hamiltonians on the whole group interpolate between the Poisson structures and Hamiltonians of Benney, KP and KdV flows. We also discuss the geometrical meaning ofW as a limit of Poisson algebrasW as 0.     

Kinematics of a Spacetime with an Infinite Cosmological Constant

A solution of the sourceless Einstein's equation with an infinite value for the cosmological constant is discussed by using Inönü–Wigner contractions of the de Sitter groups and spaces. When , spacetime becomes a four-dimensional cone, dual to Minkowski space by a spacetime inversion. This inversion relates the four-cone vertex to the infinity of Minkowski space, and the four-cone infinity to the Minkowski light-cone. The non-relativistic limit c is further considered, the kinematical group in this case being a modified Galilei group in which the space and time translations are replaced by the non-relativistic limits of the corresponding proper conformal transformations. This group presents the same abstract Lie algebra as the Galilei group and can be named the conformal Galilei group. The results may be of interest to the early Universe Cosmology.     

SÕ(n) — Symmetric field equations

Field equations satisfied by the irreducible realizations of any inhomogeneous pseudo-orthogonal group are derived. For those representations which are characterized by the vanishing of the invariants of the inhomogeneous group, the field equations are of first order, of the formS _{AB p B} =p _A . The possibility of consideringSO(q ₁,q ₂) as a higher symmetry group is discussed briefly.     

Structure of Malicious Singularities

In this paper, we investigate relativistic spacetimes, together with their singular boundaries (including the strongest singularities of the Big Bang type, called malicious singularities), as noncommutative spaces. Such a space is defined by a noncommutative algebra on the transformation groupoid = × G, where is the total space of the frame bundle over spacetime with its singular boundary, and G is its structural group. We show that there exists the bijective correspondence between unitary representations of the groupoid and the systems of imprimitivity of the group G. This allows us to apply the Mackey theorem to this case, and deduce from it some information concerning singular fibers of the groupoid . At regular points the group representation, which is a part of the corresponding system of imprimitivity, does not have discrete components, whereas at the malicious singularity such a group representation can be a single representation (in particular, an irreducible one) or a direct sum of such representations. A subgroup K G, from which—according to the Mackey theorem—the representation is induced to the whole of G, can be regarded as measuring the richness of the singularity structure. In this sense, the structure of malicious singularities is richer than those of milder ones.     

The group of automorphisms of the galilei group

The group of automorphisms of the Galilei groupG: Aut(G) is calculated. It is shown that Aut(G) has the structure of a semi-direct product byG of the group _m ^* ×_m where _m is the group of reals noted multiplicatively and _m ^* <_m is the subgroup of positive reals.     

Renormalization group study of anomalous acoustic behavior in critical percolation network

We study the acoustic behavior of critical percolation network within a real-space renormalization group framework recently proposed by Ohtsuki and Keyes. Using large cell Monte Carlo renormalization group calculations, we obtain the exponent for anomalous sound dispersion K ^1+x/v . Our estimate 2x/v0.80 is in agreement with the exponent for anomalous diffusion in percolation clusters =(–)/v.     

2+1 gravity for genus >1

In [1] we analysed the algebra of observables for the simple case of a genus 1 initial data surface ² for 2+1 De Sitter gravity. Here we extend the analysis to higher genus. We construct for genus 2 the group of automorphismsH of the homotopy group ₁ induced by the mapping class group. The groupH induces a groupD of canonical transformations on the algebra of observables which is related to the braid group for 6 threads.     

Quantization on the Virasoro group

The quantization of the Virasoro group is carried out by means of a previously established group approach to quantization. We explicitly work out the two-cocycles on the Virasoro group as a preliminary step. In our scheme the carrier space for all the Virasoro representations is made out of polarized functions on the group manifold. It is proved that this space does not contain null vector states, even forc1, although it is not irreducible. The full reduction is achieved in a striaghtforward way by just taking a well defined invariant subspace _{(c, h)} , the orbit of the enveloping algebra through the vacuum, which is irreducible for any value ofc andh. _{(c, h)} is a proper subspace of the space of polarized functions for those values ofc andh for which the Kac determinant is zero. We give the local version of these group representations as well as the associated classical phase space structures, i.e., symplectic form and Noether invariants.Research partially supported by the Conselleria de Cultura de la Generalitat Valenciana, the Plan de formación del Personal investigador, the Comision Interministerial de Ciencia y Tecnologia (CICYT) and the British Council     

Stable states of a relativistic harmonic oscillator imbedded in a random stochastic thermostat

An analysis in phase space of the behavior of a relativistic four-dimensional harmonic oscillator undergoing stochastic interactions shows that the group of linear canonical transformations in phase space which leaves invariant the Poisson brackets is anSp(12,4) group, withSp(12,4)U(6,2)SU(1,1)SO(6,2). The application of Cartan's treatment to its behavior implies a classification of its stable states characterized by a set of discrete numbers.     

Coherent states and partition function

The concept of coherent states for arbitrary Lie group is suggested as a tool for explicitly obtaining an integral representation of the partition function, whenever the Hamiltonian has a dynamical group. Two examples are thoroughly discussed: the case of the nilpotent group of Weyl related to a generic many-body problem with two-body interactions, and the case of SU(1, 1)() relevant for a superfluid system.     

Letter: Riemannian Metric of the Invariance Quantum Group of the Fermion Algebra

We calculate the bi-invariant metric of FIO(2), the inhomogeneous invariance quantum group of the fermion algebra. We find that this metric is identical to that of the bi-invariant metric of GL(2, R) ⁺ × SU (1, 1). However, the quantum group manifold is restricted to a region of the GL(2, R) manifold.