On unitary/antiunitary projective representations of groups

Unitary/antiunitary projective representations of groups (i.e., projective representations of groups where unitary as well as antiunitary operators in a separable complex Hilbert space are considered) are studied in a systematic way. Particular emphasis is put on continuous unitary/antiunitary projective representations of a Polish group G. It is shown that every continuous unitary/antiunitary projective representation of G can be lifted to a Borel unitary/antiunitary multiplier representation of G (namely, to a representation “up to a factor” which is a Borel mapping) and that this, in turn, can be derived from a continuous unitary/antiunitary (ordinary) representation of a Polish group obtained from an extension of G by the multiplicative group of all complex numbers of absolute value 1.     

On the unitary representations of N = 2 superconformal theory

The Kac formula for superconformal dimensions (generalized to N = 2) is further developed (compared to a previous article). A list of discrete values of the central charge for which unitary representations are expected to exist is proposed. For several of these, unitarity is checked by computer. For two values, unitarity is proven by providing explicit fermionic representations. For one of those values, the N = 2 theory coincides with a sub theory of one of the known unitary N = 1 theories, thus extending a similar situation between N = 0 and N = 1.     

Superconformal current algebras and their unitary representations

A natural supersymmetric extension is defined of the current (= affine Kac-Moody Lie) algebra ; it corresponds to a superconformal and chiral invariant 2-dimensional quantum field theory (QFT), and hence appears as an ingredient in superstring models. All unitary irreducible positive energy representations of are constructed. They extend to unitary representations of the semidirect sumS (G) of with the superconformal algebra of Neveu-Schwarz, for , or of Ramond, for =0.On leave of absence from the Institute for Nuclear Research and Nuclear Energy of the Bulgarian Academy of Sciences, BG-1184 Sofia, Bulgaria     

Topology,unitary representations and charged particles

The quantum mechanics of the charged particles with rigid and local symmetries propagating on the manifoldM is studied. It is shown that the classical rigid symmetries of this model may be anomalous. These anomalies are of local and global type, and they related to topological obstructions to lifting a group action of a groupG onM to a principalU(1) bundleP overM. The charged particles with local symmetries may have additional anomalies and the representation theory of the groupC is used to study these anomalies. Finally, the quantum mechanics of the supersymmetric charged particles with symmetries is examined.     

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.     

Irreducible unitary representations of quantum Lorentz group

A complete classification of irreducible unitary representations of a one parameter deformationS _q L(2,C) (0<q<1) ofSL(2,C) is given. It shows that in spite of a popular belief the representation theory forS _q L(2,C) is not a smooth deformation of the one forSL(2,C).     

Discrete degenerate representations of non-compact unitary groups

Two degenerate principal series of irreducible unitary representations of an arbitrary non-compact unitary groupU(p,1) are derived. These series are determined by the eigenvalues of the first and second-order invariant operators, which are shown to possess a discrete spectrum. The explicit form of the corresponding harmonic functions is derived and the properties of the discrete representations are discussed in detail. Moreover, in the Appendix, we derive the properties of the corresponding degenerate representations of an arbitrary compactU(p) group.On leave of absence from Institute of Nuclear Research, Warsaw, Poland.On leave of absence from Institute of Physics of the Czechoslovak Academy of Sciences, Prague, Czechoslovakia.     

The unitary irreducible representations of the quantum heisenberg algebra

The goal of this work is to describe the irreducible representations of the quantum Heisenberg algebra and the unitary irreducible representation of one of its real forms. The solution of this problem is obtained through the investigation of theleft spectrum of the quantum Heisenberg algebra using the result about spectra of generic algebras of skew differential operators (cf. [R]).     

Projective unitary representations of smooth Deligne cohomology groups

We construct projective unitary representations of the smooth Deligne cohomology group of a compact oriented Riemannian manifold of dimension 4k+1, generalizing positive energy representations of the loop group of the circle at level 2. We also classify such representations under a certain condition. The number of the equivalence classes of irreducible representations being finite is determined by the cohomology of the manifold.     

Projective unitary antiunitary representations of locally compact groups

We give here a systematic presentation of the theory of projective representations when antiunitary operators are present. In particular the imprimitivity theorem of Mackey is proved in this situation and all the unitary antiunitary representations of the extended Poincaré group are derived.     

Asymptotic behaviour of the unitary representations of the Poincaré group

The results of R. Jost and K. Hepp [2] and D. Maison [3] concerning the asymptotic behaviour of the translation operators are generalized.     

Perturbations of unitary representations of finite dimensional Lie groups

In quantum physical theories, interactions in a system of particles are commonly understood as perturbations to certain observables, including the Hamiltonian, of the corresponding interaction-free system. The manner in which observables undergo perturbations is subject to constraints imposed by the overall symmetries that the interacting system is expected to obey. Primary among these are the spacetime symmetries encoded by the unitary representations of the Galilei group and Poincaré group for the non-relativistic and relativistic systems, respectively. In this light, interactions can be more generally viewed as perturbations to unitary representations of connected Lie groups, including the non-compact groups of spacetime symmetry transformations. In this paper, we present a simple systematic procedure for introducing perturbations to (infinite dimensional) unitary representations of finite dimensional connected Lie groups. We discuss applications to relativistic and non-relativistic particle systems.     

On the study of unitary representations of the twisted Heisenberg-Virasoro algebra via highest weight modules over affine Lie algebras

In this paper, we first construct an analogue of the Sugawara operators for the twisted Heisenberg-Virasoro algebra. By using these operators, we show that every integrable highest weight module over an affine Lie algebra can be viewed as a unitary representation of the twisted Heisenberg-Virasoro algebra. As a by-product of our constructions, we give the unitary representations of the twisted Heisenberg-Virasoro algebra which have the central charges appearing in [1]. Our approach to obtain these central charges is different with that of [1].     

Degenerate representations of non-compact unitary groups. II. Continuous series

Three degenerate principal series of irreducible unitary representations of an arbitrary non-compact unitary groupU(p, q) are derived. There series are determined by the eigenvalues of the first and second-order invariant operators, the former having a discrete spectrum and the latter a continuous one. The explicit form of the corresponding harmonic functions is derived and the properties of the continuous representations are discussed.On leave of absence from Institute of Physics of the Czechoslovak Academy of Sciences, Prague, Czechoslovakia.On leave of absence from Institute of Nuclear Research, Warsaw, Poland.     

Spectra of generators in irreducible unitary representations of non-compact semi-simple groups

Several theorems concerning the spectra of elements of the complexified Lie algebra in unitary representations of non-compact semi-simple groups are proved. The principal theorem gives purely Lie algebraic sufficient conditions for the type of spectrum (point or continuous) of any element of the real Lie algebra. For elements of special self adjoint Cartan subalgebras these conditions are rephrased in terms of the basis-dependent information most readily available to the physicist, namely their hermiticity properties and the values of the structure constants: roots, etc.International Atomic Energy Agency International Center for Theoretical PhysicsOn leave of absence from New Mexico State University, NM, USA.     

A classification of the unitary irreducible representations ofSU(N, 1)

All inequivalent continuous unitary irreducible representations ofS U(N, 1) (N2) have been determined and classified. The matrix elements of the infinitesimal generators realized on a certain Hilbert space have been derived. Representations of the groups ,S U(N, 1)/Z _N+1, andU(N, 1) are classified in a similar manner.