A Monoidal Category for Perturbed Defects in Conformal Field Theory

Starting from an abelian rigid braided monoidal category C{mathcal{C}} we define an abelian rigid monoidal category C_F{mathcal{C}_F} which captures some aspects of perturbed conformal defects in two-dimensional conformal field theory. Namely, for V a rational vertex operator algebra we consider the charge-conjugation CFT constructed from V (the Cardy case). Then C = Rep(V){mathcal{C} = {rm Rep}(V)} and an object in C_F{mathcal{C}_F} corresponds to a conformal defect condition together with a direction of perturbation. We assign to each object in C_F{mathcal{C}_F} an operator on the space of states of the CFT, the perturbed defect operator, and show that the assignment factors through the Grothendieck ring of C_F{mathcal{C}_F}. This allows one to find functional relations between perturbed defect operators. Such relations are interesting because they contain information about the integrable structure of the CFT.     

Simple algebras and Drinfeld doubles

The Drinfeld double structure underlying all the Cartan series of simple Lie algebras is discussed. The two solvable algebras that allow its definition are constructed enlarging each simple algebra of rank n with a central Abelian algebra of dimension n. In these solvable algebras, isomorphic to the two Borel subalgebras, a pairing can be built. The complete machinery of Drinfeld doubles is described in all details. This offers a new approach to the explicit construction of canonical quantum deformation of simple algebras and fixes uniquely, independently and differently from known conventions, canonical bases for all of them. The Drinfeld doubles for A _n and C _n are explicitly written. The full quantization of su(3) is discussed in terms of standard commutators as the A ₂ Drinfeld double requires. The text was submitted by the authors in English.     

Weight Functions and Drinfeld Currents

A universal weight function for a quantum affine algebra is a family of functions with values in a quotient of its Borel subalgebra, satisfying certain coalgebraic properties. In representations of the quantum affine algebra it gives off-shell Bethe vectors and is used in the construction of solutions of the qKZ equations. We construct a universal weight function for each untwisted quantum affine algebra, using projections onto the intersection of Borel subalgebras of different types, and study its functional properties.     

Representation Theory of Lattice Current Algebras

Lattice current algebras were introduced as a regularization of the left- and right moving degrees of freedom in the WZNW model. They provide examples of lattice theories with a local quantum symmetry . Their representation theory is studied in detail. In particular, we construct all irreducible representations along with a lattice analogue of the fusion product for representations of the lattice current algebra. It is shown that for an arbitrary number of lattice sites, the representation categories of the lattice current algebras agree with their continuum counterparts. Received: 25 April 1996 / Accepted: 14 April 1997     

Parametric Representation of Noncommutative Field Theory

In this paper we investigate the Schwinger parametric representation for the Feynman amplitudes of the recently discovered renormalizable quantum field theory on the Moyal non commutative space. This representation involves new hyperbolic polynomials which are the non-commutative analogs of the usual “Kirchoff” or “Symanzik” polynomials of commutative field theory, but contain richer topological information. Work supported by ANR grant NT05-3-43374 “GenoPhy”.     

Spectral Representation Theory for Dielectric Behavior of Nonspherical Cell Suspensions

Recent experiments revealed that the dielectric dispersion spectrum of fission yeast cells in a suspension was mainly composed of two sub-dispersions. The low-frequency sub-dispersion depended on the cell length, while the high-frequency one was independent of it. The cell shape effect was simulated by an ellipsoidal cell model but the comparison between theory and experiment was far from being satisfactory. Prompted by the discrepancy, we proposed the use of spectral representation to analyze more realistic cell models. We adopted a shell-spheroidal model to analyze the effects of the cell membrane. It is found that the dielectric property of the cell membrane has only a minor effect on the dispersion magnitude ratio and the characteristic frequency ratio. We further included the effect of rotation of dipole induced by an external electric field, and solved the dipole-rotation spheroidal model in the spectral representation. Good agreement between theory and experiment has been obtained.     

Spectral Representation Theory for Dielectric Behavior of Nonspherical Cell Suspensions

Recent experiments revealed that the dielectric dispersion spectrum of fission yeast cells in a suspension was mainly composed of two sub-dispersions. The low-frequency sub-dispersion depended on the cell length, while the high-frequency one was independent of it. The cell shape effect was simulated by an ellipsoidal cell model but the comparison between theory and experiment was far from being satisfactory. Prompted by the discrepancy, we proposed the use of spectral representation to analyze more realistic cell models. We adopted a shell-spheroidal model to analyze the effects of the cell membrane. It is found that the dielectric property of the cell membrane has only a minor effect on the dispersion magnitude ratio and the characteristic frequency ratio. We further included the effect of rotation of dipole induced by an external electric field, and solved the dipole-rotation spheroidal model in the spectral representation.Good agreement between theory and experiment has been obtained.     

On the Representation Theory of Virasoro Nets

We discuss various aspects of the representation theory of the local nets of von Neumann algebras on the circle associated with positive energy representations of the Virasoro algebra (Virasoro nets). In particular we classify the local extensions of the c=1 Virasoro net for which the restriction of the vacuum representation to the Virasoro subnet is a direct sum of irreducible subrepresentations with finite statistical dimension (local extensions of compact type). Moreover we prove that if the central charge c is in a certain subset of (1, ), including [2, ), and h(c–1)/24, the irreducible representation with lowest weight h of the corresponding Virasoro net has infinite statistical dimension. As a consequence we show that if the central charge c is in the above set and satisfies c25 then the corresponding Virasoro net has no proper local extensions of compact type.Supported in part by the Italian MIUR and GNAMPA-INDAM.     

Drinfeld Functor and Finite-Dimensional Representations of Yangian

We extend the results of Drinfeld on the Drinfeld functor to the case . We present the character of finite-dimensional representations of the Yangian in terms of the Kazhdan–Lusztig polynomials as a consequence. Received: 4 September 1998 / Accepted: 12 February 1999     

Reflection Functor in the Representation Theory of Preprojective Algebras for Quivers and Integrable Systems

A brief overview of the representation theory of quivers and the associated (deformed) preprojective algebras, as well as of the theories of moduli spaces of these algebras, quiver varieties and a reflection functor, is given. It is proven that a bijection between moduli spaces (in particular, between quiver varieties), which is induced by a reflection function, is the isomorphism of symplectic affine varieties. The Hamiltonian systems on quiver varieties are defined, and the application of a reflection functor to them is described. The review of [1], concerning the case of a cyclic quiver is given, and a role of the reflection functor in this case is clarified. The “spin” integrable generalizations of Calogero–Moser systems and their application to the KP hierarchy generalizations are described.     

Ergodic Coactions with Large Multiplicity and Monoidal Equivalence of Quantum Groups

We construct new examples of ergodic coactions of compact quantum groups, in which the multiplicity of an irreducible corepresentation can be strictly larger than the dimension of the latter. These examples are obtained using a bijective correspondence between certain ergodic coactions on C*-algebras and unitary fiber functors on the representation category of a compact quantum group. We classify these unitary fiber functors on the universal orthogonal and unitary quantum groups. The associated C*-algebras and von Neumann algebras can be defined by generators and relations, but are not yet well understood.     

The Representation Theory of Decoherence Functionals in History Quantum Theories

In the first part of this paper the general perspective of history quantum theoriesis reviewed. History quantum theories provide a conceptual and mathematicalframework for formulating quantum theories without a globally definedHamiltonian time evolution and for introducing the concept of space-time eventinto quantum theory. On a mathematical level a history quantum theory ischaracterized by the space of histories, which represent the space-time events, andby the space of decoherence functionals, which represent the quantum mechanicalstates in the history approach. The second part of this paper is devoted to thestudy of the structure of the space of decoherence functionals for some physicallyreasonable spaces of histories in some detail. The temporal reformulation ofstandard Hamiltonian quantum theories suggests to consider the case that thespace of histories is given by (i) the lattice of projection operators on someHilbert space or, slightly more generally, (ii) the set of projection operators insome von Neumann algebra. In the case (i) the conditions are identified underwhich decoherence functionals can be represented by, respectively, trace classoperators, bounded operators, or families of trace class operators on the tensorproduct of the underlying Hilbert space by itself. Moreover, we discuss thenaturally arising representations of decoherence functionals as sesquilinear forms.The paper ends with a discussion of the consequences of the results for thegeneral axiomatic framework of history theories.