On the symmetrized cones in test function algebra

The cones K and C in the algebra of test functions involved in the notions of positivity and T-positivity are studied. We prove that the closures of symmetrized cones sK, sC and sK + sC are proper cones.     

On the algebra of test functions for field operators

It is shown that every continuous linear functional on the field algebra can be defined by a vector in the Hilbert space of some representation of the algebra. The functionals which can be written as a difference of positive ones are characterized. By an example it is shown that a positive functional on a subalgebra does not always have an extension to a positive functional on the whole algebra.     

Translationally invariant states and the spectrum ideal in the algebra of test functions for quantum fields

A class of states on Borchers' tensor algebra is constructed. These states are invariant under the translation group and fulfill the spectrum condition. This leads to a characterization of the linear span of all such states in terms of a simple continuity property.     

On closure of the light-cone gauge closed superstring algebra

We construct four-field generators to achieve closure of the ten-dimensional SUSY algebra for the heterotic and type II superstring light-cone gauge field theory. Closure for the heterotic case is achieved in a regularization independent manner, whilst that for type II needs the use of constant time and phase integration regularizations. The resulting four-field hamiltonians are briefly related to terms arising in the amplitudes for the two theories constructed from the cubic terms alone.     

On the quantum group and quantum algebra approach toq-special functions

Two interpretations ofq-special functions based on quantum groups and algebras have been presented in the literature. The connection between these approaches is explained using as an example the case whereU _q (sl(2)) is the basic structure.Supported in part by the National Sciences and Engineering Research Council (NSERC) of Canada.     

The blob algebra and the periodic Temperley-Lieb algebra

We determine the structure of two variations on the Temperley-Lieb algebra, both used for dealing with special kinds of boundary conditions in statistical mechanics models. The first is a new algebra, the blob algebra. We determine both the generic and all the exceptional structures for this two parameter algebra. The second is the periodic Temperley-Lieb algebra. The generic structure and part of the exceptional structure of this algebra have already been studied. We complete the analysis using results from the study of the blob algebra.     

A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions

A quantum deformation of the Virasoro algebra is defined. The Kac determinants at arbitrary levels are conjectured. We construct a bosonic realization of the quantum deformed Virasoro algebra. Singular vectors are expressed by the Macdonald symmetric functions. This is proved by constructing screening currents acting on the bosonic Fock space.     

An associative algebra of functions on the orbits of nilpotent Lie groups

We introduce an associative bilinear product F * G on the space of Schwartz functions of the coadjoint orbits of a large family of nilpotent Lie groups. We can then introduce an operator , for all . When the nilpotent group is the Heisenberg group N(3), the correspondence is a canonical prequantization of functions F(q, p) in phase space. Part of this work was done at the University of California, Los Angeles.     

The choice of test functions in gauge quantum field theories

We discuss the problem of the choice of test functions in gauge quantum field theories. Analysis of explicitly soluble models suggests that the test function spaces which are suitable for local and covariant formulation of gauge theories are the Gelfand and Shilov spaces , +>1. We also discuss a possible generalization of the spectral condition.     

Conformal algebra and multipoint correlation functions in 2D statistical models

Based on the conformal algebra approach, a general technique is given for the calculation of multipoint correlation functions in 2D statistical models at the critical point. Particular conformal operator algebras are found for operators of the 2D q-component Potts model (1 < q < 4), and the O(N) model (0 < N < 2) at the critical point. A number of four-point correlation functions are calculated for these models.     

Local cohomology of the algebra of C ∞ functions on a connected manifold

A multilinear version of Peetre's theorem on local operators is the key to prove the equality between the local and differentiable Hochschild cohomology on the one hand, and on the other hand the equality between the second and third local Chevalley cohomology groups and their differentiable counterpart.Aspirant du F.N.R.S.     

The BRS algebra of a free minimal differential algebra

We construct the Weil and the universal BRS algebras of theories that can have as a gauge symmetry a free minimal differential (Sullivan) algebra, the natural extension of Lie algebras allowing the definition of p-form gauge potentials (p > 1). The geometrical meaning of these p-form gauge potentials can be understood with the notion of a Quillen superconnection.     

The Virasoro algebra

Three theroems are proved. With suitable hypotheses in each case, characterizations are found for the Virasoro algebra, for some of its representations, and for the Ramond-Neveu-Schwarz superalgebra built around the Virasoro algebra.     

Dynamical Lie algebra and conditional operators for many fermion Green functions

A method is introduced to calculate thermodynamic Green functions. A powerful theorem by Masson is the motivation for expressing the Fourier-time transform of the Green function as a super space matrix element of the resolvent of the Liouville operator. The eigenvalues of the Liouville operator are then expressed in a form first suggested by Judd for atomic systems and which are shown to be members of a Lie group.     

The twisted quantum affine algebra Uq(A2(2)) and correlation functions of the Izergin-Korepin model

We derive the exchange relations of the vertex operators of U_q(A₂⁽²⁾) and show that these vertex operators give the bosonization of the Izergin-Korepin model. We give an integral expression of the correlation functions of the Izergin-Korepin model and derive the difference equations which they satisfy.     

The algebraic structure of the Onsager algebra

We study the Lie algebra structure of the Onsager algebra from the ideal theoretic point of view. A structure theorem of ideals in the Onsager algebra is obtained with the connection to the finite-dimensional representations. We also discuss the solvable algebra aspect of the Onsager algebra through the formal theory.     

The three-dimensional origin of the classifying algebra

It is known that reflection coefficients for bulk fields of a rational conformal field theory in the presence of an elementary boundary condition can be obtained as representation matrices of irreducible representations of the classifying algebra, a semisimple commutative associative complex algebra.