Quantum Riemann surfaces I. The unit disc

We construct a non-commutative ^*-algebra which is a quantum deformation of the algebra of continuous functions on the closed unit disc . is generated by the Toeplitz operators on a suitable Hilbert space of holomorphic functions onU.Supported in part by the National Science Foundation under grant DMS/PHY 88-16214     

Universal Drinfeld-Sokolov reduction and matrices of complex size

We construct affinization of the algebra of complex size matrices, that contains the algebras for integral values of the parameter. The Drinfeld-Sokolov Hamiltonian reduction of the algebra results in the quadratic Gelfand-Dickey structure on the Poisson-Lie group of all pseudodifferential operators of complex order.This construction is extended to the simultaneous deformation of orthogonal and symplectic algebras which produces self-adjoint operators, and it has a counterpart for the Toda lattices with fractional number of particles.Partially supported by NSF grant DMS 9307086.Partially supported by NSF grant DMS 9401215.     

Integrals of Vertex Operators and uantum Shuffles

Given a braided vector space , we show that iterated integrals of operator-valued functions satisfying a certain exchange relation give rise to representations of the quantum shuffle algebra built on . Using the quantum shuffle construction of the 'upper triangular part' of a quantum shuffle, this provides a simple proof of the result of Bouwknegt, MacCarthy and Pilch saying that integrals of vertex operators acting on certain Fock modules give rise to representations of .     

Staggered polarization of vertex models withU_q (\widehat{sl(}n))-Symmetry

In this paper we give an explicit formula for level 1 vertex operators related to as operators on the Fock spaces. We derive also their commutation relations. As an application we calculate with the vector representation of , thereby extending the recent work on the staggered polarization of the XXZ-model.     

Algebra of Screening Operators for the Deformed W n Algebra

We construct a family of intertwining operators (screening operators) between various Fock space modules over the deformed W _n algebra. They are given as integrals involving a product of screening currents and elliptic theta functions. We derive a set of quadratic relations among the screening operators, and use them to construct a Felder-type complex in the case of the deformed W ₃ algebra. Received: 3 March 1997 / Accepted: 20 May 1997     

The Development of a New Matrix Operator and its Application to the Study of Nonlinear Coupled Wave Equations

In this Letter, we consider Kontsevich's wheel operators for linear Poisson structures, i.e. on the dual of Lie algebras . We prove that these operators vanish on each invariant polynomial function on *. This gives a characterization of the Kontsevich star products which are deformations relative to the algebra of invariant functions.     

New quasi-exactly solvable Hamiltonians in two dimensions

Quasi-exactly solvable Schrödinger operators have the remarkable property that a part of their spectrum can be computed by algebraic methods. Such operators lie in the enveloping algebra of a finite-dimensional Lie algebra of first order differential operators—the hidden symmetry algebra. In this paper we develop some general techniques for constructing quasi-exactly solvable operators. Our methods are applied to provide a wide variety of new explicit two-dimensional examples (on both flat and curved spaces) of quasi-exactly solvable Hamiltonians, corresponding to both semisimple and more general classes of Lie algebras.Supported in Part by DGICYT Grant PS 89-0011.Supported in Part by an NSERC Grant.Supported in Part by NSF Grant DMS 92-04192.     

Free boson representation ofU_q (\widehat{sl}_3 )

A representation of the quantum affine algebra of an arbitrary levelk is constructed in the Fock module of eight boson fields. This realization reduces the Wakimoto representation in theq 1 limit. The analogues of the screening currents are also obtained. They commute with the action of modulo total differences of some fields.On leave from Department of Physics, University of Tokyo, Tokyo 113, Japan.     

Combinatorial quantization of the Hamiltonian Chern-Simons theory I

Motivated by a recent paper of Fock and Rosly [6] we describe a mathematically precise quantization of the Hamiltonian Chern-Simons theory. We introduce the Chern-Simons theory on the lattice which is expected to reproduce the results of the continuous theory exactly. The lattice model enjoys the symmetry with respect to a quantum gauge group. Using this fact we construct the algebra of observables of the Hamiltonian Chern-Simons theory equipped with a *- operation and a positive inner product.Supported by Swedish Natural Science Research Council (NFR) under the contract F-FU 06821-304 and by the Federal Ministry of Science and Research, AustriaPart of project P8916-PHY of the Fonds zur Förderung der wissenschaftlichen Forschung in ÖsterreichSupported in part by DOE Grant No DE-FG02-88ER25065;     

On the universalR-matrix ofU_q \widehat{sl}_2 at roots of unityat roots of unity

We show that the action of the universalR-matrix of the affine quantum algebra, whenq is a root of unity, can be renormalized by some scalar factor to give a well-defined nonsingular expression, satisfying the Yang-Baxter equation. It can be reduced to intertwining operators of representations, corresponding to Chiral Potts, if the parameters of these representations lie on the well-known algebraic curve.We also show that the affine forq is a root of unity from the autoquasitriangular Hopf algebra in the sense of Reshetikhin.This work is supported by NATO linkage grant LG 9303057.     

Complex Geometry and Dirac Equation

Complex geometry represents a fundamentalingredient in the formulation of the Dirac equation bythe Clifford algebra. The choice of appropriate complexgeometries is strictly related to the geometricinterpretation of the complex imaginary unit . We discuss two possibilities which appearin the multivector algebra approach: the₁₂₃ and ₂₁ complexgeometries. Our formalism provides a set of rules which allows an immediate translation between thecomplex standard Dirac theory and its version withingeometric algebra. The problem concerning a doublegeometric interpretation for the complex imaginary unit is also discussed.     

BV Structure of the Cohomology of Nilpotent Subalgebras and the Geometry of(W) Strings

Given a simple, simply laced, complex Lie algebra corresponding to the Lie group G, let be thesubalgebra generated by the positive roots. In this Letter we construct aBV algebra whose underlying graded commutative algebra is given by the cohomology, with respect to , of the algebra of regular functions on G with values in . We conjecture that describes the algebra of allphysical (i.e., BRST invariant) operators of the noncritical string. The conjecture is verified in the two explicitly known cases, ₂ (the Virasoro string) and ₃ (the string).     

On Modules over Matrix Quantum Pseudo-Differential Operators

We classify all the quasifinite highest-weight modules over the central extension of the Lie algebra of matrix quantum pseudo-differential operators, and obtain them in terms of representation theory of the Lie algebra (, R _m ) of infinite matrices with only finitely many nonzero diagonals over the algebra R _m = [t]/(t ^m+1). We also classify the unitary ones.     

Discrete Miura Opers and Solutions of the Bethe Ansatz Equations

Solutions of the Bethe ansatz equations associated to the XXX model of a simple Lie algebra come in families called the populations. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra The proof is based on the correspondence between the solutions of the Bethe ansatz equations and special difference operators which we call the discrete Miura opers. The notion of a discrete Miura oper is one of the main results of the paper.For a discrete Miura oper D, associated to a point of a population, we show that all solutions of the difference equation DY=0 are rational functions, and the solutions can be written explicitly in terms of points composing the population.Supported in part by NSF grant DMS-0140460Supported in part by NSF grant DMS-0244579     

ℤ n 3 -graded colored supersymmetry

We build generalizations of the Grassmann algebras from a few following simple assumptions: the algebras are graded, maximally symmetric and contain an ordinary Grassmann algebra as a subalgebra. These algebras are graded by and display surprising properties that indicate their possible application to the modeling of quark fields. We build the generalized supersymmetry generators based on these algebras and their derivation operators. These generators are cubic roots of the usual supersymmetry generators.     

Differential Groupoids and Their Application to the Theory of Spacetime Singularities

The transformation groupoid = × G, where is the total space of the generalized frame G-bundle over spacetime with a singular boundary, is not a Lie groupoid but a differential groupoid, i.e., a smooth groupoid in the category of structured spaces. We define this concept and use it to investigate spacetimes with various kinds of singularities. Any differential transformation groupoid can be represented by an algebra of operators on a bundle of Hilbert spaces defined on the groupoid fibers. This algebra reflects the structure of a given fiber even if it is a fiber over a singularity. It is also shown that any spacetime with singularities can be regarded as a noncommutative space. Its geometry is done in terms of a noncommutative algebra defined on the corresponding differential transformation groupoid. We focus on the structure of malicious singularities such as the ones appearing in the beginning and in the end of the closed Friedman universe.     

Unbounded derivations commuting with compact group actions

Let be a closed * derivation in aC* algebra which commutes with an ergodic action of a compact group on . Then generates aC* dynamics of . Similar results are obtained for non-ergodic actions on abelianC* algebras and on the algebra of compact operators.Research supported by N.S.F.     

A Construction of Representations and Quantum Homogeneous Spaces

A simplified construction of representations is presented for the quantized enveloping algebra q ( ), with being a simple complex Lie algebra belonging to one of the four principal series A\ell, B\ell, C\ell or D\ell. The carrier representation space is the quantized algebra of polynomials in antiholomorphic coordinate functions on the big cell of a coadjoint orbit of K where K is the compact simple Lie group with the Lie algebra – the compact form of .     

Nonabelian orbifolds and the boson-fermion correspondence

For a positive integerl divisible by 8 there is a (bosonic) holomorphic vertex operator algebra (VOA) associated to the spin lattice _l . For a broad class of finite groupsG of automorphisms of we prove the existence and uniqueness of irreducibleg-twisted -modules and establish the modular-invariance of the partition functionsZ(g, h, ) for commuting elements inG. In particular, for any finite group there are infinitely many holomorphic VOAs admittingG for which these properties hold. The proof is facilitated by a boson-fermion correspondence which gives a VOA isomorphism between and a certain fermionic construction, and which extends work of Frenkel and others.Supported by NSA grant MDA904-92-H-3099.Supported by NSF grant DMS-9122030.