Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials

Theq=0 combinatorics for is studied in connection with solvable lattice models. Crystal bases of highest weight representations of are labelled by paths which were introduced as labels of corner transfer matrix eigenvectors atq=0. It is shown that the crystal graphs for finite tensor products ofl-th symmetric tensor representations of approximate the crystal graphs of levell representations of . The identification is made between restricted paths for the RSOS models and highest weight vectors in the crystal graphs of tensor modules for .Partially supported by NSF grant MDA904-90-H-4039     

Spectral decomposition of path space in solvable lattice model

We give thespectral decomposition of the path space of the vertex model with respect to the local energy functions. The result suggests the hidden Yangian module structure on the levell integrable modules, which is consistent with the earlier work [1] in the level one case. Also we prove the fermionic character formula of the levell integrable representations in consequence.     

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.     

Quantum principal commutative subalgebra in the nilpotent part ofU_q \widehat{sl}_2 and lattice KdV variables

We propose a quantum lattice version of B. Feigin and E. Frenkel's constructions, identifying the KdV differential polynomials with functions on a homogeneous space under the nilpotent part of . We construct an action of the nilpotent part of on their lattice counterparts, and embed the lattice variables in a , coinduced from a quantum version of the principal commutative subalgebra, which is defined using the identification of with its dual algebra.     

Isomorphism of two realizations of quantum affine algebraU_q (\widehat{\mathfrak{g}\mathfrak{l}{\text{(}}n{\text{)}}})

We establish an explicit isomorphism between two realizations of the quantum affine algebra given previously by Drinfeld and Reshetikhin-Semenov-Tian-Shansky. Our result can be considered as an affine version of the isomorphism between the Drinfield/Jimbo and the Faddeev-Reshetikhin-Takhtajan constructions of the quantum algebra .     

Level-rank duality of WZW models in conformal field theory

We consider the decomposition of the conformal blocks under the conformal embeddings. The case (â is an affine extension of the abelian subalgebra of the central elements ofgl(lr)) is studied in detal. The reciprocal decompositions of -modules induce a pairing between the spaces of conformal blocks of and Wess-Zumino-Witten models on the Riemann sphere. The completeness of the pairing is shown. Hence it defines aduality between two spaces.Dedicated to Professor Masahisa Adachi on his 60th birthday     

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.     

Jackson integrals of Jordan-Pochhammer type and quantum Knizhnik-Zamolodchikov equations

We show that theq-difference systems satisfied by Jackson integrals of Jordan-Pochhammer type give a class of the quantum Knizhnik-Zamolodchikov equation for in the sense of Frenkel and Reshetikhin.     

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     

Quantum affine algebras

We classify the finite-dimensional irreducible representations of the quantum affine algebra in terms of highest weights (this result has a straightforward generalization for arbitrary quantum affine algebras). We also give an explicit construction of all such representations by means of an evaluation homomorphism , first introduced by M. Jimbo. This is used to compute the trigonometricR-matrices associated to finite-dimensional representations of .     

Modules over\mathfrak{U}_q (\mathfrak{s}\mathfrak{l}_2 )

The restricted quantum universal enveloping algebra decomposes in a canonical way into a direct sum of indecomposable left (or right) ideals. They are useful for determining the direct summands which occur in the tensor product of two simple . The indecomposable finite-dimensional are classified and located in the Auslander-Reiten quiver.     

On the equilibrium states in quantum statistical mechanics

Representations of theC*-algebra of observables corresponding to thermal equilibrium of a system at given temperatureT and chemical potential are studied. Both for finite and for infinite systems it is shown that the representation is reducible and that there exists a conjugation in the representation space, which maps the von Neumann algebra spanned by the representative of onto its commutant. This means that there is an equivalent anti-linear representation of in the commutant. The relation of these properties with the Kubo-Martin-Schwinger boundary condition is discussed.     

Covariance algebras in field theory and statistical mechanics

Starting from aC*-algebra and a locally compact groupT of automorphisms of we construct a covariance algebra with the property that the corresponding *-representations are in one-to-one correspondence with covariant representations of i.e. *-representations of in which the automorphisms are continuously unitarily implemented. We further construct for relativistic field theory an algebra yielding the *-representations of in which the space time translations have their spectrum contained inV. The problem of denumerable occurence of superselection sectors is formulated as a condition on the spectrum of . Finally we consider the covariance algebra built with space translations alone and show its relevance for the discussion of equilibrium states in statistical mechanics, namely we restore in this framework the equivalence of uniqueness of the vacuum, irreducibility and a weak clustering property.On leave of absence from Istituto di Fisica G. Marconi — Roma.     

Bethe subalgebras in twisted Yangians

We study analogues of the Yangian of the Lie algebra for the other classical Lie algebras and . We call them twisted Yangians. They are coideal subalgebras in the Yangian of and admit homomorphisms onto the universal enveloping algebras U( ) and U( ) respectively. In every twisted Yangian we construct a family of maximal commutative subalgebras parametrized by the regular semisimple elements of the corresponding classical Lie algebra. The images in U( ) and U( ) of these subalgebras are also maximal commutative.     

On affine yangians

A Yangian , a deformation of the universal enveloping algebra of the two-dimensional loop algebra sl(2) C [t ^–1,t;u], is constructed. This deformation is an analogue of a Yangian which was constructed by V. Drinfeld for any simple Lie algebra. The PBW theorem for is proved and some representations are constructed. Like usual Yangians, possesses a one-dimensional group of auto- morphisms and at zero level - a two-dimensional group of automorphisms. This observation allows one to conjecture that the representation theory of should give rise to new solutions of QYBE.Yangians of other affine algebras can be constructed similarly and they enjoy similar properties.     

Vertex operators for the central extension of the quantum affine algebra 1203-11203-11203-1

In this paper we study the vertex (intertwining) operators for certain infinite dimensional representations of , which is a central extension of . We present bosonized expressions for intertwining operators at level 1.Presented at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–22 June 1996.The first author acknowledges partial support by the Czech Republic Grant Agency (No. 202/96/0218).     

N=2 structures on solvable Lie algebras: Thec=9 classification

Let be a finite-dimensional Lie algebra (not necessarily semisimple). It is known that if is self-dual (that is, if it possesses an invariant metric) then it admits anN=1 (affine) Sugawara construction. Under certain additional hypotheses, thisN=1 structure admits anN=2 extension. If this is the case, is said to possess anN=2 structure. It is also known that anN=2 structure on a self-dual Lie algebra is equivalent to a vector space decomposition , where are isotropic Lie subalgebras. In other words,N=2 structures on in one-to-one correspondence with Manin triples . In this paper we exploit this correspondence to obtain a classification of thec=9N=2 structures on solvable Lie algebras. In the process we also give some simple proofs for a variety of Lie algebras. In the process we also give some simple proofs for a variety of Lie algebraic results concerning self-dual Lie algebras admitting symplectic or Kähler structures.     

Bicross-Product Structure of Affine Quantum Groups

We show that the affine quantum group is isomorphic to a bicross-product central extension of the quantum loop group by a quantum cocycle in R-matrix form.     

Quantum algebra structure of certain Jackson integrals

Theq-difference system satisfied by Jackson integrals with a configuration ofA-type root system is studied. We explicitly construct some linear combination of Jackson integrals, which satisfies the quantum Knizhnik-Zamolodchikov equation for the 2-point correlation function ofq-vertex operators, introduced by Frenkel and Reshetikhin, for the quantum affine algebra . The expression of integrands for then-point case is conjectured, and a set of linear relations for the corresponding Jackson integrals is proved.