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     

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).     

Statistical mechanics of quantum spin systems. III

In the algebraic formulation the thermodynamic pressure, or free energy, of a spin system is a convex continuous functionP defined on a Banach space of translationally invariant interactions. We prove that each tangent functional to the graph ofP defines a set of translationally invariant thermodynamic expectation values. More precisely each tangent functional defines a translationally invariant state over a suitably chosen algebra of observables, i. e., an equilibrium state. Properties of the set of equilibrium states are analysed and it is shown that they form a dense set in the set of all invariant states over . With suitable restrictions on the interactions, each equilibrium state is invariant under time-translations and satisfies the Kubo-Martin-Schwinger boundary condition. Finally we demonstrate that the mean entropy is invariant under time-translations.     

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.     

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.     

$$\mathfrak{s}\mathfrak{u} $$ q(2)-Invariant Schrödinger Equation of the Three-Dimensional Harmonic Oscillator

We propose a q-deformation of the -invariant Schrödinger equation of a spinless particle in a central potential, which allows us not only to determine a deformed spectrum and the corresponding eigenstates, as in other approaches, but also to calculate the expectation values of some physically-relevant operators. Here we consider the case of the isotropic harmonic oscillator and of the quadrupole operator governing its interaction with an external field. We obtain the spectrum and wave functions both for and generic , and study the effects of the q-value range and of the arbitrariness in the Casimir operator choice. We then show that the quadrupole operator in l=0 states provides a good measure of the deformation influence on the wave functions and on the Hilbert space spanned by them.     

On some groups of automorphisms of physical observables

Given a weakly continuous one-parameter group of automorphisms of aC*-algebra of operators on a Hilbert space we show that it is implementable by a strongly continuous one-parameter group of unitary operators belonging to the weak closure of , provided that a certain condition — akin to the boundedness from below of the spectrum of the generators — is satisfied.On leave from the Istituto di Fisica Teorica, Universitá di Napoli.     

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.     

A quantum dual pair \mathfraks\mathfrakl2 ,\mathfrakon \mathfrak{s}\mathfrak{l}_2 ,\mathfrak{o}_n and the associated Capelli identity

A quantum analogue of the dual pair is introduced in terms of the oscillator representation of U _q . Its commutant and the associated identity of Capelli type are discussed.     

A q-Analogue of the Berezin Quantization Method

We use Berezin's quantization procedure to obtain a formal -invariant deformation of the quantum disc. Explicit formulae for the associated bidifferential operators are produced.     

On the Relation Between U q (sl(2)) Vertex Operators and q-Zonal Functions

We show how the states constructed from the action of the modes of bosonized vertex operators that intertwine U modules are related toq -zonal functions.     

Aq-deformation of Wakimoto modules,primary fields and screening operators

Theq-vertex operators of Frenkel and Reshetikhin are studied by means of aq-deformation of the Wakimoto module for the quantum affine algebraU _q at an arbitrary levelk0, –2. A Fock-module version of theq-deformed primary field of spinj is introduced, as well as the screening operators which (anti-)commute with the action ofU _q up to a total difference of a field. A proof of the intertwining property is given for theq-vertex operators corresponding to the primary fields of spinj1/2Z ₀. A sample calculation of the correlation function is also given.This is a revised version of the preprint distributed in December, 1992, with the title Free Field Realization ofq-deformed Primary Fields forU _q (sl ₂)     

On a Theorem of Høegh-Krohn

A theorem proved by R. Høegh-Krohn in Comm. Math. Phys. 38(1974), 195–224, which yields a possibility to define states of systems of quantum particles by their values on the products , where \mathfraka _t , t are time automorphisms and F _j are multiplication operators, is generalized and extended. In particular, it is shown that the algebras generated by such products with F _j taken from the families of multiplication operators satisfying certain conditions are dense in the algebras of observables in the -weak topology, in which normal states are continuous. This result was obtained for the systems with two types of kinetic energy: the usual one expressed by means of the Laplacian; the relativistic kinetic energy defined by a pseudo-differential operator.     

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.     

Classical Dynamical r-Matrices,Poisson Homogeneous Spaces,and Lagrangian Subalgebras

Lu has shown that any dynamical r-matrix for the pair ( , ) naturally induces a Poisson homogeneous structure on G/U. She also proved that if is complex simple, is its Cartan subalgebra and r is quasitriangular, then this correspondence is in fact one-to-one. In this Letter we find some general conditions under which the Lu correspondence is one-to-one. Then we apply this result to describe all triangular Poisson homogeneous structures on G/U for a simple complex group G and its reductive subgroup U containing a Cartan subgroup.     

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.     

Non-compact quantum groups associated with Abelian subgroups

LetG be a Lie group. For any Abelian subalgebra of the Lie algebra g ofG, and any , the difference of the left and right translates ofr gives a compatible Poisson bracket onG. We show how to construct the corresponding quantum group, in theC ^*-algebra setting. The main tool used is the general deformation quantization construction developed earlier by the author for actions of vector groups onC ^*-algebras.The research reported on here was supported in part by National Science Foundation grant DMS-9303386.     

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.     

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.     

Coordinates of the quantum plane as q-tensor operators in (\mathfrak{s}\mathfrak{u}(2)*\mathfrak{s}\mathfrak{u}(2))

The relation between the set of transformations of the quantum plane and the quantum universal enveloping algebra U _q (u(2)) is investigated by constructing representations of the factor algebra U _q (u(2))* . The noncommuting coordinates of , on which U _q (2) * U _q (2) acts, are realized as q-spinors with respect to each U _q (u(2)) algebra. The representation matrices of U _q (2) are constructed as polynomials in these spinor components. This construction allows a derivation of the commutation relations of the noncommuting coordinates of directly from properties of U _q (u(2)). The generalization of these results to U _q (u(n)) and is also discussed.