Quantum-algebras and elliptic algebras

We define a quantum-algebra associated to as an associative algebra depending on two parameters. For special values of the parameters, this algebra becomes the ordinary-algebra of , or theq-deformed classical-algebra algebra of . We construct free field realizations of the quantum-algebra and the screening currents. We also point out some interesting elliptic structures arising in these algebras. In particular, we show that the screening currents satisfy elliptic analogues of the Drinfeld relations in.The research of the second author was partially supported by NSF grant DMS-9501414.     

Free boson realization of

We construct a realization of the quantum affine algebra of an arbitrary level k in terms of free boson fields. In the q1 limit this realization becomes the Wakimoto realization of. The screening currents and the vertex operators (primary fields) are also constructed; the former commutes with modulo total difference, and the latter creates the highest weight state from the vacuum state of the boson Fock space.fellow of Soryushi ShogakukaiAddress after June 1: Department of Physics, Faculty of Liberal Arts, Shinshu University, Matsumoto 390, Japan     

On the thermodynamic formalism for the Gauss map

We study the generalized transfer operator of the Gauss mapTx=(1/x) mod 1 on the unit interval. This operator, which for =1 is the familiar Perron-Frobenius operator ofT, can be defined for Re >1/2 as a nuclear operator either on the Banach spaceA (D) of holomorphic functions over a certain discD or on the Hilbert space of functions belonging to some Hardy class of functions over the half planeH _–1/2. The spectra of on the two spaces are identical. On the space is isomorphic to an integral operator with kernel the Bessel function and hence to some generalized Hankel transform. This shows that has real spectrum for real >1/2. On the spaceA (D) the operator can be analytically continued to the entire -plane with simple poles at and residue the rank 1 operator . From this similar analyticity properties for the Fredholm determinant of and hence also for Ruelle's zeta function follow. Another application is to the function , where [n] denotes the irrational[n]=(n+(n ²+4)^1/2)/2. M() extends to a meromorphic function in the -plane with the only poles at =±1 both with residue 1.     

Super loop groups,Hamiltonian actions and super Virasoro algebras

The quotient of a super loop group by the subgroup of constant loops is given a supersymplectic structure and identified through a moment map embedding with a coadjoint orbit of the centrally extended super loop algebra. The algebra of super-conformal vector fields on the circle is shown to have a natural representation as Hamiltonian vector fields on generated by an equivariant moment map. This map is obtained by composition of315-8 with a super Poisson map defining a supersymmetric extension of the classical Sugawara formula. Upon quantization, this yields the corresponding formula of Kac and Todorov on unitary highest weight representations. For any homomorphism :u(1)G, an associated twisted moment map is also derived, generating a super Poisson bracket realization of a super Virasoro subalgebra of the semi-direct sum. The corresponding super Poisson map is interpreted as a nonabelian generalization of the super Miura map and applied to two super KdV hierarchies to derive corresponding integrable generalized super MKdV hierarchies in.Research supported in part by the Natural Sciences and Engineering Research Council of Canada and the National Science Foundation (USA)     

On two integrable cellular automata

A representation-free approach to theq-analog of the quantum central limit theorem for is presented. It is shown that for certain functionals one can derive a version of a quantum central limit theorem (qclt) with as a scaling parameter, which may be viewed as aq-analog of qclt.     

Quantum symmetry and braid group statistics inG-spin models

In two-dimensional lattice spin systems in which the spins take values in a finite groupG we find a non-Abelian parafermion field of the formorder x disorder that carries an action of the Hopf algebra, the double ofG. This field leads to a quantization of the Cuntz algebra and allows one to define amplifying homomorphisms on the subalgebra that create the and generalize the endomorphisms in the Doplicher-Haag-Roberts program. The so-obtained category of representations of the observable algebra is shown to be equivalent to the representation category of. The representation of the braid group generated by the statistics operator and the corresponding statistics parameter are calculated in each sector.     

Extensions of automorphisms and gauge symmetries

We characterize the automorphisms of aC ^*-algebra which extend to automorphisms of the crossed product by a compact group dual. The case where the inclusion is equipped with a group of automorphisms commuting with the dual action is also treated. These results are applied to the analysis of broken gauge symmetries in Quantum Field Theory to draw conclusions on the structure of the degenerate vacua on the field algebra.Dedicated to Huzihiro ArakiResearch supported by MURST and CNR, GNAFA     

with central chargeN

We study representations of the central extension of the Lie algebra of differential operators on the circle, the algebra. We obtain complete and specialized character formulas for a large class of representations, which we call primitive; these include all quasi-finite irreducible unitary representations. We show that any primitive representation with central chargeN has a canonical structure of an irreducible representation of the with the same central charge and that all irreducible representations of with central chargeN arise in this way. We also establish a duality between integral modules of and finite-dimensional irreducible modules ofgl _N , and conjecture their fusion rules.Supported by a Junior Fellowship from Harvard Society of Fellows and in part by NSF grant DMS-9205303.Supported in part by NSF grant DMS-9103792.     

The relation between quantumW algebras and Lie algebras

By quantizing the generalized Drinfeld-Sokolov reduction scheme for arbitrarysl ₂ embeddings we show that a large set of quantumW algebras can be viewed as (BRST) cohomologies of affine Lie algebras. The set contains many knownW algebras such asW _N andW ₃ ⁽²⁾ . Our formalism yields a completely algorithmic method for calculating theW algebra generators and their operator product expansions, replacing the cumbersome construction ofW algebras as commutants of screening operators. By generalizing and quantizing the Miura transformation we show that anyW algebra in can be embedded into the universal enveloping algebra of a semisimple affine Lie algebra which is, up to shifts in level, isomorphic to a subalgebra of the original affine algebra. Thereforeany realization of this semisimple affine Lie algebra leads to a realization of theW algebra. In particular, one obtains in this way a general and explicit method for constructing the free field realizations and Fock resolusions for all algebras in. Some examples are explicitly worked out.     

On the relationship between Monstrous Moonshine and the uniqueness of the Moonshine Module

We consider the relationship between the conjectured uniqueness of the Moonshine Module,, and Monstrous Moonshine, the genus zero property of the modular invariance group for each Monster group Thompson series. We first discuss a family of possibleZ _n meromorphic orbifold constructions of based on automorphisms of the Leech lattice compactified bosonic string. We reproduce the Thompson series for all 51 non-Fricke classes of the Monster groupM together with a new relationship between the centralisers of these classes and 51 corresponding Conway group centralisers (generalising a well-known relationship for 5 such classes). Assuming that is unique, we consider meromorphic orbifoldings of and show that Monstrous Moonshine holds if and onlyZ _r if the only meromorphic orbifoldings of are itself or the Leech theory. This constraint on the meromorphic orbifoldings of therefore relates Monstrous Moonshine to the uniqueness of in a new way.     

Geometric prequantization on the spin bundle based onN-symplectic geometry: The dirac equation

We present preliminary results for a prequantization procedure that leads in a natural way to the Dirac equation. The starting point is the recently introducedn-symplectic geometry on the bundle of linear framesLM of ann-dimensional manifoldM in which the ⁿ-valued soldering 1-form onLM plays the role of then-symplectic potential. On a 4-dimensional spacetime manifold we consider the tensorial ⁴⁴valued function onLM determined by the spacetime metric tensor g as the Hamiltonian for free observers and determine the associated ⁴-valued Hamiltonian vector field , Integration of theX ⁱ yields the dynamics of free observers on spacetime, namely parallel transport of linear frames along spacetime geodesies. In order to obtain a vector field on the spin bundleSM which is a lift of and which is induced by a vector field for an appropriate mapping , it is useful to define a prolongation of some bundleL ^o M of oriented frames ofM. IfGL ⁺(4, ) denotes the identity component ofGL(4, ), thenGL ⁺(4, ) is the structure group ofL ^o M and its double cover is the structure group of. We show that the lift of onL ^o M to induces a natural 4-symplectic potential on. If is the lift of g to, then we find the ⁴-valued Hamiltonian vector field on determined by and show that the vector fieldsX _g ⁱ on are tangent to the subbundleSM. Integration of the restriction of theX ⁱ toSM now yields parallel transport of spin frames and thus tetrads along spacetime geodesies of g. We consider a naive prequantization operator assignment acting on ⁴-spinors in the standard representation ofSL(2, ). The eigenvalue equation for the system of new Hilbert space operators yields the Dirac equation.     

Action of truncated quantum groups on quasi-quantum planes and a quasi-associative differential geometry and calculus

Ifq is ap ^th root of unity there exists a quasi-coassociative truncated quantum group algebra whose indecomposable representations are the physical representations ofU _q (sl ₂), whose coproduct yields the truncated tensor product of physical representations ofU _q (sl ₂), and whoseR-matrix satisfies quasi-Yang Baxter equations. These truncated quantum group algebras are examples of weak quasitriangular quasi-Hopf algebras (quasi-quantum group algebras). We describe a space of functions on the quasi quantum plane, i.e. of polynomials in noncommuting complex coordinate functionsz _a , on which multiplication operatorsZ _a and the elements of can act, so thatz _a will transform according to some representation ^f of can be made into a quasi-associative graded algebra on which elements of act as generalized derivations. In the special case of the truncatedU _q (sl ₂) algebra we show that the subspaces of monomials inz _a ofn ^th degree vanish fornp–1, and that carries the 2J+ 1 dimensional irreducible representation of ifn=2J, J=0,1/2, ..., 1/2(p–2). Assuming that the representation ^f of the quasi-quantum group algebra gives rise to anR-matrix with two eigenvalues, we develop a quasi-associative differential calculus on. This implies construction of an exterior differentiation, a graded algebra of forms and partial derivatives. A quasi-associative generalization of noncommutative differential geometry is introduced by defining a covariant exterior differentiation of forms. It is covariant under gauge transformations.     

Lie algebra cohomology and the fusion rules

We prove a vanishing theorem for Lie algebra cohomology which constitutes a loop group analogue of Kostant's Lie algebra version of the Borel-Weil-Bott theorem. Consider a complex semi-simple Lie algebra and an integrable, irreducible, negative energy representation of. Givenn distinct pointsz _k in , with a finite-dimensional irreducible representationV _k of assigned to each, the Lie algebra of-valued polynomials acts on eachV _k , via evaluation atz _k . Then, the relative Lie algebra cohomologyH ^* is concentrated in one degree. As an application, based on an idea of G. Segal's, we prove that a certain homolorphic induction map from representations ofG to representations ofLG at a given level takes the ordinary tensor product into the fusion product. This result had been conjectured by R. Bott.     

Universality in orbit spaces of compact linear groups

If {p ¹(x),...,p ^q (x)} is a minimal integrity basis of the ideal of polynomial invariants of a compact coregular linear groupG, the orbit map, yields a diffeomorphic image of the orbit spaceR ⁿ /G. Starting from this fact, we point out some properties which are common to the orbit space of all the compact coregular linear groups of transformations ofR ⁿ . In particular we show that a contravariant metric matrix can be defined in the interior of, as a polynomial function of (p ¹,...,p ^q ). We prove that the matrix , which characterizes the set, as it is positive semi-definite only forp, can be determined as a solution of a canonical differential equation, which, for every compact coregular linear group, depends only on the numberq and on the degrees of the elements of the minimal integrity bases. This allows to determine all the isomorphism classes of the orbit spaces of the compact coregular linear groups through a determination of the equivalence classes of the corresponding matrices . Forq3 (orbit spaces with dimensions 3), the solutions of the canonical equation are explicitly determined and the number of their equivalence classes is shown to be finite. It is also shown that, with a convenient choice of the minimal integrity basis, the polynomial matrix elements of have only integer coefficients. Arguments are given in favour of the conjecture that our conclusions hold true for all values ofq. Our results are relevant and lead to universality properties in the physics of spontaneous symmetry breaking.Partially supported by INFN and Ministero della Pubblica Istruzione     

Classification of the indecomposable bounded admissible modules over the Virasoro Lie algebra with weightspaces of dimension not exceeding two

In view of [1,2] any bounded admissible moduleA over the Virasoro Lie algebra is a finite length extension of irreducible modules with one-dimensional weightspaces. To each extension of finite lengthn are associatedn+1 invariants (a₁, ₁, ..., _n ). We prove that we have _i – _j {0, 1, ... 6(n – 1b)} for all (i, j) with 1ijn. In the casen=2 this result allows us to construct all the indecomposable bounded admissible modules, where the dimensions of the weightspaces are less than or equal to two. In particular we obtain all the extensions of two irreducible bounded-modules.     

Deformation of a Kac algebra by an abelian subgroup

Some examples of quantum groups in literature arise as deformations of a locally compact group by a dual 2-cocycle. We make this construction in the framework of Kac algebras; we show that these deformations are still Kac algebras; using this construction, we give new quantizations of the Heisenberg group. From this point of view, we analyse the dimension 8 non-trivial example of Kac and Paljutkin, and give a new example of non-trivial dimension 12 semi-simple *-Hopf algebras (a dimension 12 Kac algebra).Dedicated to the memory of George Kac ()Research of the second author was supported in part by the Ukrainian Foundation for Fundamental Studies and by the International Science Foundation.     

On the spectra of Schrödinger operators

We give two formulas for the lowest point in the spectrum of the Schrödinger operatorL=–(d/dt)p(d/dt)+q, where the coefficientsp andq are real-valued, bounded, uniformly continuous functions on the real line. We determine whether or not is an eigenvalue forL in terms of a set of probability measures on the maximal ideal space of theC ^*-algebra generated by the translations ofp andq.Research supported in part by the National Science Foundation under Grant DMS-910496     

A global formalism for nonlinear waves in conservation laws

We introduce a unifying framework for treating all of the fundamental waves occurring in general systems ofn conservation laws. Fundamental waves are represented as pairs of states statisfying the Rankine-Hugoniot conditions; after trivial solutions have been eliminated by means of a blow-up procedure, these pairs form an (n+1)-dimensional manifold, the fundamental wave manifold. There is a distinguishedn-dimensional submanifold of containing a single one-dimensonal foliation that represents the rarefaction curves for all families. Similarly, there is a foliation of itself that represent shock curves. We identify othern-dimensional submanifolds of that are naturally interpreted as boundaries of regions of admissible shock waves. These submanifolds also have one-dimensional foliations, which represent curves of composite waves. This geometric framework promises to simplify greatly the study of the stability and bifurcation propertiesThis work was supported in part by: the NSF/CNP_q U.S.-Latin America Cooperative Science Program under Grant INT-8612605; the Institute for Mathematics and its Applications with funds provided by the National Science Foundation; the Air Force Office of Scientific Research under Grant AFOSR 90-0075; the National Science Foundation under Grant 8901884; the U.S. Department of Energy under Grant DE-FG02-90ER25084; the U.S. Army Research Office under Grant DAAL03-89-K-0017; the Financiadora de Estudos e Projetos; the Conselho Nacional de Desenvolvimento Científico e Tecnológica (CNP_q); the Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ); the Coordenação de Aperfeiçamento de Pessoal de Ensino Superior (CAPES); and the Sociedade Brasileira de Matemática (SBM)     

Space-time correlation of the amplitude and phase of the field in a waveguide having a statistically rough boundary

The perturbation theory method in the Born approximation is used to solve the problem of the space-time correlation of the amplitude and phase of the field in a plane waveguide having a statistically rough wall. With respect to the boundary roughnesses it is assumed that their height is small in comparison with a wavelength and that, andl _y are the correlation radii of the roughnesses along and across the propagation routes, respectively; D is the length of the route; is the longitudinal wave number of the n-th normal wave;, and also that the variation of the boundary shape with time takes place considerably more slowly than a period of the electromagnetic oscillations. It is shown that the coefficient of space-time correlation has an identical form for the amplitude and phase of the field, does not depend on the specific waveguide model, and is determined solely by the statistical characteristics of the boundary roughnesses. The expressions are analyzed for the dispersions of the amplitude and phase fluctuations, their space correlation functions, and their time spectra in a series of limiting cases.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 15, No. 1, pp. 73–82, January, 1972.In conclusion we consider it our pleasant duty to thank P. V. Bliokh and G. I. Makarov for discussing the results of the work and for their valuable remarks.