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.     

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.     

Global existence and boundedness of large solutions to nonlinear equations of viscoelasticity with hardening

For the solutions of an initial-boundary value problem for the equations of viscoelasticity with isotropic hardening we derive a uniform bound under a growth condition for the nonlinearities in the case of one-space dimension. Global-in-time existence of solutions to large initial data is a consequence of the existence of this bound. In the most simple form, the equations we consider are with suitable functionsg ₁,g ₂,h satisfyingg ₂0 andXDedicated to Erhard Meister on the occasion of his sixtyfifth birthdayThis research was partially supported by the DFG-Forschergruppe Mathematische und Ingenieurwissenschaftliche Analyse bruchmechanischer und inelastischer Probleme     

On geometrical interpretation of thep-adic Maslov index

A set of selfdual lattices in a two-dimensionalp-adic symplectic space is provided by an integer valued metricd. A realization of the metric space (,d) as a graph is suggested and this graph has been linked to the Bruhat-Tits tree. An action of symplectic group on a set of cycles of length three of the graph is considered and a geometrical interpretation of thep-adic Maslov index is given in terms of this action.     

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.     

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.     

Conformal quantum field theory and half-sided modular inclusions of von-Neumann-Algebras

LetN, be von-Neumann-Algebras on a Hilbert space , a comon cyclic and separarting vector. Assume to be cyclic and separating also forN . Denote byJ , J _N the modular conjugations to (, ), and _N the associated modular operators. If and these data define in a canonical way a conformal quantum field theory in a cricle. Conversely, the chiral part of a conformal quantum field theory in two dimensions always yields such data in a natural way.Partly supported by the DFG, SFB 288 Differentiageometrie und Quantenphysik     

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.     

Spectral properties of a class of operators associated with conformal maps in two dimensions

Iff is a rational map of the Riemann sphere, define the transfer operator by Let also be the Banach space of functions for which the second derivatives are measures. Ifg andg satisfies a simple integrability condition (implying thatg vanishes at critical points and multiple poles off) then is a bounded linear operator on . The essential spectral radius of can be estimated and, under suitable conditions, proved to be strictly less than the spectral radius. Similar estimates for more general operators are also obtained.     

Quantum affine algebras and holonomic difference equations

We derive new holonomicq-difference equations for the matrix coefficients of the products of intertwining operators for quantum affine algebra representations of levelk. We study the connection opertors between the solutions with different asymptotics and show that they are given by products of elliptic theta functions. We prove that the connection operators automatically provide elliptic solutions of Yang-Baxter equations in the face formulation for any type of Lie algebra and arbitrary finite-dimensional representations of. We conjecture that these solutions of the Yang-Baxter equations cover all elliptic solutions known in the contexts of IRF models of statistical mechanics. We also conjecture that in a special limit whenq1 these solutions degenerate again into solutions with . We also study the simples examples of solutions of our holonomic difference equations associated to and find their expressions in terms of basic (orq–)-hypergeometric series. In the special case of spin –1/2 representations, we demonstrate that the connection matrix yields a famous Baxter solution of the Yang-Baxter equation corresponding to the solid-on-solid model of statistical mechanics.     

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     

Navier and stokes meet the wavelet

We work in the space = of divergence-free measurable vector fields onR ³ complete in the norm , where     

Long range scattering for nonlinear Schrödinger equations in one space dimension

We consider the scattering problem for the nonlinear Schrödinger equation in 1+1 dimensions: where = /x,R{0},R,p>3. We show that modified wave operators for (*) exist on a dense set of a neighborhood of zero in the Lebesgue spaceL ²(R) or in the Sobolev spaceH ¹(R)., The modified wave operators are introduced in order to control the long range nonlinearity |u|² u.Laboratoire associé au Centre National de la Recherche Scientifique     

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)     

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     

On the pointwise behavior of semi-classical measures

In this paper we concern ourselves with the small asymptotics of the inner products of the eigenfunctions of a Schrödinger-type operator with a coherent state. More precisely, let _j and E _j denote the eigenfunctions and eigenvalues of a Schrödinger-type operator H with discrete spectrum. Let _(x,) be a coherent state centered at the point (x, ) in phase space. We estimate as 0 the averages of the squares of the inner products ( _a ^(x,) , _j ) over an energy interval of size around a fixed energy, E. This follows from asymptotic expansions of the form for certain test function and Schwartz amplitudes a of the coherent state. We compute the leading coefficient in the expansion, which depends on whether the classical trajectory through (x, ) is periodic or not. In the periodic case the iterates of the trajectory contribute to the leading coefficient. We also discuss the case of the Laplacian on a compact Riemannian manifold.Research supported in part by NSF grant DMS-9303778     

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.     

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.     

Convergence of Nelson fiffusions

Let _t, _t ⁿ ,n1, be solutions of Schrödinger equations with potentials form-bounded by –1/2 and initial data inH ¹( ^d ). LetP, P ⁿ ,n1, be the probability measures on the path space =C(₊, ^d ) given by the corresponding Nelson diffusions. We show that if { _t ⁿ } _n1 converges to _t inH ¹( ^d ), uniformly int over compact intervals, then converges to in total variation t0. Moreover, if the potentials are in the Kato classK _d , we show that the above result follows fromH ¹-convergence of initial data, andK _d -convergence of potentials.