title: Static Electric Dipole Polarizabilities of Na Clusters

abstract:
  The static electric dipole polarizability of $\mathrm{Na_N}$ clusters with
even N has been calculated in a collective, axially averaged and a
three-dimensional, finite-field approach for $2\le N \le 20$, including the
ionic structure of the clusters. The validity of a collective model for the
static response of small systems is demonstrated. Our density functional
calculations verify the trends and fine structure seen in a recent experiment.
A pseudopotential that reproduces the experimental bulk bond length and atomic
energy levels leads to a substantial increase in the calculated
polarizabilities, in better agreement with experiment. We relate remaining
differences in the magnitude of the theoretical and experimental
polarizabilities to the finite temperature present in the experiments.


title: Compendium of vector analysis with applications to continuum mechanics

abstract:
  1. Connection between integration and differentiation/ Gauss theorem.
Coordinate-free definitions: gradient, divergence, curl. Stokes theorem. 2.
Elements of continuum mechanics/ Continuity equation. Stress tensor. Euler
equation. Hydrodynamics. Elasticity.


title: Determination of meteor showers on other planets using comet ephemerides

abstract:
  Meteor showers on the Earth occur at well known times, and are associated
with the decay of comets or other minor bodies whose orbital paths pass close
to the Earth's trajectory. On the surface, determining the closest proximity of
two orbital paths appears to be a computationally intensive procedure. This
paper describes a simple geometric method for determining the proximity between
the orbital paths of two bodies (i.e., a comet and a planet) in the solar
system from the known ephemerides of the objects. The method is used to
determine whether or not meteor showers on other planets in the solar system
could be associated with any of 250 known comets.


title: Coulomb crystals in the harmonic lattice approximation

abstract:
  The dynamic structure factor ${\tilde S}({\bf k},\omega)$ and the
two-particle distribution function $g({\bf r},t)$ of ions in a Coulomb crystal
are obtained in a closed analytic form using the harmonic lattice (HL)
approximation which takes into account all processes of multi-phonon excitation
and absorption. The static radial two-particle distribution function $g(r)$ is
calculated for classical ($T \gtrsim \hbar \omega_p$, where $\omega_p$ is the
ion plasma frequency) and quantum ($T \ll \hbar \omega_p$) body-centered cubic
(bcc) crystals. The results for the classical crystal are in a very good
agreement with extensive Monte Carlo (MC) calculations at $1.5 \lesssim r/a
\lesssim 7$, where $a$ is the ion-sphere radius. The HL Coulomb energy is
calculated for classical and quantum bcc and face-centered cubic crystals, and
anharmonic corrections are discussed. The inelastic part of the HL static
structure factor $S''(k)$, averaged over orientations of wave-vector {\bf k},
is shown to contain pronounced singularities at Bragg diffraction positions.
The type of the singularities is different in classical and quantum cases. The
HL method can serve as a useful tool complementary to MC and other numerical
methods.


title: Exploring a rheonomic system

abstract:
  A simple and illustrative rheonomic system is explored in the Lagrangian
formalism. The difference between Jacobi's integral and energy is highlighted.
A sharp contrast with remarks found in the literature is pointed out. The
non-conservative system possess a Lagrangian not explicitly dependent on time
and consequently there is a Jacobi's integral. The Lagrange undetermined
multiplier method is used as a complement to obtain a few interesting
conclusions


title: Differential criterion of a bubble collapse in viscous liquids

abstract:
  The present work is devoted to a model of bubble collapse in a Newtonian
viscous liquid caused by an initial bubble wall motion. The obtained bubble
dynamics described by an analytic solution significantly depends on the liquid
and bubble parameters. The theory gives two types of bubble behavior: collapse
and viscous damping. This results in a general collapse condition proposed as
the sufficient differential criterion. The suggested criterion is discussed and
successfully applied to the analysis of the void and gas bubble collapses.


title: Rutherford Scattering with Retardation

abstract:
  Numerical solutions for Sommerfeld model in nonrelativistic case are
presented for the scattering of a spinless extended charged body in a static
Coulomb field of a fixed point charge. It is shown that differential cross
section for extended body preserves the form of the Rutherford result with
multiplier, not equal to one (as in classical case), but depending on the size
of Sommerfeld particle. Also the effect of capture by attractive center is
found out for Sommerfeld particle. The origin of this effect lies in radiation
damping.


title: Variational methods, multiprecision and nonrelativistic energies

abstract:
  It is known that the variational methods are the most powerful tool for
studying the Coulomb three-body bound state problem. However, they often suffer
from loss of stability when the number of basis functions increases. This
problem can be cured by applying the multiprecision package designed by D.H.
Bailey. We consider the variational basis functions of the type exp(-a_n r_1 -
b_n r_2 - g_n r_12) with complex exponents. The method yields the best
available energies for the ground states of the helium atom and the positive
hydrogen ion as well as many other known atomic and molecular systems.


title: Effective interaction between helical bio-molecules

abstract:
  The effective interaction between two parallel strands of helical
bio-molecules, such as deoxyribose nucleic acids (DNA), is calculated using
computer simulations of the "primitive" model of electrolytes. In particular we
study a simple model for B-DNA incorporating explicitly its charge pattern as a
double-helix structure. The effective force and the effective torque exerted
onto the molecules depend on the central distance and on the relative
orientation. The contributions of nonlinear screening by monovalent counterions
to these forces and torques are analyzed and calculated for different salt
concentrations. As a result, we find that the sign of the force depends
sensitively on the relative orientation. For intermolecular distances smaller
than $6\AA$ it can be both attractive and repulsive. Furthermore we report a
nonmonotonic behaviour of the effective force for increasing salt
concentration. Both features cannot be described within linear screening
theories. For large distances, on the other hand, the results agree with linear
screening theories provided the charge of the bio-molecules is suitably
renormalized.

categories: physics.bio-ph physics.chem-ph physics.comp-ph q-bio


title: Atom-optics hologram in the time domain

abstract:
  The temporal evolution of an atomic wave packet interacting with object and
reference electromagnetic waves is investigated beyond the weak perturbation of
the initial state. It is shown that the diffraction of an ultracold atomic beam
by the inhomogeneous laser field can be interpreted as if the beam passes
through a three-dimensional hologram, whose thickness is proportional to the
interaction time. It is found that the diffraction efficiency of such a
hologram may reach 100% and is determined by the duration of laser pulses. On
this basis a method for reconstruction of the object image with matter waves is
offered.


title:
A Second-Order Stochastic Leap-Frog Algorithm for Multiplicative Noise
  Brownian Motion

abstract:
  A stochastic leap-frog algorithm for the numerical integration of Brownian
motion stochastic differential equations with multiplicative noise is proposed
and tested. The algorithm has a second-order convergence of moments in a finite
time interval and requires the sampling of only one uniformly distributed
random variable per time step. The noise may be white or colored. We apply the
algorithm to a study of the approach towards equilibrium of an oscillator
coupled nonlinearly to a heat bath and investigate the effect of the
multiplicative noise (arising from the nonlinear coupling) on the relaxation
time. This allows us to test the regime of validity of the energy-envelope
approximation method.


title:
Analytical Structure Matching and Very Precise Approach to the Coulombic
  Quantum Three-Body Problem

abstract:
  A powerful approach to solve the Coulombic quantum three-body problem is
proposed. The approach is exponentially convergent and more efficient than the
Hyperspherical Coordinate(HC) method and the Correlation Function
Hyperspherical Harmonic(CFHH) method. This approach is numerically competitive
with the variational methods, such as that using the Hylleraas-type basis
functions. Numerical comparisons are made to demonstrate them, by calculating
the non-relativistic and infinite-nuclear-mass limit of the ground state energy
of the helium atom. The exponentially convergency of this approach is due to
the full matching between the analytical structure of the basis functions that
I use and the true wave function. This full matching was not reached by almost
any other methods. For example, the variational method using the Hylleraas-type
basis does not reflects the logarithmic singularity of the true wave function
at the origin as predicted by Bartlett and Fock. Two important approaches are
proposed in this work to reach this full matching: the coordinate
transformation method and the asymptotic series method. Besides these, this
work makes use of the least square method to substitute complicated numerical
integrations in solving the Schr\"{o}dinger equation, without much loss of
accuracy; this method is routinely used by people to fit a theoretical curve
with discrete experimental data, but I use it here to simplify the computation.


title: Conductivity of a Relativistic Plasma

abstract:
  The collision operator for a relativistic plasma is reformulated in terms of
an expansion in spherical harmonics. In this formulation the collision operator
is expressed in terms of five scalar potentials which are given by
one-dimensional integrals over the distribution function. This formulation is
used to calculate the electrical conductivity of a uniform electron-ion plasma
with infinitely massive ions.


title: Stellarator News for January, 1995

abstract:
  The Stellarator News is a bi-monthly newsletter covering the international
stellarator community.


title: Excitation of Large-ktheta Ion-Berstein Waves in Tokamaks

abstract:
  The mode-converted ion-Bernstein wave excited in tokamaks is shown to exhibit
certain very interesting behavior, including the attainment of very small
poloidal phase velocities, the reversal of poloidal direction, and up-down
asymmetries in propagation and damping. Because of these effects, this wave
holds promise for channeling {$\alpha$-particle}\ power to ions, something that
would make a tokamak fusion reactor far more attractive than presently
envisioned.


title: Stellarator News, Issue 38, March 1995

abstract:
  Stellarator News, an international journal of the stellarator community, is
Published by Fusion Energy Division, Oak Ridge National Laboratory, James A.
Rome, Editor
  In the March 1995 issue . . .
  **** Exerpts from the U.S. Congress Office of Technology Assment report on
TPX and Alternate Concepts.
  **** Edge transport and turbulence studies on U-3M
  The turbulent-driven particle flow is shown to be comparable with the
equilibrium flow at the boundary of the configuration under Alfven-heating
conditions in the U-3M torsatron.
  **** Topological aspects of island divertor studies on W7-AS
  The structure of the edge plasma in W7-AS, observed with probes, television
camera, and H-alpha light agrees at low beta with vacuum field calculations:
the low-valued resonances at iotabar=5/m are resolved for m = 8 to 11; external
perturbations are not significant at the edge, even for iotabar = 5/10.
  **** 140-GHz second harmonic O-mode electron cyclotron heating at W7-AS
  First experimental results are presented of 140-GHz second harmonic O-mode
(0.7-MW power) ECRH at a high electron density of 1.8 x 10^20 m^-3 on W7-AS.
  **** Equilibrium plasma currents in quasi-symmetric stellarators with a
helical magnetic axis
  The plasma currents in quasi-symmetric stellarators are calculated to determe
the quasi-symmetric conditions with the help of analytical equations.


title: Narrowing of Bennett Hole in Collisional Plasma

abstract:
  The profile of a Bennett hole induced by laser field in ionic distribution in
collisional plasma is calculated. Influence of Chandrasekhar's dependence of
coefficients of velocity space transport on the profile is included into the
calculation for the first time. It is found that the hole narrows down as the
field detuning frequency increases. Physical cause of the effect is the falling
dependence of Coulomb collision frequency on the ionic velocity. Estimations
show that the effect is quite observable under conditions of high-current
gas-discharge plasma.


title: Stellarator News, Issue 39, May 1995

abstract:
  Special Issue: New Stellarators
  New stellarators are being constructed throughout the world as part of a
well-coordinated international program. This issue concentrates on the
engineering and construction details of
  Large Helical Device (LHD), Toki, Japan
  Wendelstein 7-X (W7-X), Greifswald, Germany
  Helias Stellarator Experiment (HSX), Madison Wisconsin, USA
  TJ-II, Madrid, Spain


title: Radiation Rates for Low Z Impurities in Edge Plasmas

abstract:
  The role of impurity radiation in the reduction of heat loads on divertor
plates in present experiments such as DIII-D, JET, JT-60, ASDEX, and Alcator
C-Mod, and in planned experiments such as ITER and TPX places a new degree of
importance on the accuracy of impurity radiation emission rates for electron
temperatures below 250 eV for ITER and below 150 eV for present experiments. We
have calculated the radiated power loss using a collisional radiative model for
Be, B, C, Ne and Ar using a multiple configuration interaction model which
includes density dependent effects, as well as a very detailed treatment of the
energy levels and meta-stable levels. The "collisional radiative" effects are
very important for Be at temperatures below 10 eV. The same effects are present
for higher Z impurities, but not as strongly. For some of the lower Z elements,
the new rates are about a factor of two lower than those from a widely used,
simpler average-ion package (ADPAK) developed for high Z ions and for higher
temperatures. Following the approach of Lengyel for the case where electron
heat conduction is the dominant mechanism for heat transport along field lines,
our analysis indicates that significant enhancements of the radiation losses
above collisional radiative model rates due to such effects as rapid recycling
and charge exchange recombination will be necessary for impurity radiation to
reduce the peak heat loads on divertor plates for high heat flux experiments
such as ITER.


title:
Analytic Criteria for Power Exhaust in Divertors due to Impurity
  Radiation

abstract:
  Present divertor concepts for next step experiments such ITER and TPX rely
upon impurity and hydrogen radiation to transfer the energy from the edge
plasma to the main chamber and divertor chamber walls. The efficiency of these
processes depends strongly on the heat flux, the impurity species, and the
connection length. Using a database for impurity radiation rates constructed
from the ADPAK code package, we have developed criteria for the required
impurity fraction, impurity species, connection length and electron temperature
and density at the mid-plane. Consistent with previous work, we find that the
impurity radiation from coronal equilibrium rates is, in general, not adequate
to exhaust the highest expected heating powers in present and future
experiments. As suggested by others, we examine the effects of enhancing the
radiation rates with charge exchange recombination and impurity recycling, and
develop criteria for the minimum neutral fraction and impurity recycling rate
that is required to exhaust a specified power. We also use this criteria to
find the optimum impurity for divertor power exhaust.


title:
A Review of Recent Developments in Atomic Processes for Divertors and
  Edge Plasmas

abstract:
  The most promising concepts for power and particle control in tokamaks and
other fusion experiments rely upon atomic processes to transfer the power and
momentum from the edge plasma to the plasma chamber walls. This places a new
emphasis on processes at low temperatures (1-200 eV) and high densities
(10^20-10^22 m^-3). The most important atomic processes are impurity and
hydrogen radiation, ionization, excitation, recombination, charge exchange,
radiation transport, molecular collisions, and elastic scattering of atoms,
molecules and ions. Important new developments have occurred in each of these
areas. The best available data for these processes and an assessment of their
role in plasma wall interactions are summarized, and the major areas where
improved data are needed are reviewed.


title:
Calculations of Energy Losses due to Atomic Processes in Tokamaks with
  Applications to the ITER Divertor

abstract:
  Reduction of the peak heat loads on the plasma facing components is essential
for the success of the next generation of high fusion power tokamaks such as
the International Thermonuclear Experimental Reactor (ITER) 1 . Many present
concepts for accomplishing this involve the use of atomic processes to transfer
the heat from the plasma to the main chamber and divertor chamber walls and
much of the experimental and theoretical physics research in the fusion program
is directed toward this issue. The results of these experiments and
calculations are the result of a complex interplay of many processes. In order
to identify the key features of these experiments and calculations and the
relative role of the primary atomic processes, simple quasi-analytic models and
the latest atomic physics rate coefficients and cross sections have been used
to assess the relative roles of central radiation losses through
bremsstrahlung, impurity radiation losses from the plasma edge, charge exchange
and hydrogen radiation losses from the scrape-off layer and divertor plasma and
impurity radiation losses from the divertor plasma. This anaysis indicates that
bremsstrahlung from the plasma center and impurity radiation from the plasma
edge and divertor plasma can each play a significant role in reducing the power
to the divertor plates, and identifies many of the factors which determine the
relative role of each process. For instance, for radiation losses in the
divertor to be large enough to radiate the power in the divertor for high power
experiments, a neutral fraction of 10-3 to 10-2 and an impurity recycling rate
of netrecycle of ~ 10^16 s m^-3 will be required in the divertor.


title:
Studies of Plasma Detachment Using a One Dimensional Model for Divertor
  Operation

abstract:
  To characterize the conditions required to reach advanced divertor regimes, a
one-dimensional computational model has been developed based on a coordinate
transformation to incorporate two-dimensional effects. This model includes
transport of ions, two species each of atoms and molecules, momentum, and ion
and electron energy both within and across the flux surfaces. Impurity
radiation is calculated using a coronal equilibrium model which includes the
effects of charge-exchange recombination. Numerical results indicate that
impurity radiation acts to facilitate plasma detachment and enhances the power
lost from the divertor channel in escaping neutral atoms by cooling the
electrons and suppressing ionization. As divertor particle densities increase,
cold and thermal molecules become increasingly important in cooling the plasma,
with molecular densities dominating electron and atomic densities under some
conditions.


title: Detailed Radiative Transport Modeling of a Radiative Divertor

abstract:
  An effective radiative divertor maximizes the utilization of atomic processes
to spread out the energy deposition to the divertor chamber walls and to reduce
the peak heat flux. Because the mixture of neutral atoms and ions in the
divertor can be optically thick to a portion of radiated power, it is necessary
to accurately model the magnitude and distribution of line radiation in this
complex region. To assess their importance we calculate the effects of
radiation transport using CRETIN, a multi-dimensional, non-local thermodynamic
equilibrium simulation code that includes the atomic kinetics and radiative
transport processes necessary to model the complex environment of a radiative
divertor. We also include neutral transport to model radiation from recycling
neutral atoms. This paper presents a case study of a high-recycling radiative
divertor with a typical large neutral pressure at the divertor plate to
estimate the impact of H line radiation on the overall power balance in the
divertor region with consideration for line opacities and atomic kinetics.


title:
A New Look at the Landau's Theory of Spreading and Damping of Waves in
  Collisionless Plasmas

abstract:
  The theory of plasma waves and Landau damping in Maxwellian plasmas, Landau's
``rule of pass around poles'' include doubtful statements, particularly related
to an artificial ``constructing'' of the dispersion equation, what should allow
the possibility of its solution otherwise not existing at all, and the
possibility of analytical continuations of corresponding very specific ruptured
functions in the one-dimensional Laplace transformation, used by Landau, what
is the base of his theory.
  We represent, as an accessible variant, a more general alternative theory
based on a two-dimensional Laplace transformation, leading to an asymptotical
in time and space solution as a complicated superposition of coupled damping
and {\em non-damping \/} plane waves and oscillations with different dispersion
laws for every constituent mode. This theory naturally and very simply explains
paradoxes of the phenomenon of plasma echo. We propose for discussion a new
ideology of plasma waves (both electron and ion-acoustic waves) qualitatively
different from the traditional theory of Landau damping for non-collisional as
well as for low-collisional plasmas.


title:
On the conical refraction of hydromagnetic waves in plasma with
  anisotropic thermal pressure

abstract:
  A phenomenon analogous to the conical refraction widely known in the
crystalooptics and crystaloacoustics is discovered for the
magnetohydrodynamical waves in the collisionless plasma with anisotropic
thermal pressure. Angle of the conical refraction is calculated for the medium
under study which is predicted to be $18^{\circ}26^{\prime}$. Possible
experimental corroborating of the discovered phenomenon is discussed.


title: Power Balance in the ITER Plasma and Divertor

abstract:
  It is planned to use atomic processes to spread out most of the heating power
over the first wall and side walls to reduce the heat loads on the plasma
facing components in ITER to ~ 50 MW. Calculations indicate that there will be
100 MW in bremstrahlung radiation from the plasma center, 50 MW of radiation
from the plasma edge inside the separatrix and 100 MW of radiation from the
scrape-off layer and divertor plasma, leaving 50 MW of power to be deposited on
the divertor plates. The radiation losses are enhanced by the injection of
impurities such as Neon or Argon at acceptably low levels (~0.1 % Argon, etc.)


title: Experimental Divertor Similarity Database Parameters

abstract:
  A set of experimentally-determined dimensionless parameters is proposed for
characterizing the regime of divertor operation. The objective is to be able to
compare as unambiguously as possible the operation of different divertors and
to understand what physical similarities and differences they represent.
Examples from Alcator C-Mod are given.


title:
Influence of wave frequency variation on anomalous cyclotron resonance
  interaction of energetic electrons with finite amplitutude ducted
  whistler-mode wave

abstract:
  The influence of wave frequency variation on the anomalous cyclotron
resonance $\omega=\omega_{Be}+kv_{\|}$ interaction (ACRI) of energetic
electrons with a ducted finite amplitude whistler-mode wave propagating through
the so-called transient plasma layer (TPL) in the magnetosphere or in the
ionosphere is studied both analytically and numerically. The anomalous
cyclotron resonance interaction takes place in the case when the whistler-mode
wave amplitude $B_{W}$ is consistent with the gradient of magnetic field
$\overrightarrow{B_0}$. The region of phase space occupied by anomalously
interacting energetic electrons (synchronous particles) is determined. The
efficiencies of both the pitch-angle scattering of resonant electrons and their
transverse acceleration are studied and the efficiencies dependence on the
magnitude and sign of the wave frequency drift is considered. It has been shown
that in the case of ACRI occuring under conditions relevant to VLF-emission in
the magnetosphere, the energy and pitch-angle changes of synchronous electrons
may be enchanced by a factor $10^2 \div 10^3$ in comparison with ones for
nonsynchronous resonant electrons. So the small in density group of synchronous
particles may give significant contribution to a whistler-mode wave damping in
TPL.


title:
Negative-Energy Perturbations in Circularly Cylindrical Equilibria
  within the Framework of Maxwell-Drift Kinetic Theory

abstract:
  The conditions for the existence of negative-energy perturbations (which
could be nonlinearly unstable and cause anomalous transport) are investigated
in the framework of linearized collisionless Maxwell-drift kinetic theory for
the case of equilibria of magnetically confined, circularly cylindrical plasmas
and vanishing initial field perturbations. For wave vectors with a
non-vanishing component parallel to the magnetic field, the plane equilibrium
conditions (derived by Throumoulopoulos and Pfirsch [Phys Rev. E {\bf 49}, 3290
(1994)]) are shown to remain valid, while the condition for perpendicular
perturbations (which are found to be the most important modes) is modified.
Consequently, besides the tokamak equilibrium regime in which the existence of
negative-energy perturbations is related to the threshold value of 2/3 of the
quantity $\eta_\nu = \frac {\partial \ln T_\nu} {\partial \ln N_\nu}$, a new
regime appears, not present in plane equilibria, in which negative-energy
perturbations exist for {\em any} value of $\eta_\nu$. For various analytic
cold-ion tokamak equilibria a substantial fraction of thermal electrons are
associated with negative-energy perturbations (active particles). In
particular, for linearly stable equilibria of a paramagnetic plasma with flat
electron temperature profile ($\eta_e=0$), the entire velocity space is
occupied by active electrons. The part of the velocity space occupied by active
particles increases from the center to the plasma edge and is larger in a
paramagnetic plasma than in a diamagnetic plasma with the same pressure
profile. It is also shown that, unlike in plane equilibria, negative-energy
perturbations exist in force-free reversed-field pinch equilibria with a
substantial fraction of active particles.


title: Mode Coupling in Space and Time Varying Anisotropic Absorbing Plasmas

abstract:
  A four dimensional systematic mathematical approach for investigating
propagation and coupling of wave modes in a slowly varying (in all space
directions and time) anisotropic, absorbing plasma is represented. The
formalism is especially useful for energy considerations of the waves. It is
applicable to general cases of mode conversion in plasmas with general
geometries of space-time and magnetic field configurations. A simple example of
how this formalism can be applied to practical cases is given.


title: Electrostatic Ion Perturbations in Unmagnetized Plasma Shear Flow

abstract:
  The electrostatic perturbations in an unmagnetized, non-isothermal
($T_i{\ll}T_e$) electron-ion plasma shear flow are considered. New physical
effects, arising due to the non-normality of linear dynamics, are described. It
is shown that the velocity shear induces the extraction of the mean flow energy
by the acoustic perturbations (ion-sound waves). The influence of the medium
dispersion, rising due to the violation of quasineutrality for perturbations,
is examined. It is shown that in the course of the evolution ion-sound waves
turn into ion plasma oscillations. New class of nonperiodic, electrostatic
perturbations (with vortical motion of ion component), characterized by the
intense energy exchange with the mean flow, is also described.


title:
Relativistic ponderomotive force, uphill acceleration, and transition to
  chaos

abstract:
  Starting from a covariant cycle-averaged Lagrangian the relativistic
oscillation center equation of motion of a point charge is deduced and
analytical formulae for the ponderomotive force in a travelling wave of
arbitrary strength are presented. It is further shown that the ponderomotive
forces for transverse and longitudinal waves are different; in the latter,
uphill acceleration can occur. In a standing wave there exists a threshold
intensity above which, owing to transition to chaos, the secular motion can no
longer be described by a regular ponderomotive force.
  PACS number(s): 52.20.Dq,05.45.+b,52.35.Mw,52.60.+h


title: Special-relativistic model flows of viscous fluid

abstract:
  Two, the most simple cases of special-relativistic flows of a viscous,
incompressible fluid are considered: plane Couette flow and plane Poiseuille
flow. Considering only the regular motion of the fluid we found the
distribution of velocity in the fluid (velocity profiles) and the friction
force, acting on immovable wall. The results are expressed through simple
analytical functions for the Couette flow, while for the Poiseiulle flow they
are expressed by higher transcendental functions (Jacobi's elliptic functions).


title: Ionization equilibrium of hot hydrogen plasma

abstract:
  The hydrogen plasma is studied at temperatures T ~ 10^4 - 10^6 K using the
free energy minimization method. A simple analytic free energy model is
proposed which is accurate at densities up to 1 g/cc and yields convergent
internal partition function of atoms. The occupation probability formalism is
modified for solving the ionization equilibrium problem. The ionization degree
and equation of state are calculated and compared with the results of other
models.


title:
Localized structures of electromagnetic waves in hot
  electron-positronplasmas

abstract:
  The dynamics of relativistically strong electromagnetic (EM) wave propagation
in hot electron-positron plasma is investigated. The possibility of finding
localized stationary structures of EM waves is explored. It is shown that under
certain conditions the EM wave forms a stable localized soliton-like structures
where plasma is completely expelled from the region of EM field location.


title:
On the conical refraction of hydromagnetic waves in plasma with
  anisotropic thermal pressure: general consideration

abstract:
  A phenomenon analogous to the conical refraction well-known in the
crystalooptics and crystaloacoustics is considered for the
magnetohydrodynamical waves in a collisionless plasma with anisotropic thermal
pressure. Imposing the most general (generalization of Tsiklauri, 1996, Phys.
Plasmas, 3, 800) condition for the existence of the phenomenon, angle of the
conical refraction is calculated which appeared to be dependent on the ratio of
the Alfven velocity and sound speed measured in the perpendicular direction in
respect to the external magnetic field. Feasible ways of experimental
demonstration of the phenomenon are discussed and a novelty brought by the
general consideration is outlined.


title:
Magnetohydrodynamic equilibria of a cylindrical plasma with poloidal
  mass flow and arbitrary cross section shape

abstract:
  The equilibrium of a cylindrical plasma with purely poloidal mass flow and
cross section of arbitrary shape is investigated within the framework of the
ideal MHD theory. For the system under consideration it is shown that only
incompressible flows are possible and, conscequently, the general two
dimensional flow equilibrium equations reduce to a single second-order
quasilinear partial differential equation for the poloidal magnetic flux
function $\psi$, in which four profile functionals of $\psi$ appear. Apart from
a singularity occuring when the modulus of Mach number associated with the
Alfv\'en velocity for the poloidal magnetic field is unity, this equation is
always elliptic and permits the construction of several classes of analytic
solutions. Specific exact equlibria for a plasma confined within a perfectly
conducting circular cylindrical boundary and having i) a flat current density
and ii) a peaked current density are obtained and studied.


title: Theory of the Spatio-Temporal Dynamics of Transport Bifurcations

abstract:
  The development and time evolution of a transport barrier in a magnetically
confined plasma with non-monotonic, nonlinear dependence of the anomalous flux
on mean gradients is analyzed. Upon consideration of both the spatial
inhomogeneity and the gradient nonlinearity of the transport coefficient, we
find that the transition develops as a bifurcation front with radially
propagating discontinuity in local gradient. The spatial location of the
transport barrier as a function of input flux is calculated. The analysis
indicates that for powers slightly above threshold, the barrier location
$x_b(t) \sim ( D_n t (P-P_c)/P_c)^{1/2},$ where $P_c$ is the local transition
power threshold and $D_n$ is the neoclassical diffusivity . This result
suggests a simple explanation of the high disruptivity observed in reversed
shear plasmas. The basic conclusions of this theory are insensitive to the
details of the local transport model.


title:
Iso-topological relaxation, coherent structures, and Gaussian turbulence
  in two dimensional magnetohydrodynamics

abstract:
  The long-time relaxation of ideal two dimensional magnetohydrodynamic
turbulence subject to the conservation of two infinite families of constants of
motion---the magnetic and the "cross" topology invariants--is examined. The
analysis of the Gibbs ensemble, where all integrals of motion are respected,
predicts the initial state to evolve into an equilibrium, stable coherent
structure (the most probable state) and decaying Gaussian turbulence
(fluctuations) with a vanishing, but always positive temperature. The
non-dissipative turbulence decay is accompanied by decrease in both the
amplitude and the length scale of the fluctuations, so that the fluctuation
energy remains finite. The coherent structure represents a set of singular
magnetic islands with plasma flow whose magnetic topology is identical to that
of the initial state, while the energy and the cross topology invariants are
shared between the coherent structure and the Gaussian turbulence. These
conservation laws suggest the variational principle of iso-topological
relaxation which allows us to predict the appearance of the final state from a
given initial state. For a generic initial condition having X points in the
magnetic field, the coherent structure has universal types of singularities:
current sheets terminating at Y points. These structures, which are similar to
those resulting from the 2D relaxation of magnetic field frozen into an ideally
conducting viscous fluid, are observed in the numerical experiment of Biskamp
and Welter. The Gibbs ensemble method developed in this work admits extension
to other Hamiltonian systems with invariants not higher than quadratic in the
highest-order-derivative variables. The turbulence in two dimensional Euler
fluid is of a different nature: there the coherent structures are also formed,
but the fluctuations about these structures are non-Gaussian.

categories: plasm-ph chao-dyn nlin.CD physics.plasm-ph


title:
Self-Diagonal Tensor Powers of Quantum Groups and R-Matrices for Tensor
  Products of Representations

abstract:
  Twisted tensor powers of quasitriangular Hopf algebras with diagonal
sub-Hopf-algebras (self-diagonal tensor powers) are introduced together with
their duals and their mutual *-structures as generalizations of the Drinfel'd
double as given by Reshetikhin and Semenov-Tian-Shansky. R-Matrices for tensor
products of representations are derived.


title: Clifford Algebras and Spinors for Arbitrary Braids

abstract:
  General braided counterparts of classical Clifford algebras are introduced
and investigated. Braided Clifford algebras are defined as Chevalley-Kahler
deformations of the corresponding braided exterior algebras. Analogs of the
spinor representations are studied, generalizing classical Cartan's approach.
It is shown that, under certain assumptions concerning the braiding, the spinor
representation is faithful and irreducible, as in the classical theory.


title: On Braided Quantum Groups

abstract:
  A braided generalization of the concept of Hopf algebra (quantum group) is
presented. The generalization overcomes an inherent geometrical inhomogeneity
of quantum groups, in the sense of allowing completely pointless objects. All
braid-type equations appear as a consequence of initial axioms. Braided
counterparts of basic algebraic relations between fundamental entities of the
standard theory are found.


title: On Differential Structures on Quantum Principal Bundles

abstract:
  A constructive approach to differential calculus on quantum principal bundles
is presented. The calculus on the bundle is built in an intrinsic manner,
starting from given graded (differential) *-algebras representing horizontal
forms on the bundle and differential forms on the base manifold, together with
a family of antiderivations acting on horizontal forms, playing the role of
covariant derivatives of regular connections. In this conceptual framework, a
natural differential calculus on the structure quantum group is described.


title: Geometry of Quantum Principal Bundles II-Extended Version

abstract:
  A general noncommutative-geometric theory of principal bundles is presented.
Quantum groups play the role of structure groups. General quantum spaces play
the role of base manifolds. A differential calculus on quantum principal
bundles is studied. In particular, algebras of horizontal and verticalized
differential forms on the bundle are introduced and investigated. The formalism
of connections is developed. Operators of horizontal projection, covariant
derivative and curvature are constructed and analyzed. A quantum generalization
of classical Weil's theory of characteristic classes is sketched. Quantum
analogs of infinitesimal gauge transformations are studied. Illustrative
examples and constructions are presented.


title: Classical Spinor Structures on Quantum Spaces

abstract:
  A noncommutative-geometric generalization of the classical concept of spinor
structure is presented. This is done in the framework of the formalism of
quantum principal bundles. In particular, analogs of the Dirac operator and the
Laplacian are introduced and analyzed. A general construction of examples of
quantum spaces with a spinor structure is presented.


title:
Central elements for quantum affine algebras and affine Macdonald's
  operators

abstract:
  We describe a generalization of Drinfeld's description of the center of a
quantum group to the case of quantum affine algebras. We use the obtained
central elements to construct the affine analogue of Macdonald's difference
operators.


title: Projective Systems of Imprimitivity

abstract:
  We apply Mackey procedure of classifying projective systems of imprimitivity
to a thorough study of the projective unitary irreducible representations of
the Galilei group in 1+3 and 1+2 dimensions.


title: Classification of the GL(3) Quantum Matrix Groups

abstract:
  We define quantum matrix groups GL(3) by their coaction on appropriate
quantum planes and the requirement that the Poincare series coincides with the
classical one. It is shown that this implies the existence of a Yang-Baxter
operator. Exploiting stronger equations arising at degree four of the algebra,
we classify all quantum matrix groups GL(3). We find 26 classes of solutions,
two of which do not admit a normal ordering. The corresponding R-matrices are
given.


title:
Differential Calculus on the Quantum Sphere and Deformed Self-Duality
  Equation

abstract:
  We discuss the left-covariant 3-dimensional differential calculus on the
quantum sphere $SU_q (2)/U(1) $. The $SU_q (2)$-spinor harmonics are treated as
coordinates of the quantum sphere. We consider the gauge theory for the quantum
group $SU_q (2)\times U(1) $ on the deformed Euclidean space $E_q (4)$. A
$q$-generalization of the harmonic-gauge-field formalism is suggested . This
formalism is applied for the harmonic (twistor) interpretation of the
quantum-group self-duality equation (QGSDE). We consider the zero-curvature
representation and the general construction of QGSDE-solutions in terms of the
analytic prepotential.


title: Highest Weight Representations of Quantum Current Algebras

abstract:
  We study the highest weight and continuous tensor product representations of
q-deformed Lie algebras through the mappings of a manifold into a locally
compact group. As an example the highest weight representation of the
q-deformed algebra $sl_q(2,\Cb)$ is calculated in detail.


title:
The McKay-Thompson series associated with the irreducible characters of
  the Monster

abstract:
  Let $ \Bbb V = \coprod_{h = 0}^{\infty} \Bbb V_h$ be the graded monster
module of the monster simple group $\Bbb M$ and let $\chi_k$ be an irreducible
representation of $\Bbb M$. The generating function of $c_{hk}$ (the
multiplicity of $\chi_k$ in $\Bbb V_h$) is determined. Furthermore, the
invariance group of the modular function associated with the generating
function is also determined in this paper.


title: Modular forms associated with the Monster module

abstract:
  We investigate the relation between the generating functions of the highest
weight vectors of the Monster module and the McKay-Thompson series.


title:
Vertex--IRF correspondence and factorized L-operators for an elliptic
  R-operator

abstract:
  As for an elliptic $R$-operator which satisfies the Yang--Baxter equation,
the incoming and outgoing intertwining vectors are constructed, and the
vertex--IRF correspondence for the elliptic $R$-operator is obtained. The
vertex--IRF correspondence implies that the Boltzmann weights of the IRF model
satisfy the star--triangle relation. By means of these intertwining vectors,
the factorized L-operators for the elliptic $R$-operator are also constructed.
The vertex--IRF correspondence and the factorized L-operators for Belavin's
$R$-matrix are reproduced from those of the elliptic $R$-operator.


title: Macdonald's Evaluation Conjectures and Difference Fourier Transform

abstract:
  In the previous author's paper the Macdonald norm conjecture (including the
famous constant term conjecture) was proved. This paper contains the proof of
the remaining two (the duality and evaluation conjectures). The evaluation
theorem is in fact a $q,t$-generalization of the classic Weyl dimension
formula. As to the duality theorem, it states that the generalized
trigonometric-difference zonal Fourier transform is self-dual (at least
formally). We define this transform in terms of double affine Hecke algebras
related to elliptic braid groups. The duality appeared to be directly connected
with the transposition of the periods of an elliptic curve.


title:
Localization of $\frak{u}$-modules. II. Configuration spaces and quantum
  groups

abstract:
  This paper is a sequel to "Localization of $\frak{u}$-modules. I",
hep-th/9411050. We are starting here the geometric study of the tensor category
$\cal{C}$ associated with a quantum group (corresponding to a Cartan matrix of
finite type) at a root of unity. The main results establish isomorphisms
between homogeneous components of irreducible objects in $\cal{C}$ and spaces
of vanishing cycles at the origin of certain Goresky-MacPherson sheaves on
configuration spaces; establish isomorphisms of the stalks at the origin of the
above GM sheaves with certain Hochschild complexes (which compute the
Hochschild homology of a certain "triangular" subalgebra of our quantum group
with coefficients in the coresponding irreducible representation); establish
the analogous results for tensor products of irreducibles. In geometry, the
tensor product of representations corresponds to a "fusion" of sheaves on
configuration spaces --- operation defined using the functor of nearby cycles.


title: On the deformation of abelian integrals

abstract:
  We consider the deformation of abelian integrals which arose from the study
of SG form factors. Besides the known properties they are shown to satisfy
Riemann bilinear identity. The deformation of intersection number of cycles on
hyperelliptic curve is introduced.


title:
A Littlewood-Richardson filtration at roots of 1 for multiparameter
  deformations of skew Schur modules.

abstract:
  Let R be a commutative ring, q a unit of R and P a multiplicatively
antisymmetric matrix with coefficients which are integers powers of q. Denote
by SE(q,P) the multiparameter quantum matrix bialgebra associated to q and
P.Slightly generalizing [Hashimoto-Hayashi,Tohoku Math.Tohoku Math.J.
44(1992)],we define a multiparameter deformation $L_{\l/\mu}V_P$ of the
classical skew Schur module.In case R is a field and q is not a root of 1,
arguments like those given in [H-H] show that $L_{\l/\mu}V_P$ is irreducible
and its decomposition into irreducibles is $\sum_\nu c(\l/\mu;\nu)L_\nu V_P$
where the coefficients are the usual Littlewood-Richardson ones. When R is any
ring and q is allowed to be a root of 1, we construct a filtration of
$L_{\l/\mu}V_P$ as an SE(q,P)-comodule, such that its associated graded object
is precisely $\sum_\nu c(\l/\mu;\nu)L_\nu V_P$.


title: Quantum affine algebras and affine Hecke algebras

abstract:
  We describe a connection between finite--dimensional representations of
quantum affine algebras and affine Hecke algebras.


title: Invariants for 1-dimensional cohomology classes arising from TQFT

abstract:
  Let $(V,Z)$ be a Topological Quantum Field Theory over a field $f$ defined on
a cobordism category whose morphisms are oriented $n+1$-manifolds perhaps with
extra structure. Let $(M,\chi)$ be a closed oriented $n+1$-manifold $M$ with
this extra structure together with $\chi \in H^1(M).$ Let $M_{\infty}$ denote
the infinite cyclic cover of $M$ given by $\chi.$ Consider a fundamental domain
$E$ for the action of the integers on $M_{\infty}$ bounded by lifts of a
surface $\Sigma$ dual to $\chi,$ and in general position. $E$ can be viewed as
a cobordism from $\Sigma$ to itself. We give Turaev and Viro's proof of their
theorem that the similarity class of the non-nilpotent part of $Z(E)$ is an
invariant. We give a method to calculate this invariant for the $(V_p,Z_p)$
theories of Blanchet,Habegger, Masbaum and Vogel when $M$ is zero framed
surgery to $S^3$ along a knot K. We give a formula for this invariant when $K$
is a twisted double of another knot. We obtain formulas for the quantum
invariants of branched covers of knots, and unbranched covers of 0-surgery to
$S^3$ along knots.


title:
q-Bosonization of the quantum group $ GL_q(2) $ based on the Gauss
  decomposition

abstract:
  The new method of q-bosonization for quantum groups based on the Gauss
decomposition of a transfer matrix of generators is suggested. The simplest
example of the quantum group $GL_q(2)$ is considered in some details.


title: Quantum Groups,Deformed Oscillators and their Interrelations

abstract:
  The main notions of the quantum groups: coproduct, action and coaction,
representation and corepresentation are discussed using simplest examples:
$GL_q(2)$, $sl_q(2)$, $q$-oscillator algebra ${\cal A}(q)$, and reflection
equation algebra. The Gauss decompositions of quantum groups and their
realizations in terms of\, ${\cal A}(q)$ are given.


title: Quantization of Poisson pencils and generalized Lie algebras

abstract:
  We describe two types of Poisson pencils generated by a linear bracket and a
quadratic one arising from a classical R-matrix. A quantization scheme is
discussed for each. The quantum algebras are represented as the enveloping
algebras of ``generalized Lie algebras''.


title:
Covariant Differential and Integral Calculi for Lattice (l,q)-deformed
  Fields

abstract:
  Using the Hecke $\hat R$-matrix, we give a definition of the lattice
$(l,q)$-deformed $n$-component boson and Grassmann fields. Here $l$ is a
deformation parameter for the commutation relations of "values" of these fields
in two arbitrary lattice sites and $q$ is a deformation parameter for
$n$-component $q$-boson or $q$-Grassmann variable. In framework of the
Wess-Zumino approach to the noncommutative differential calculus the
commutation relations between differentials and derivatives of these fields are
determined. The $SL_q(n,C)$-invariant generalization of the Berezin integration
for the lattice $n$-component $(l,q)$-Grassmann field is suggested. We show
that the Gaussian functional integral for this field is expressed through the
$(l,q)$-deformed counterpart of the Pfaffian.


title: The Constrained KP Hierarchy and the Generalised Miura Transformation

abstract:
  Recently much attention has been paid to the restriction of KP to the
submanifold of operators which can be represented as a ratio of two purely
differential operators L=AB^{-1}. Whereas most of the aspects concerning this
reduced hierarchy, like the Lax flows and the Hamiltonians, are by now well
understood, there still lacks a clear and conclusive statement about the
associated Poisson structure. We fill this gap by placing the problem in a more
general framework and then showing how the required result follows from an
interesting property of the second Gelfand-Dickey brackets under multiplication
and inversion of Lax operators. As a byproduct we give an elegant and simple
proof of the generalised Kupershmidt-Wilson theorem.


title:
Bicrossproduct Hopf Superalgebras and $D=4$ $\kappa$-deformed Poincar\'e
  Supergroup.

abstract:
  The general framework of bicrossproduct Hopf algebras given by Majid is
extended to $Z_2$-graded bicrossproduct Hopf superalgebras. As examples of
bicrossproduct Hopf superalgebras we provide the graded algebras of functions
on undeformed as well as $\kappa$-deformed $D=4$ Poincar\'e supergroups.


title: Differential calculus on deformed E(2) group

abstract:
  Fourdimensional bicovariant differential calculus on quantum E(2) group is
constructed.


title: Representations of generalized oscillator algebra

abstract:
  The representations of the oscillator algebra introduced by Brzezinski et al.
(Phys. Lett. B 311 (1993), 202) are classified.


title: Non-Standard KP Evolution and Quantum $\tau$-function

abstract:
  One possible way to fix partly a ``canonical definition'' of $\tau$-functions
beyond the conventional KP/Toda framework could be to postulate that evolution
operators are {\it always} group elements. We discuss implications of this
postulate for the first non-trivial case: fundamental representations of
quantum groups $SL_q(N)$. It appears that the most suited (simple) for quantum
deformation framework is some non-standard formulation of KP/Toda systems. It
turns out that the postulate needs to be slightly modified to take into account
that no ``nilpotent subgroups'' exist in $SL_q(N)$ for $q\neq 1$. This has some
definite and simple implications for $q$-determinant-like representations of
quantum $\tau$-functions.


title: Commutative Quantum Operator Algebras

abstract:
  A key notion bridging the gap between {\it quantum operator algebras}
\cite{LZ10} and {\it vertex operator algebras} \cite{Bor}\cite{FLM} is the
definition of the commutativity of a pair of quantum operators (see section 2
below). This is not commutativity in any ordinary sense, but it is clearly the
correct generalization to the quantum context. The main purpose of the current
paper is to begin laying the foundations for a complete mathematical theory of
{\it commutative quantum operator algebras.} We give proofs of most of the
relevant results announced in \cite{LZ10}, and we carry out some calculations
with sufficient detail to enable the interested reader to become proficient
with the algebra of commuting quantum operators.


title: Moonshine Cohomology

abstract:
  We construct a new cohomology functor from the a certain category of {\it
quantum operator algebras} to the category of {\it Batalin-Vilkovisky
algebras}. This {\it Moonshine cohomology} has, as a group of natural
automorphisms, the Fischer-Griess Monster finite group. We prove a general
vanishing theorem for this cohomology. For a certain commutative QOA attached
to a rank two hyperbolic lattice, we show that the degree one cohomology is
isomorphic to the so-called Lie algebra of physical states. In the case of a
rank two unimodular lattice, the degree one cohomology gives a new construction
of Borcherd's Monster Lie algebra. As applications, we compute the graded
dimensions and signatures of this cohomology as a hermitean Lie algebra graded
by a hyperbolic lattice. In the first half of this paper, we give as
preparations an exposition of the theory of quantum operator algebras. Some of
the results here were announced in lectures given by the first author at the
Research Institute for Mathematical Sciences in Kyoto in September 94.


title: On the universal $R$-matrix of $U_q\hat{sl}_2$ at roots of unity

abstract:
  We show that the action of universal $R$-matrix of affine $U_qsl_2$ quantum
algebra, when $q$ is a root of unity, can be renormalized by some scalar factor
to give a well defined nonsingular expression, satisfying Yang-Baxter equation.
It reduced to intertwining operators of all representations, corresponding to
Chiral Potts, if the parameters of these representations lie on well known
algebraic curve. We also show that affine $U_qsl_2$ for $q$ is a root of unity
form the autoquasitriangular Hopf algebra in the sence of Reshetikhin.


title: Boundary values as Hamiltonian variables. II. Graded Structures

abstract:
  It is shown that the new Poisson brackets proposed in Part I of this work (J.
Math. Phys. 34, 5747(hep-th/9305133)) arise naturally in an extension of the
formal variational calculus incorporating divergences. The linear spaces of
local functionals, evolutionary vector fields, functional forms, multi-vectors
and differential operators become graded with respect to divergences. Bilinear
operations, such as action of vector fields onto functionals, commutator of
vector fields, interior product of forms and vectors and the Schouten-Nijenhuis
bracket are compatible with the grading. A definition of the adjoint graded
operator is proposed and skew-adjoint operators are constructed with the help
of boundary terms. Fulfilment of the Jacobi identity for the new Poisson
brackets is shown to be equivalent to vanishing of the Schouten-Nijenhuis
bracket for Poisson bivector with itself. The simple procedure for testing this
condition proposed by P. Olver is applicable with a minimal modification. It is
demonstrated that the second structure of the Korteweg--de Vries equation is
not Hamiltonian with respect to the new brackets.


title: Coordinate Calculi on Associative Algebras

abstract:
  A new notion of an optimal algebra for a first order coordinate differential
was introduced in \cite{BKO}. Some relevant examples are indicated. Quadratic
identities in the optimal algebras and calculi on quadratic algebras are
studied. Canonical construction of a quantum de Rham complex for the coordinate
differential is proposed. The relations between calculi and various
generalizations of the Yang--Baxter equation are established.


title: Quadratic Poisson brackets and Drinfel'd theory for associative algebras

abstract:
  Quadratic Poisson brackets on associative algebras are studied. Such a
bracket compatible with the multiplication is related to a differentiation in
tensor square of the underlying algebra. Jacobi identity means that this
differentiation satisfies a classical Yang--Baxter equation. Corresponding Lie
groups are canonically equipped with a Poisson Lie structure. A way to quantize
such structures is suggested.


title:
The quantum superalgebra $U_q[osp(1/2n)]$: deformed para-Bose operators
  and root of unity representations

abstract:
  We recall the relation between the Lie superalgebra $osp(1/2n)$ and para-Bose
operators. The quantum superalgebra $U_q[osp(1/2n)]$, defined as usual in terms
of its Chevalley generators, is shown to be isomorphic to an associative
algebra generated by so-called pre-oscillator operators satisfying a number of
relations. From these relations, and the analogue with the non-deformed case,
one can interpret these pre-oscillator operators as deformed para-Bose
operators. Some consequences for $U_q[osp(1/2n)]$ (Cartan-Weyl basis,
Poincar\'e-Birkhoff-Witt basis) and its Hopf subalgebra $U_q[gl(n)]$ are
pointed out. Finally, using a realization in terms of ``$q$-commuting''
$q$-bosons, we construct an irreducible finite-dimensional unitary Fock
representation of $U_q[osp(1/2n)]$ and its decomposition in terms of
$U_q[gl(n)]$ representations when $q$ is a root of unity.


title:
The fundamental invariant of the Hecke algebra $H_n(q)$ characterizes
  the representations of $H_n(q)$, $S_n$, $SU_q(N)$ and $SU(N)$

abstract:
  The irreducible representations (irreps) of the Hecke algebra $H_n(q)$ are
shown to be completely characterized by the fundamental invariant of this
algebra, $C_n$. This fundamental invariant is related to the quadratic Casimir
operator, ${\cal{C}}_2$, of $SU_q(N)$, and reduces to the transposition
class-sum, $[(2)]_n$, of $S_n$ when $q\rightarrow 1$. The projection operators
constructed in terms of $C_n$ for the various irreps of $H_n(q)$ are
well-behaved in the limit $q\rightarrow 1$, even when approaching degenerate
eigenvalues of $[(2)]_n$. In the latter case, for which the irreps of $S_n$ are
not fully characterized by the corresponding eigenvalue of the transposition
class-sum, the limiting form of the projection operator constructed in terms of
$C_n$ gives rise to factors that depend on higher class-sums of $S_n$, which
effect the desired characterization. Expanding this limiting form of the
projection operator into a linear combination of class-sums of $S_n$, the
coefficients constitute the corresponding row in the character table of $S_n$.
The properties of the fundamental invariant are used to formulate a simple and
efficient recursive procedure for the evaluation of the traces of the Hecke
algebra. The closely related quadratic Casimir operator of $SU_q(N)$ plays a
similar role, providing a complete characterization of the irreps of $SU_q(N)$
and - by constructing appropriate projection operators and then taking the
$q\rightarrow 1$ limit - those of $SU(N)$ as well, even when the quadratic
Casimir operator of the latter does not suffice to specify its irreps.


title: Is a Knot Classification possible?

abstract:
  The goal of this paper is to discuss the possibility of finding an algorithm
that can give all distinct knots up to a desired complexity. Two such
algorithms are presented, one based on projections on a plane, the other on
closed self-avoiding walks.


title:
$Fun(SO_q(N))$-Isotropic Harmonic Oscillator on the Quantum Euclidean
  Space $R_q^N$

abstract:
  We briefly describe the construction of a consistent $q$-deformation of the
quantum mechanical isotropic harmonic oscillator on ordinary $\rn^N$ space.


title: First Order Optimum Calculi

abstract:
  A new notion of an optimum first order calculi was introduced in [Borowiec,
Kharchenko and Oziewicz, 1993]. A module of vector fields for a coordinate
differential is defined. Some examples of optimal algebras for homogeneous
bimodule commutations are presented. Classification theorem for homogeneous
calculi with commutative optimal algebras in two variables is proved.


title: Quantum Groups from Path Integrals

abstract:
  Lecture notes from the 1994 CRM-CAP Summer School ``Particles and Fields
'94''. Covers material written elsewhere in a more leisurely fashion, including
many exercises. Describes derivation of quantum groups from the Chern-Simons
lagrangian for the case of a finite gauge group.


title: Lorentz Transformations as Lie-Poisson Symmetries

abstract:
  We write down the Poisson structure for a relativistic particle where the
Lorentz group does not act canonically, but instead as a Poisson-Lie group. In
so doing we obtain the classical limit of a particle moving on a noncommutative
space possessing $SL_q(2,C)$ invariance. We show that if the standard mass
shell constraint is chosen for the Hamiltonian function, then the particle
interacts with the space-time. We solve for the trajectory and find that it
originates and terminates at singularities.


title: Lie-Poisson groups and the Miura transformation

abstract:
  We point out that the recent proof of the Kupershmidt-Wilson theorem by Cheng
and Mas-Ramos is underpinned by the Lie-Poisson property of the second
Gel'fand-Dickey bracket. The supersymmetric Kupershmidt-Wilson theorem is also
proved along these same lines. Finally we comment on the possible repercussions
in the problem of the coproduct for W-algebras.


title: Quantum Deformation of igl(n) Algebra on Quantum Space

abstract:
  We study quantum deformed $gl(n)$ and $igl(n)$ algebras on a quantum space
discussing multi-parametric extension. We realize elements of deformed $gl(n)$
and $igl(n)$ algebras by a quantum fermionic space. We investigate a map
between deformed $igl(2)$ algebras of our basis and other basis.


title: Non-standard quantum (1+1) Poincar\'e group: a $T$--matrix approach

abstract:
  The Hopf algebra dual form for the non--standard uniparametric deformation of
the (1+1) Poincar\'e algebra $iso(1,1)$ is deduced. In this framework, the
quantum coordinates that generate $Fun_w(ISO(1,1))$ define an infinite
dimensional Lie algebra. A change in the basis of the dual form is obtained in
order to compare this deformation to the standard one. Finally, a non--standard
quantum Heisenberg group acting on a quantum Galilean plane is obtained.


title: Universal $R$--matrices for non-standard (1+1) quantum groups

abstract:
  A universal quasitriangular $R$--matrix for the non-standard quantum (1+1)
Poincar\'e algebra $U_ziso(1,1)$ is deduced by imposing analyticity in the
deformation parameter $z$. A family $g_\mu$ of ``quantum graded contractions"
of the algebra $U_ziso(1,1)\oplus U_{-z}iso(1,1)$ is obtained; this set of
quantum algebras contains as Hopf subalgebras with two primitive translations
quantum analogues of the two dimensional Euclidean, Poincar\'e and Galilei
algebras enlarged with dilations. Universal $R$--matrices
  for these quantum Weyl algebras and their associated quantum groups are
constructed.


title:
Generalized Hirota bilinear identity and integrable q-difference and
  lattice hierarchies.

abstract:
  Hirota bilinear identity for Cauchy-Baker-Akhieser (CBA) kernel is introduced
as a basic tool to construct integrable hierarchies containing lattice and
q-difference times. Determinant formula for the action of meromorphic function
on CBA kernel is derived. This formula gives opportunity to construct generic
solutions for integrable lattice equations.


title: Sacler lectures

abstract:
  The series of three lectures given at Tel-Aviv University in 1992:
  1. Tensor categories.
  2. Quantum groups.
  3. Topological (quantum) field theories.
  Published as the preprint IAS 897-92 of Tel-Aviv University and The Mortimer
and Raymond Sacler Institute of Advanced Studies.


title:
Representation Theory Approach to the Polynomial Solutions of q -
  Difference Equations : U_q(sl(3)) and Beyond,

abstract:
  A new approach to the theory of polynomial solutions of q - difference
equations is proposed. The approach is based on the representation theory of
simple Lie algebras and their q - deformations and is presented here for
U_q(sl(n)). First a q - difference realization of U_q(sl(n)) in terms of
n(n-1)/2 commuting variables and depending on n-1 complex representation
parameters r_i, is constructed. From this realization lowest weight modules
(LWM) are obtained which are studied in detail for the case n=3 (the well known
n=2 case is also recovered). All reducible LWM are found and the polynomial
bases of their invariant irreducible subrepresentations are explicitly given.
This also gives a classification of the quasi-exactly solvable operators in the
present setting. The invariant subspaces are obtained as solutions of certain
invariant q - difference equations, i.e., these are kernels of invariant q -
difference operators, which are also explicitly given. Such operators were not
used until now in the theory of polynomial solutions. Finally the states in all
subrepresentations are depicted graphically via the so called Newton diagrams.


title:
Poincare'-Birkhoff-Witt property for bicovariant differential algebras
  on simple quantum groups

abstract:
  We investigate the possibility to construct bicovariant differential calculi
on quantum groups SO_q(N) and Sp_q(N) as a quantization of an underlying
bicovariant bracket.We show that, opposite to GL(N) and SL(N)-cases, neither of
possible graded SO- and Sp- bicovariant brackets (associated with a
quasitriangular r-matrices) obey the Jacobi identity when the differential
forms are Lie algebra- valued. The absence of a classical Poisson structure
gives an indication that differential algebras describing bicovariant
differential calculi on quantum orthogonal and symplectic groups are not of
Poincar'e- Birkhoff- -Witt type.


abstract:
  An introduction to quantum groups and non-commutative differential calculus
(Lecture at the III Workshop on Differential Geometry, Granada, September 1994)


title: Symmetries of the Kac-Peterson Modular Matrices of Affine Algebras

abstract:
  The characters $\chi_\mu$ of nontwisted affine algebras at fixed level define
in a natural way a representation $R$ of the modular group $SL_2(Z)$. The
matrices in the image $R(SL_2(Z))$ are called the Kac-Peterson modular
matrices, and describe the modular behaviour of the characters. In this paper
we consider all levels of $(A_{r_1}\oplus\cdots\oplus A_{r_s})^{(1)}$, and for
each of these find all permutations of the highest weights which commute with
the corresponding Kac-Peterson matrices. This problem is equivalent to the
classification of automorphism invariants of conformal field theories, and its
solution, especially considering its simplicity, is a major step toward the
classification of all Wess-Zumino-Witten conformal field theories.


title:
Realization of Vector fields for Quantum Groups as Pseudodifferential
  Operators on Quantum Spaces

abstract:
  The vector fields of the quantum Lie algebra are described for the quantum
groups $GL_q(N), SL_q(N)$ and $SO_q(N)$ as pseudodifferential operators on the
linear quantum spaces covariant under the corresponding quantum group. Their
expressions are simple and compact. It is pointed out that these vector fields
satisfy certain characteristic polynomial identities. The real forms $SU_q(N)$
and $SO_q(N,R)$ are discussed in detail.


title: Cohomology and Deformation of Leibniz Pairs

abstract:
  Cohomology and deformation theories are developed for Poisson algebras
starting with the more general concept of a Leibniz pair, namely of an
associative algebra $A$ together with a Lie algebra $L$ mapped into the
derivations of $A$. A bicomplex (with both Hochschild and Chevalley-Eilenberg
cohomologies) is essential.


title: Dynamical Symmetries in q-deformed Quantum Mechanics

abstract:
  The dynamical algebra of the q-deformed harmonic oscillator is constructed.
As a result, we find the free deformed Hamiltonian as well as the Hamiltonian
of the deformed oscillator as a complicated, momentum dependent interaction
Hamiltonian in terms of the usual canonical variables. Furthermore we construct
a well-defined algebra SU$_q$(1,1) with consistent conjugation properties and
comultiplication. We obtain non lowest weight representations of this algebra.


title: Representations of Yangians with Gelfand-Zetlin Bases

abstract:
  We study certain family of finite-dimensional modules over the Yangian
$Y(gl_N)$. The algebra $Y(gl_N)$ comes equipped with a distinguished maximal
commutative subalgebra $A(gl_n)$ generated by the centres of all algebras in
the chain $Y(gl_1)\subset Y(gl_2)\subset...\subset Y(gl_N)$. We study the
finite-dimensional $Y(gl_N)$-modules with a semisimple action of the subalgebra
$A(gl_N)$. We call these modules tame.
  We provide a characterization of irreducible tame modules in terms of their
Drinfeld polynomials. We prove that every irreducible tame module splits into a
tensor product of modules corresponding to the skew Young diagrams and some
one-dimensional module.
  The eigenbases of $A(gl_N)$ in irreducible tame modules are called
Gelfand-Zetlin bases. We provide explicit formulas for the action of the
Drinfeld generators of the algebra $Y(gl_N)$ on the vectors of Gelfand-Zetlin
bases.


title: Quantum Cohomology of a Product

abstract:
  The operation of tensor product of Cohomological Field Theories (or algebras
over genus zero moduli operad) introduced in an earlier paper by the authors is
described in full detail, and the proof of a theorem on additive relations
between strata classes is given. This operation is a version of the Kuenneth
formula for quantum cohomology. In addition, rank one CohFT's are studied, and
a generalization of Zograf's formula for Weil-Petersson volumes is suggested.


title: On the relation between two quantum group invariants of 3-cobordisms

abstract:
  We prove in the context of quantum groups at even roots of unity that a
Turaev-Viro type invariant of a 3-dimensional cobordism M equals the tensor
product of the Reshetikhin-Turaev invariants of M and M*, where M* denotes M
with orientation reversed.


title:
Weight Vectors of the Basic A_1^(1)-Module and the Littlewood-Richardson
  Rule

abstract:
  The basic representation of $\A$ is studied. The weight vectors are
represented in terms of Schur functions. A suitable base of any weight space is
given. Littlewood-Richardson rule appears in the linear relations among weight
vectors.


title: Characteristic Relations for Quantum Matrices

abstract:
  General algebraic properties of the algebras of vector fields over quantum
linear groups $GL_q(N)$ and $SL_q(N)$ are studied. These quantum algebras
appears to be quite similar to the classical matrix algebra. In particular,
quantum analogues of the characteristic polynomial and characteristic identity
are obtained for them. The $q$-analogues of the Newton relations connecting two
different generating sets of central elements of these algebras (the
determinant-like and the trace-like ones) are derived. This allows one to
express the $q$-determinant of quantized vector fields in terms of their
$q$-traces.


title: Langlands Reciprocity for Algebraic Surfaces

abstract:
  This note is an attempt to extend "Geometric Langlands Conjecture" from
algebraic curves to algebraic surfaces. We introduce certain Hecke-type
operators on vector bundles on an algebraic surface. The crucial observation is
that the algebra generated by the Hecke operators turns out to be a homomorphic
image of the {\it quantum toroidal algebra}. The latter is a quantization, in
the spirit of Drinfeld-Jimbo, of the universal enveloping algebra of the
universal central extension of a "double-loop" Lie algebra. This yields, in
particular, a new geometric construction of affine quantum groups of types A, D
E in terms of Hecke operators for an elliptic surface.


title: Some remarks on the {$q$}-Poincare algebra in R-matrix form

abstract:
  The braided approach to q-deformation (due to the author and collaborators)
gives natural algebras $R_{21}u_1Ru_2=u_2R_{21}u_1R$ and $R_{21}x_1x_2=x_2x_1R$
for q-Minkowski and q-Euclidean spaces respectively. These algebras are
covariant under a corresponding background `rotation' quantum group. Semidirect
product by this according to the bosonisation procedure (also due to the
author) gives the corresponding Poincar\'e quantum groups. We review the
construction and collect the resulting R-matrix formulae for both Euclidean and
Minkowski cases in both enveloping algebra and function algebra form, and the
duality between them. Axioms for the Poincar\'e quantum group $*$-structure and
the dilaton problem are discussed.


title: Some Rational Vertex Algebras

abstract:
  Let $L((n-\tfrac 3 2)\Lambda_0)$, $n \in \Bbb N$, be a vertex operator
algebra associated to the irreducible highest weight module $L((n-\tfrac 3
2)\Lambda_0)$ for a symplectic affine Lie algebra. We find a complete set of
irreducible modules for $L((n-\tfrac 3 2)\Lambda_0)$ and show that every module
for $L((n-\tfrac 3 2)\Lambda_0)$ from the category $\Cal O$ is completely
reducible.


title: Hecke algebras, $U_qsl_n$, and the Donald--Flanigan conjecture for $S_n$

abstract:
  To each partition $\frak p$ of $n$ we associate in a canonical way a simple
$S_n$ module with an orthogonal basis indexed by Young diagrams in a way which
carries over immediately to the quantized case. With this we show that the
Hecke algebra of $S_n$ is a global solution to the Donald--Flanigan problem for
$S_n.$ The procedure gives ``canonical'' primitive idempotents different from
the classical ones of Frobenius--Young and makes some number--theoretic
statements.


title: Groups of ribbon knots

abstract:
  Given a positive integer $n$, we say that two knots are $V_n$-equivalent if
they have the same Vassiliev invariants of order $\le n$. We showed that the
$V_n$-equivalence classes of ribbon knots form a group, the operation being
induced by the connected sums of knots. The group is free abelian and its rank
is equal to the dimension of the space of primitive Vassiliev invariants of
order $\le n$. The $n$-equivalence classes (following Gusarov's notion) of
ribbon knots also constitute a group. As a corollary, any non-ribbon knot whose
Arf invariant istrivial cannot be distinguished from ribbon knots by finitely
many Vassiliev invariants. Furthermore, all additive knot cobordism invariants
are not of finite type. As a bi-product, we prove that the number of
independent Vassiliev invariants of order $n$ is bounded above by $(n-2)!/2$
for $n>5$.


title:
The character table of the Hecke algebra $H_n(q)$ in terms of traces of
  products of Murphy operators

abstract:
  The traces of the Murphy operators of the Hecke algebra $H_n(q)$, and of
products of sets of Murphy operators with non-consecutive indices, can be
evaluated by a straightforward recursive procedure. These traces are shown to
determine all the reduced traces in this algebra, which, in turn, determine all
other traces. To illustrate the procedure we obtain the set of reduced traces
for $H_7(q)$ - the lowest order Hecke algebra whose character table has not
hitherto been reported. This is preceded by the presentation of an explicit
algorithm for the reduction of the trace of an arbitrary element of the Hecke
algebra into a linear combination of traces of elements consisting of
appropriately defined disjoint cycles; and of a proof, presented in order to
make the present article reasonably self-contained, that a reduced trace
depends only on the set of lengths of the disjoint cycles that it consists of.


title: A New Null-Plane Quantum Poincare Algebra

abstract:
  A new quantum deformation, which we call null-plane, of the (3+1) Poincar\'e
algebra is obtained. The algebraic properties of the classical null-plane
description are generalized to this quantum deformation. In particular, the
classical isotopy subalgebra of the null-plane is deformed into a Hopf
subalgebra, and deformed spin operators having classical commutation rules can
be defined. Quantum Hamiltonian, mass and position operators are studied, and
the null-plane evolution is expressed in terms of a deformed Schr\"odinger
equation.


title:
Crystal Graphs and $q$-Analogues of Weight Multiplicities for the Root
  System $A_n$

abstract:
  We give an expression of the $q$-analogues of the multiplicities of weights
in irreducible $\sl_{n+1}$-modules in terms of the geometry of the crystal
graph attached to the corresponding $U_q(\sl_{n+1})$-modules. As an
application, we describe multivariate polynomial analogues of the
multiplicities of the zero weight, refining Kostant's generalized exponents.


title: Higher-dimensional Algebra and Topological Quantum Field Theory

abstract:
  The study of topological quantum field theories increasingly relies upon
concepts from higher-dimensional algebra such as n-categories and n-vector
spaces. We review progress towards a definition of n-category suited for this
purpose, and outline a program in which n-dimensional TQFTs are to be described
as n-category representations. First we describe a "suspension" operation on
n-categories, and hypothesize that the k-fold suspension of a weak n-category
stabilizes for k >= n+2. We give evidence for this hypothesis and describe its
relation to stable homotopy theory. We then propose a description of
n-dimensional unitary extended TQFTs as weak n-functors from the "free stable
weak n-category with duals on one object" to the n-category of "n-Hilbert
spaces". We conclude by describing n-categorical generalizations of deformation
quantization and the quantum double construction.


title: Lie Algebroids Associated to Poisson Actions

abstract:
  This work is motivated by a result of Drinfeld on Poisson homogeneous spaces.
For each Poisson manifold $P$ with a Poisson action by a Poisson Lie group $G$,
we describe a Lie algebroid structure on the direct sum vector bundle $P \times
{\frak g} \oplus T^*P$, where ${\frak g}$ is the Lie algebra of $G$. It is
built out of the transformation Lie algebroid $P \times {\frak g}$ and the
cotangent bundle Lie algebroid $T^*P$ together with a pair of representations
of them on each other. When the action of $G$ on $P$ is transitive, the kernel
of the anchor map of this Lie algebroid gives a Lie algebra bundle over $P$,
the fibers of which are given by Drinfeld. As applications, we describe the
symplectic leaves and the $G$-invariant Poisson cohomology of Poisson
homogeneous $G$-spaces.


title:
Phase Space Reduction for Star-Products: An Explicit Construction for
  CP^n

abstract:
  We derive a closed formula for a star-product on complex projective space and
on the domain $SU(n+1)/S(U(1)\times U(n))$ using a completely elementary
construction: Starting from the standard star-product of Wick type on $C^{n+1}
\setminus \{ 0 \}$ and performing a quantum analogue of Marsden-Weinstein
reduction, we can give an easy algebraic description of this star-product.
Moreover, going over to a modified star-product on $C^{n+1} \setminus \{ 0 \}$,
obtained by an equivalence transformation, this description can be even further
simplified, allowing the explicit computation of a closed formula for the
star-product on $\CP^n$ which can easily transferred to the domain
$SU(n+1)/S(U(1)\times U(n))$.


title: Heisenberg Double and Pentagon Relation

abstract:
  It is shown that the Heisenberg double has a canonical element, satisfying
the pentagon relation. From a given invertible constant solution to the
pentagon relation one can restore the structure of the underlying algebras.
Drinfeld double can be realized as a subalgebra in the tensor square of the
Heisenberg double. This enables one to write down solutions to the Yang-Baxter
relation in terms of solutions to the pentagon relation.


title:
Macdonald's Evaluation Conjectures, Difference Fourier Transform, and
  Applications

abstract:
  This paper contains the proof of Macdonald's duality and evaluation
conjectures, the definition of the difference Fourier transform, the recurrence
theorem generalizing Pieri rules, and the action of GL(2,Z) on the Macdonald
polynomials at roots of unity.


title: Field Theory on $q=-1$ Quantum Plane

abstract:
  We build the $q=-1$ defomation of plane on a product of two copies of
algebras of functions on the plane. This algebra constains a subalgebra of
functions on the plane. We present general scheme (which could be used as well
to construct quaternion from pairs of complex numbers) and we use it to derive
differential structures, metric and discuss sample field theoretical models.


title: q-Deformation of the Lorentz Group

abstract:
  We describe a q-deformation of the Lorentz group in terms of a q-deformation
of the van der Waerden spinor algebra.


title: On some nonlinear extensions of the angular momentum algebra

abstract:
  Deformations of the Lie algebras so(4), so(3,1), and e(3) that leave their
so(3) subalgebra undeformed and preserve their coset structure are considered.
It is shown that such deformed algebras are associative for any choice of the
deformation parameters. Their Casimir operators are obtained and some of their
unitary irreducible representations are constructed. For vanishing deformation,
the latter go over into those of the corresponding Lie algebras that contain
each of the so(3) unitary irreducible representations at most once. It is also
proved that similar deformations of the Lie algebras su(3), sl(3,R), and of the
semidirect sum of an abelian algebra t(5) and so(3) do not lead to associative
algebras.


title: Graph Invariants of Vassiliev Type and Application to 4D Quantum Gravity

abstract:
  We consider a special class of Kauffman's graph invariants of rigid vertex
isotopy (graph invariants of Vassiliev type). They are given by a functor from
a category of colored and oriented graphs embedded into a 3-space to a category
of representations of the quasi-triangular ribbon Hopf algebra $U_q(sl(2,\bf
C))$. Coefficients in expansions of them with respect to $x$ ($q=e^x$) are
known as the Vassiliev invariants of finite type. In the present paper, we
construct two types of tangle operators of vertices. One of them corresponds to
a Casimir operator insertion at a transverse double point of Wilson loops. This
paper proposes a non-perturbative generalization of Kauffman's recent result
based on a perturbative analysis of the Chern-Simons quantum field theory. As a
result, a quantum group analog of Penrose's spin network is established taking
into account of the orientation. We also deal with the 4-dimensional canonical
quantum gravity of Ashtekar. It is verified that the graph invariants of
Vassiliev type are compatible with constraints of the quantum gravity in the
loop space representation of Rovelli and Smolin.


title:
The Trivial Connection Contribution to Witten's Invariant and Finite
  Type Invariants of Rational Homology Spheres

abstract:
  We derive an analog of Melvin-Morton bound on the power series expansion of
Jones polynomial of algebraically split links and boundary links. This allows
us to produce a simple formula for the trivial connection contribution to
Witten's invariant of rational homology spheres. We show that the n-th term in
the 1/K expansion of the logarithm of this contribution is a finite type
invariant of Ohtsuki order 3n and of at most Garoufalidis order n. This result
is a manifold counterpart of the statement that n-th derivative of the Jones
polynomial is Vassiliev's invariant of order n.


title: Traces of intertwining operators and Macdonald's polynomials

abstract:
  Let $\Phi:V\to V\otimes U$ be an intertwining operator between
representations of a simple Lie algebra (quantum group, affine Lie algebra). We
define its generalized character to be the following function on the Cartan
subalgebra with values in $U$:
  $\chi_\Phi(h)=\Tr_V (\Phi e^h)$.
  These generalized characters are a rich source of special functions,
possessing many interesting properties; for example, they are common
eigenfunctions of a family of commuting differential (difference) operators. We
show that the special functions that can be obtained this way include
Macdonald's polynomials of type $A$, and this technique allows to prove inner
product and symmetry identities for these polynomials (though proved earlier by
other methods).
  Generalized characters for affine Lie algebras are closely related with
so-called correlation functions on the torus in the Wess-Zumino-Witten (WZW)
model of conformal field theory. We derive differential equations satisfied by
these correlation functions (elliptic Knizhnik-Zamolodchikov, or
Knizhnik-Zamolodchikov-Bernard equations) and study their monodromies.


title:
Localization of $\frak{u}$-modules. III. Tensor categories arising from
  configuration spaces

abstract:
  This article is a sequel to hep-th/9411050, q-alg/9412017.
  In Chapter 1 we associate with every Cartan matrix of finite type and a
non-zero complex number $\zeta$ an abelian artinian category $\FS$. We call its
objects {\em finite factorizable sheaves}. They are certain infinite
collections of perverse sheaves on configuration spaces, subject to a
compatibility ("factorization") and finiteness conditions.
  In Chapter 2 the tensor structure on $\FS$ is defined using functors of
nearby cycles. It makes $\FS$ a braided tensor category.
  In Chapter 3 we define, using vanishing cycles functors, an exact tensor
functor $$\Phi:\FS\lra\CC$$ to the category $\CC$ connected with the
corresponding quantum group.
  In Chapter 4 we show that $\Phi$ is an equivalence. Some proofs are only
sketched.


title: Quasi-{$*$} Structure on {$q$}-Poincare algebras

abstract:
  We use braided groups to introduce a theory of $*$-structures on general
inhomogeneous quantum groups, which we formulate as {\em quasi-$*$} Hopf
algebras. This allows the construction of the tensor product of unitary
representations up to a quantum cocycle isomorphism, which is a novel feature
of the inhomogeneous case. Examples include $q$-Poincar\'e quantum group
enveloping algebras in $R$-matrix form appropriate to the previous
$q$-Euclidean and $q$-Minkowski spacetime algebras $R_{21}x_1x_2=x_2x_1R$ and
$R_{21}u_1Ru_2=u_2R_{21}u_1R$. We obtain unitarity of the fundamental
differential representations. We show further that the Euclidean and Minkowski
Poincar\'e quantum groups are twisting equivalent by a another quantum cocycle.


title: Representations of knot groups and Vassiliev invariants

abstract:
  We show that the number of homomorphisms from a knot group to a finite group
$G$ cannot be a Vassiliev invariant, unless it is constant on the set of
$(2,2p+1)$ torus knots. In several cases, such as when $G$ is a dihedral or
symmetric group, this implies that the number of homomorphisms is not a
Vassiliev invariant.


title: Representation Theory of Chern Simons Observables

abstract:
  Recently we suggested a new quantum algebra, the moduli algebra, which was
conjectured to be a quantum algebra of observables of the Hamiltonian Chern
Simons theory. This algebra provides the quantization of the algebra of
functions on the moduli space of flat connections on a 2-dimensional surface.
In this paper we classify unitary representations of this new algebra and
identify the corresponding representation spaces with the spaces of conformal
blocks of the WZW model. The mapping class group of the surface is proved to
act on the moduli algebra by inner automorphisms. The generators of these
automorphisms are unitary elements of the moduli algebra. They are constructed
explicitly and proved to satisfy the relations of the (unique) central
extension of the mapping class group.


title: Geometry of Deformed Boson Algebras

abstract:
  Phase-space realisations of an infinite parameter family of quantum
deformations of the boson algebra in which the $q$-- and the $qp$--deformed
algebras arise as special cases are studied. Quantum and classical models for
the corresponding deformed oscillators are provided. The deformation parameters
are identified with coefficients of non-linear terms in the normal forms
expansion of a family of classical Hamiltonian systems. These quantum
deformations are trivial in the sense that they correspond to non-unitary
transformations of the Weyl algebra. They are non-trivial in the sense that the
deformed commutators consistently quantise a class of non-canonical classical
Poisson structures.


title: Finite Group Factorisations and Braiding

abstract:
  We compute the quantum double, braiding and other canonical Hopf algebra
constructions for the bicrossproduct Hopf algebra $H$ associated to the
factorization of a finite group into two subgroups. The representations of the
quantum double are described by a notion of bicrossed bimodules, generalising
the cross modules of Whitehead. We also show that self-duality structures for
the bicrossproduct Hopf algebras are in one-one correspondence with
factor-reversing group isomorphisms. The example $Z_6Z_6$ is given in detail.
  We show further that the quantum double $D(H)$ is the twisting of $D(X)$ by a
non-trivial quantum cocycle, where $X$ is the associated double cross product
group.


title: Quadratic Poisson brackets and Drinfeld theory for associative algebras

abstract:
  The paper is devoted to the Poisson brackets compatible with multiplication
in associative algebras. These brackets are shown to be quadratic and their
relations with the classical Yang--Baxter equation are revealed. The paper also
contains a description of Poisson Lie structures on Lie groups whose Lie
algebras are adjacent to an associative structure.


title: Connections on central bimodules

abstract:
  We define and study the theory of derivation-based connections on a recently
introduced class of bimodules over an algebra which reduces to the category of
modules whenever the algebra is commutative. This theory contains, in
particular, a noncommutative generalization of linear connections. We also
discuss the different noncommutative versions of differential forms based on
derivations. Then we investigate reality conditions and a noncommutative
generalization of pseudo-riemannian structures.


title:
Classical and Quantum sl(1|2) Superalgebras, Casimir Operators and
  Quantum Chain Hamiltonians

abstract:
  We examine the two parameter deformed superalgebra $U_{qs}(sl(1|2))$ and use
the results in the construction of quantum chain Hamiltonians. This study is
done both in the framework of the Serre presentation and in the $R$-matrix
scheme of Faddeev, Reshetikhin and Takhtajan (FRT). We show that there exists
an infinite number of Casimir operators, indexed by integers $p > 1$ in the
undeformed case and by $p \in Z$ in the deformed case, which obey quadratic
relations. The construction of the dual superalgebra of functions on
$SL_{qs}(1|2)$ is also given and higher tensor product representations are
discussed. Finally, we construct quantum chain Hamiltonians based on the
Casimir operators. In the deformed case we find two Hamiltonians which describe
deformed $t-J$ models.


title: Representations of Quantum Affine Algebras

abstract:
  In this paper we define a quantum version of the ``fusion'' tensor product of
two representations of an affine Kac-Moody algebra.It is replaced by what we
call fusion action of the category of finite-dimensional representations of
quantum affine algebra on its highest weight representations. We construct a
quantum version of the associativity constraint. We give categorical treatment
of the subject and related questions ( like quantum Knizhnik-Zamolodchikov
equations).


title: Reconstruction of Weak Quasi Hopf Algebras

abstract:
  All rational semisimple braided tensor categories are representation
categories of weak quasi Hopf algebras. To proof this result we construct for
any given category of this kind a weak quasi tensor functor to the category of
finite dimensional vector spaces. This allows to reconstruct a weak quasi Hopf
algebra with the given category as its representation category.


title:
Quantization of quadratic Poisson brackets on a polynomial algebra of
  three variables

abstract:
  Poisson brackets (P.b) are the natural initial terms for the deformation
quantization of commutative algebras. There is an open problem whether any
Poisson bracket on the polynomial algebra of $n$ variables can be quantized. It
is known (Poincare-Birkhoff-Witt theorem) that any linear P.b. for all $n$ can
be quantized. On the other hand, it is easy to show that in case $n=2$ any P.b.
is quantizable as well. Quadratic P.b. appear as the unitial terms for the
quantization of polynomial algebras as quadratic algebras. The problem of the
quantization of quadratic P.b. is also open. In the paper we show that in case
$n=3$ any quadratic P.b can be quantized. Moreover, the quantization is given
as the quotient algebra of tensor algebra of three variables by relations which
are similar to those in the Poincare-Birkhoff-Witt theorem. The proof uses a
classification of all quadratic Poisson brackets of three variables, which we
also give in the paper. In Appendix we give explicit algebraic constructions of
the quantized algebras appeared here to show that they are related to algebras
of global dimension three considered by M.Artin, W.Schelter, J.Tate, and M.Van
Den Bergh from a different point of view.


title: The quantum 2-sphere as a complex quantum manifold

abstract:
  We describe the quantum sphere of Podle\'{s} for $c=0$ by means of a
stereographic projection which is analogous to that which exhibits the
classical sphere as a complex manifold. We show that the algebra of functions
and the differential calculus on the sphere are covariant under the coaction of
fractional transformations with \SU coefficients as well as under the action of
\SU vector fields. Going to the classical limit we obtain the Poisson sphere.
Finally, we study the invariant integration of functions on the sphere and find
its relation with the translationally invariant integration on the complex
quantum plane.


title:
The Robinson-Schensted correspondence as the quantum straightening at
  $q=0$

abstract:
  We show that the quantum straightening algorithm for Young tableaux and Young
bitableaux reduces in the crystal limit $q \mapsto 0$ to the Robinson-Schensted
algorithm.


title: Yangian Construction of the Virasoro Algebra

abstract:
  We show that a Yangian construction based on the algebra of an infinite
number of harmonic oscillators (i.e. a vibrating string) terminates after one
step, yielding the Virasoro algebra.


title: Representations of U_q(sl(N)) at Roots of Unity

abstract:
  The Gelfand--Zetlin basis for representations of $U_q(sl(N))$ is improved to
fit better the case when $q$ is a root of unity. The usual $q$-deformed
representations, as well as the nilpotent, periodic (cyclic), semi-periodic
(semi-cyclic) and also some atypical representations are now described with the
same formalism.


title: Comments on Bosonisation and Biproducts

abstract:
  We clarify the relation between the `bosonisation' construction (due to the
author) which can be used to turn a Hopf algebra $B$ in the category of
$H$-modules or $H$-comodules into an equivalent ordinary Hopf algebra, and a
version of Radfords theorem (also due in this form to the author) which does
the same for $B$ in the category of crossed $H$-modules. We also comment on
reconstruction from the category of $B$-comodules.


title: Simple currents and extensions of vertex operator algebras

abstract:
  We consider how a vertex operator algebra can be extended to an abelian
intertwining algebra by a family of weak twisted modules which are {\em simple
currents} associated with semisimple weight one primary vectors. In the case
that the extension is again a vertex operator algebra, the rationality of the
extended algebra is discussed. These results are applied to affine Kac-Moody
algebras in order to construct all the simple currents explicitly (except for
$E_8$) and to get various extensions of the vertex operator algebras associated
with integrable representations.


title: A Quantum Analogue of the ${\cal Z}$ Algebra

abstract:
  We define a natural quantum analogue for the ${\cal Z}$ algebra, and which we
refer to as the ${\cal Z}_q$ algebra, by modding out the Heisenberg algebra
from the quantum affine algebra $U_q(\hat{sl(2)})$ with level $k$. We discuss
the representation theory of this ${\cal Z}_q$ algebra. In particular, we
exhibit its reduction to a group algebra, and to a tensor product of a group
algebra with a quantum Clifford algebra when $k=1$, and $k=2$, and thus, we
recover the explicit constructions of $\uq$-standard modules as achieved by
Frenkel-Jing and Bernard, respectively. Moreover, for arbitrary nonzero level
$k$, we show that the explicit basis for the simplest ${\cal Z}$-generalized
Verma module as constructed by Lepowsky and primc is also a basis for its
corresponding ${\cal Z}_q$-module, i.e., it is invariant under the
q-deformation for generic q. We expect this ${\cal Z}_q$ algebra (associated
with $\uq$ at level $k$), to play the role of a dynamical symmetry in the
off-critical $ Z_k$ statistical models.


title: Cycles for asymptotic solutions and Weyl group

abstract:
  In this paper cycles for asymptotic solutions for Heckman-Opdam
hypergeometric system of type $A_n$ are described. Cycles are enumerated by
elements of symmetric group. Leading asymptotic and leading coefficient are
calculated. Value of certain multiple integral over special cycles is
calculated with the help of result of Opdam.


title:
Bases of Bethe Vectors and Difference Equations with Regular Singular
  Points

abstract:
  We prove that Bethe vectors generically form a base in a tensor product of
irreducible heighest weight $gl_2$-modules or $U_q(gl_2)$-modules. We apply
this result to difference equations with regular singular points. We show that
if such an equation has local solutionss at each of its singular point, then
generically it has a polynomial solution.


title: On the diagonalization of difference Calogero-Sutherland systems

abstract:
  We discuss the simultaneous diagonalization of a family of commuting
difference operators by Koornwinder's multivariable generalization of the
Askey-Wilson polynomials. The operators constitute a complete set of quantum
integrals for a difference version of the $n$-particle quantum
Calogero-Sutherland system related to the root system $BC_n$.