Steven Duplij

Weak Hopf Algebras and Singular Solutions of Quantum Yang-Baxter Equation

Abstract: We investigate a generalization of Hopf algebra by weakening the invertibility of the generator K, i.e. exchanging its invertibility KK− 1= 1 to the regularity . This leads to a weak Hopf algebra and a J-weak Hopf algebra which are studied in detail. It is shown that the monoids of group-like elements of and are regular monoids, which supports the general conjucture on the connection betweek weak Hopf algebras and regular monoids. Moreover, from a quasi-braided weak Hopf algebra is constructed and it is shown that the corresponding quasi-R-matrix is regular .

Noncommutative Structures in Mathematics and Physics

Preface. Gauge Theories Beyond Gauge Theory J. Wess. Symmetries Wider Than Supersymmetry D. Leites, V. Serganova. Tensions in Supergravity Braneworlds K. Stelle. An Unconventional Supergravity P. Grozman, D. Leites. Supersymmetry of RS Bulk and Brane E. Bergshoeff, et al. D-Branes and Vacuum Periodicity D. Galtsov, V. Dyadichev. Quantum Deformations of Space-Time SUSY and Noncommutative Superfield Theory P. Kosinski, et al. The Howe Duality and Lie Superalgebras D. Leites, I. Shchepochkina. Enveloping Algebra of GL(3) and Orthogonal Polynomials A. Sergeev. Noninvertibility, Semisupermanifolds and Categories Regularization S. Duplij, W. Marcinek. An Overview of New Supersymmetric Gauge Theories With 2-Form Gauge Potentials F. Brandt. Supersymmetric R4 Actions and Quantum Corrections to Superspace Torsion Constraints K. Peeters, et al. Massive Superparticle With Spinorial Central Charges S. Fedoruk, V.G. Zima. Rotating Super Black Hole as Spinning Particle A. Burinskii. Classifying N-extended 1-dimensional Super Systems F. Toppan. Para, Pseudo, and Orthosupersymmetric Quantum Mechanics and Their Bosonization C. Quesne. Supersymmetric Odd Mechanical Systems and Hilbert Q-module Quantization A. Frydryszak. Locally Anisotropic Supergravity and Gauge Gravity on Noncommutative Spaces S. Vacaru, et al. Finiteness in Conventional N = 1 GUTs T. Kobayashi, et al. World Volume Realization of Automorphisms J. Simon. Some Metrics on the Manin Plane G. Fiore, et al. Coherence Isomorphisms for a Hopf Category V. Lyubashenko. Fusion Rings and Tensor Categories A. Ganchev. On Categories of Gelfand-Zelin Modules V. Mazorchuk. Hidden Symmetry of SomeAlgebras Of q-differential Operators D. Shklyarov, et al. A Family Of -Algebras Allowing Wick Ordering: Fock Representations and Universal Enveloping C -Algebras P. Jorgensen, et al. Nonstandard Quantization of the Enveloping Algebra U(so(n)) and its Applications A.U. Klimyk. Can the Cabibbo mixing originate from noncommutative extra dimensions? A. Gavrilik. Nonclassical Type Representations of Nonstandard Quantization of Enveloping Algebras U(so(n)), U(so(n,1)) and U(iso(n)) N. Iorgov. Quasiparticles in Non-commutative Field Theory K. Landsteiner. Time Dependence and (Non)Commutativity of Symmetries of Evolution Equations A. Sergyeyev. p-Adic Strings and Noncommuntativity B. Dragovich, I.V. Volovich. Adelic Quantum Mechanics: Nonarchimedean and Noncommutative Aspects G. Djordjevic, et al. Gibbs States of a Lattice System of Quantum Anharmonic Oscillators Y. Kozitsky. A Metric-Affine Field Model for the Neutrino D. Vassiliev. Generalized Taub-NUT Metrics and Killing-Yano Tensors M. Visinescu. An Effective Model of the Spacetime Foam V. Dzhunushaliev. Possible Constraints on String Theory in Closed Space with Symmetries A. Higuchi. Semiclassical Dynamics of SU (2) Models A. Alscher, H. Grabert. List of Speakers and their E-Prints.

On the semigroup nature of superconformal symmetry

A semigroup of N=1 superconformal transformations is introduced and analyzed. Noninvertible ones can describe transitions from body to soul and form a proper ideal containing a set of nilpotent transformations. The projective superspace is also considered. Transformations twisting the parity of a tangent space are brought in. They can be a nonsuperconformal ‘‘square root’’ of noninvertible superconformal transformations and the analogs of the Poincare metric and conformal invariance are suggested for them.

Bi-Element Representations of Ternary Groups#

General properties of ternary semigroups and groups are consi-dered. The bi-element representation theory in which every representation matrix corresponds to a pair of elements is built up. Connections with the standard theory are considered and several concrete examples are constructed. For the sake of clarity and completeness the shortened versions of classical Gluskin-Hosszú and Post theorems are also presented.

Some Abstract Properties of Semigroups Appearing in Superconformal Theories

A new type of semigroup which appears while dealing withN = 1 superconformal symmetry in superstring theories is considered. The ideal series having unusual abstract properties is constructed. Various idealisers are introduced and studied. The ideal quasicharacter is defined. Green’s relations are found and their connection with the ideal quasicharacter is established.

Regular solutions of quantum Yang-Baxter equation from weak hopf algebras

AbstractGeneralization of Hopf algebraslq (2) by weakening the invertibility of the generatorK, i.e., exchanging its invertibilityKK−1=1 to the regularity K

Nilpotent Marsh and SUSY QM

\bar K

Dependence of nucleotide physical properties on their placement in codons and determinative degree

K=K is studied. Two weak Hopf algebras are introduced: a weak Hopf algebrawslq (2) and aJ-weak Hopf algebravslq (2) which are investigated in detail. The monoids of group-like elements ofwslq (2) andvslq (2) are regular monoids, which supports the general conjucture on the connection betweek weak Hopf algebras and regular monoids. A quasi-braided weak Hopf algebraŪqw is constructed fromwslq (2). It is shown that the corresponding quasi-R-matrix is regular Rw