Hamiltonian structure of multi component integrable systems

In this work we have generalized the super KdV equation into a multicomponent super KdV equation. It is shown that the system is bi-super Hamiltonian. The third super Hamiltonian in the multicomponent super KdV hierarchy is obtained and the corresponding first and second members of evolution equations are given. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. This work is supported by Yildiz Technical University Foundation under project No. 21-01-01-02.     

Lie symmetry, discrete symmetry and supersymmetry of the Pauli Hamiltonian

Starting from the full group of symmetries of a system we select a discrete subset of transformations which allows to introduce the Clifford algebra of operators generating new supercharges of extended supersymmetry. The system defined by the Pauli Hamiltonian is discussed. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001 Partially supported by the KBN-Grant # 5 P03B056 20.     

Fractional sequential mechanics — models with symmetric fractional derivative

The symmetric fractional derivative is introduced and its properties are studied. The Euler-Lagrange equations for models depending on sequential derivatives of type are derived using minimal action principle. The Hamiltonian for such systems is introduced following methods of classical generalized mechanics and the Hamilton’s equations are obtained. It is explicitly shown that models of fractional sequential mechanics are non-conservative. The limiting procedure recovers classical generalized mechanics of systems depending on higher order derivatives. The method is applied to fractional deformation of harmonic oscillator and to the case of classical frictional force proportional to velocity. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001.     

Quasigraded deformations of loop algebras and classical integrable systems

We construct a family of infinite-dimensional quasigraded Lie algebras, that could be viewed as deformation of the graded loop algebras. Using them we obtain new series of integrable Hamiltonian systems on semisimple Lie algebras and their extensions. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

Integrals of motion for some equations onq-minkowski space

The conservation laws for second order linear equation with constant coefficients on braided linear space are derived. As an example we study conserved currents connected with symmetry operators for scalar and spinor wave equations onq-Minkowski space. Then we formulate set of conditions for integral over spatial dimensions and use the postulated integral in construction of constant of motion for arbitrary conserved current onq-Minkowski space. Presented at the 6th International Colloquium on Quantum Groups “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997.     

Nonlocal charges in susy integrable models

We study systematically the general properties of theB-extension of any integrable model and its properties as Hamiltonian structures etc. We clarify the origin of “exotic” changes in such models. We show that in such models there exist at least two sets of non-local conserved charges and that the “exotic” charges are part of this non-local charge hierarchy. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

Operadic deformations as a tool for cogravity

The deformation equation and its integrability condition (Bianchi identity) of a non-(co)associative deformation in operad algebra are found. Based on physical analogies, two ideas ofcogravity equations are proposed. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

Lagrangean and Hamiltonian fractional sequential mechanics

The models described by fractional order derivatives of Riemann-Liouville type in sequential form are discussed in Lagrangean and Hamiltonian formalism. The Euler-Lagrange equations are derived using the minimum action principle. Then the methods of generalized mechanics are applied to obtain the Hamilton’s equations. As an example free motion in fractional picture is studied. The respective fractional differential equations are explicitly solved and it is shown that the limitα→1⁺ recovers classical model with linear trajectories and constant velocity. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

Hamilton—Jacobi quantization of constrained systems

The path integral quantization of contrained systems is analysed using Hamilton-Jacobi formalism. The integrability conditions are investigated and the results are in agreement with those obtained by Dirac’s method. Presented at the 10th Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June, 2001.     

Yangians, quantum groups and solutions of the quantum dynamical Yang-Baxter equation

We construct Drinfel’d twists that define deformed Hopf structures. In particular, we obtain deformed double Yangians and dynamical double Yangians. Presented by D. Arnaudon at the 10th Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June, 2001.     

ψ-extensions ofq-hermite andq-laguerre polynomials — properties and principal statements

Spectral theorem, reccurence relations and difference eqations for Shefferψ-polynomials are derived. These includeq-Hermite andq-Laguerre polynomials and many others — as special cases. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001.     

A new method forq-hypergeometric series

We reintroduce a notation of Heine, which leads to a new method for computations and classifications ofq-special functions. The main topic of the new method is an involution operator on the set of allq-shifted factorials. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001.     

Anyonic lie algebras

We introduce anyonic Lie algebras in terms of structure constants. We provide the simplest examples and formulate some open problems. Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997. This paper is in final form and no version of it will be published elsewhere.     

Towardsψ-extension of rota calculus in mizar language

Some cases of extended Umbral calculus provide an underpining for deformed quantum oscillator models. The Umbral calculus has been already formulated in Maple package. We shall present the first stages of the Mizar System usage in formulating and checking the first principal statements of extended Umbral calculus. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001.     

An algebraic analog of the virasoro group

The group of diffeomorphisms of a circle is not an infinite-dimensional algebraic group, though in many ways it acts as if it were. Here we construct an algebraic model for this object, and discuss some of its representations, which appear in the Kontsevich-Witten theory of 2D topological gravity. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. Supported in part by the NSF.     

Three short distance structures from quantum algebras

We review known and we present new results on three types of short distance structures of observables which typically appear in studies of quantum group related algebras. In particular, one of the short distance structures is shown to suggest a new mechanism for the introduction of internal symmetries. Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997.     

Operads for x-physics

The essential parts of the operad algebra are presented, which should be useful when confronted with the operadic physics. It is also clarified how the Gerstenhaber algebras can be associated with the linear pre-operads (comp algebras). Their relation to mechanics is concisely discussed. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. Research supported by the ESF grant 3654.     

Deformed yangians and integrable models

Twisted Hopf algebraslξ(2) gives rise to a deformation of the Yangiany(sl(2)). The corresponding deformations of the integrableXXX-spin chain and the Gaudin model are discussed. Presented at 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrables Systems”, Prague, 19–21 June 1997. This research was supported by the Swedish National Research Council and RFFI grant 96-01-00851.     

Reflection equation- and FRT-type algebras

We introduce two types of algebras which include respectively the well known reflection equation (RE) and Faddeev-Reshetikhin-Takhtayan algebras associated with a quasitriangular Hopf algebraH. We show that these two types of algebras are twist-equivalent. It follows that a RE algebra is a module algebra over a twisted tensor square ofH. We present some applications to the equivariant quantization. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.