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J. Lukierski

Publications and source records attributed to J. Lukierski.

68 records · Page 4Linked to original sources
arXiv2000

Unified description of quantum affine (super)algebras U_q(A_{1}^{(1)}) and U_q(C(2)^{(2)})

We show that the quantum affine algebra U_{q}(A_{1}^{(1)}) and the quantum affine superalgebra U_{q}(C(2)^{(2)}) admit unified description. The difference between them consists in the phase factor which is equal to 1 for U_{q}(A_{1}^{(1)}) and is equal to -1 for U_{q}(C(2)^{(2)}). We present such a description for the construction of Cartan-Weyl generators and their commutation relations, as well for the universal R-matrices.

math.QA↗
arXiv2000

Quantum Affine (Super)Algebras $U_q(A_{1}^{(1)})$ and $U_q(C(2)^{(2)})$

We show that the quantum affine algebra $U_{q}(A_{1}^{(1)})$ and the quantum affine superalgebra $U_{q}(C(2)^{(2)})$ admit a unified description. The difference between them consists in the phase factor which is equal to 1 for $U_{q}(A_{1}^{(1)})$ and it is equal to -1 for $U_{q}(C(2)^{(2)})$. We present such a description for the actions of the braid group, for the construction of Cartan-Weyl generators and their commutation relations, as well for the extremal projector and the universal R-matrix. We give also a unified description for the 'new realizations' of these algebras together with explicit calculations of corresponding R-matrices.

math.QA↗
arXiv2000

Translational Chern--Simons Action and New Planar Particle Dynamics

We consider a nonstandard $D=2+1$ gravity described by a translational Chern--Simons action, and couple it to the nonrelativistic point particles. We fix the asymptotic coordinate transformations in such a way that the space part of the metric becomes asymptotically Euclidean. The residual symmetries are (local in time) translations and rigid rotations. The phase space Hamiltonian $H$ describing two-body interactions satisfies a nonlinear equation $H={\cal H}(\vec{x},\vec{p};H)$ what implies, after quantization, a nonstandard form of the Schrödinger equation with energy-dependent fractional angular momentum eigenvalues. Quantum solutions of the two-body problem are discussed. The bound states with discrete energy levels correspond to a confined classical motion (for the planar distance between two particles $r\leq r_0$) and the scattering states with continuous energy correspond to classical motion for $r>r_0$.

hep-th↗
arXiv1999

Local D=4 Field Theory on $κ$--Deformed Minkowski Space

We describe the local D=4 field theory on $κ$--deformed Minkowski space as nonlocal relativistic field theory on standard Minkowski space--time. For simplicity the case of $κ$-deformed scalar field $ϕ$ with the interaction $λϕ^{4}$ is considered, and the $κ$--deformed interaction vertex is described. It appears that fundamental mass parameter $κ$ plays a role of regularizing imaginary Pauli--Villars mass in $κ$--deformed propagator.

hep-th↗
arXiv1998

Deformed Quantum Relativistic Phase Spaces -- an Overview

We describe three ways of modifying the relativistic Heisenberg algebra - first one not linked with quantum symmetries, second and third related with the formalism of quantum groups. The third way is based on the identification of generalized deformed phase space with the semidirect product of two dual Hopf algebras describing quantum group of motions and the corresponding quantum Lie algebra. As an example the $κ$-deformation of relativistic Heisenberg algebra is given, determined by $κ$-deformed D=4 Poincaré symmetries.

hep-th↗
arXiv1998

On Quantum Deformations of D=4 Conformal Algebra

Three classes of classical r-matrices for sl(4,C) algebra are constructed in quasi-Frobenius algebra approach. They satisfy CYBE and are spanned respectively on 8,10,12 generators. The o(4,2) reality condition can be imposed only on the eight dimensional r matrices with dimension-full deformation parameters. Contrary to the Poincare algebra case, it appears that all deformations with a mass-like deformation parameter (kappa- deformations) are described by classical r-matrices satisfying CYBE.

math-ph↗
arXiv1994

The classical basis for $κ$-deformed Poincaré (super)algebra and the second $κ$-deformed supersymmetric Casimir

We present here the general solution describing generators of \kdef \poin algebra as the functions of classical \poin algebra generators as well as the inverse formulae. Further we present analogous relations for the generators of N=1 D=4 \kdef \poin superalgebra expressed by the classical \poin superalgebra generators. In such a way we obtain the \kdef \poin (super)algebras with all the quantum deformation present only in the coalgebra sector. Using the classical basis of \kdef \poin superalgebra we obtain as a new result the $\k$-deformation of supersymmetric covariant spin square Casimir.

hep-th↗
arXiv1994

$κ$-Deformation of Poincaré Superalgebra with Classical Lorentz Subalgebra and its Graded Bicrossproduct Structure

The $κ$-deformed $D=4$ Poincar{é} superalgebra written in Hopf superalgebra form is transformed to the basis with classical Lorentz subalgebra generators. We show that in such a basis the $κ$-deformed $D=4$ Poincare superalgebra can be written as graded bicrossproduct. We show that the $κ$-deformed $D=4$ superalgebra acts covariantly on $κ$-deformed chiral superspace.

hep-th↗
arXiv1994

Quantum Deformation of the Poincare Supergroup and $κ$-deformed Superspace

The classical $r$-matrix for $N=1$ superPoincar{é} algebra, given by Lukierski, Nowicki and Sobczyk is used to describe the graded Poisson structure on the $N=1$ Poincar{é} supergroup. The standard correspondence principle between the even (odd) Poisson brackets and (anti)commutators leads to the consistent quantum deformation of the superPoincar{é} group with the deformation parameter $q$ described by fundamental mass parameter $κ\quad (κ^{-1}=\ln{q})$. The $κ$-deformation of $N=1$ superspace as dual to the $κ$-deformed supersymmetry algebra is discussed.

hep-th↗
arXiv1993

Classical and Quantum Mechanics of Free \k Relativistic Systems

We consider the Hamiltonian and Lagrangian formalism describing free \k-relativistic particles with their four-momenta constrained to the \k-deformed mass shell. We study the modifications of the formalism which follow from the introduction of space coordinates with nonvanishing Poisson brackets and from the redefinitions of the energy operator. The quantum mechanics of free \k-relativistic particles and of the free \k-relativistic oscillator is also presented. It is shown that the \k-relativistic oscillator describes a quantum statistical ensemble with finite Hagedorn temperature. The relation to a \k-deformed Schrödinger quantum mechanics in which the time derivative is replaced by a finite difference derivative is also discussed.

hep-th↗
arXiv1992

Quantum Group $σ$ Models

Field-theoretic models for fields taking values in quantum groups are investigated. First we consider $SU_q(2)$ $σ$ model ($q$ real) expressed in terms of basic notions of noncommutative differential geometry. We discuss the case in which the $σ$ models fields are represented as products of conventional $σ$ fields and of the coordinate-independent algebra. An explicit example is provided by the $U_q(2)$ $σ$ model with $q\sp{N}=1$, in which case quantum matrices $U_q(2)$ are realised as $2N\times 2N$ unitary matrices. Open problems are pointed out.

hep-th↗
arXiv1991

Real Forms of Complex Quantum Anti de Sitter Algebra $U_q (Sp(4,C))$ and their Contraction Schemes

We describe four types of inner involutions of the Cartan-Weyl basis providing (for $ |q|=1$ and $q$ real) three types of real quantum Lie algebras: $U_{q}(O(3,2))$ (quantum D=4 anti-de-Sitter), $U_{q}(O(4,1))$ (quantum D=4 de-Sitter) and $U_{q}(O(5))$. We give also two types of inner involutions of the Cartan-Chevalley basis of $U_{q}(Sp(4;C))$ which can not be extended to inner involutions of the Cartan-Weyl basis. We outline twelve contraction schemes for quantum D=4 anti-de-Sitter algebra. All these contractions provide four commuting translation generators, but only two (one for $ |q|=1$, second for $q$ real) lead to the quantum \po algebra with an undeformed space rotations O(3) subalgebra.

hep-th↗