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P. P. Kulish

Publications and source records attributed to P. P. Kulish.

At least 37 records · Page 2Linked to original sources
arXiv1999

Inhomogeneous quantum Lie algebras

We study quantization of a class of inhomogeneous Lie bialgebras which are crossproducts in dual sectors with Abelian invariant parts. This class forms a category stable under dualization and the double operations. The quantization turns out to be a functor commuting with them. The Hopf operations and the universal R-matrices are given in terms of generators. The quantum algebras obtained appear to be isomorphic to the universal enveloping Poisson-Lie algebras on the dual groups.

math.QA↗
arXiv1999

Twist-related geometries on q-Minkowski space

The role of the quantum universal enveloping algebras of symmetries in constructing non-commutative geometry of the space-time including vector bundles, measure, equations of motion and their solutions is discussed. In the framework of the twist theory the Klein-Gordon-Fock and Dirac equations on the quantum Minkowski space are studied from this point of view for the simplest quantum deformation of the Lorentz algebra induced by its Cartan subalgebra twist.

math.QA↗
arXiv1998

On twisting solutions to the Yang-Baxter equation

Sufficient conditions for an invertible two-tensor $F$ to relate two solutions to the Yang-Baxter equation via the transformation $R\to F^{-1}_{21} R F$ are formulated. Those conditions include relations arising from twisting of certain quasitriangular bialgebras.

math.QA↗
arXiv1998

Classical and quantum duality in jordanian quantizations

The limiting transitions between different types of quantizations are studied by the deformation theory methods. We prove that for the first order coboundary deformation (g,g*_1 + x g*_2) of a Lie bialgebra (g,g*) one can always get the quantized Lie bialgebra (g,g*_2) as a limit of the sequence of quantizations of the type A(g,g*_1). The obtained results are illustrated by some low-dimensional examples of quantized Lie algebras and superalgebras.

math.QA↗
arXiv1998

Super-jordanian deformation of the orthosymplectic Lie superalgebras

The recently proposed jordanian quantization of the Lie superalgebra $osp(1|2)$ due to the embedding $sl(2) \subset osp(1|2)$, is extended including odd generators into the twisting element $\cal F$. This deformation is obtained as a contraction of the quantum superalgebra ${\cal U}_{q}(osp(1|2))$.

math.QA↗
arXiv1998

Extended jordanian twists for Lie algebras

Jordanian quantizations of Lie algebras are studied using the factorizable twists. For a restricted Borel subalgebras ${\bf B}^{\vee}$ of $sl(N)$ the explicit expressions are obtained for the twist element ${\cal F}$, universal ${\cal R}$-matrix and the corresponding canonical element ${\cal T}$. It is shown that the twisted Hopf algebra ${\cal U}_{\cal F} ({\bf B}^{\vee})$ is self dual. The cohomological properties of the involved Lie bialgebras are studied to justify the existence of a contraction from the Dinfeld-Jimbo quantization to the jordanian one. The construction of the twist is generalized to a certain type of inhomogenious Lie algebras.

math.QA↗
arXiv1998

Universal R-matrix for esoteric quantum group

The universal $R$-matrix for a class of esoteric (non-standard) quantum groups ${\cal U}_q(gl(2N+1))$ is constructed as a twisting of the universal $R$-matrix ${\cal R}_S$ of the Drinfeld-Jimbo quantum algebras. The main part of the twisting element ${\cal F}$ is chosen to be the canonical element of appropriate pair of separated Hopf subalgebras (quantized Borel's ${\cal B}(N) \subset {\cal U}_q(gl(2N+1))$), providing the factorization property of ${\cal F}$. As a result, the esoteric quantum group generators can be expressed in terms of the Drinfeld-Jimbo ones.

math.QA↗
arXiv1998

On position operator spectral measure for deformed oscillator in the case of indetermine Hamburger moment problem

The spectral measure of the position (momentum) operator $X$ for $q$-deformed oscillator is calculated in the case of the indetermine Hamburger moment problem. The exposition is given for concrete choice of generators for $q$-oscillator algebra, although developed technique apply for every other cases with indetermine moment problem. The Stieltjes transformation $m(z)$ of spectral measure is expressed in terms of the entries of Jacobi matrix $X$ only. The direct connection between values of parameters labeling the spectral measures and related selfadjoint extensions of $X$ is established.

math.QA↗
arXiv1997

Twist Deformation of the rank one Lie Superalgebra

The Drinfeld twist is applyed to deforme the rank one orthosymplectic Lie superalgebra $osp(1|2)$. The twist element is the same as for the $sl(2)$ Lie algebra due to the embedding of the $sl(2)$ into the superalgebra $osp(1|2)$. The R-matrix has the direct sum structure in the irreducible representations of $osp(1|2)$. The dual quantum group is defined using the FRT-formalism. It includes the Jordanian quantum group $SL_ξ(2)$ as subalgebra and Grassmann generators as well.

q-alg↗
arXiv1997

Deformed Yangians and Integrable Models

Twisted Hopf algebra $sl_ξ(2)$ gives rise to a deformation of the Yangian ${\cal Y}(sl(2))$. The corresponding deformations of the integrable XXX-spin chain and the Gaudin model are discussed.

q-alg↗
arXiv1997

Twisted h-spacetimes and invariant equations

We analyze the h-deformations of the Lorentz group and their associated spacetimes. We prove that they have a twisted character and give explicitly the twisting matrices. After studying the representations of one of the deformed spacetime algebras, we discuss the Klein-Gordon operator. It is found that the h-deformed d'Alembertian has plane wave solutions of the same form as the standard ones. We also give explicit expressions for the h-gamma matrices defining the associated Dirac equations.

q-alg↗
arXiv1997

Quasiclassical Limit in q-Deformed Systems, Noncommutativity and the q-Path Integral

Different analogs of quasiclassical limit for a q-oscillator which result in different (commutative and non-commutative) algebras of ``classical'' observables are derived. In particular, this gives the q-deformed Poisson brackets in terms of variables on the quantum planes. We consider the Hamiltonian made of special combination of operators (the analog of even operators in Grassmann algebra) and discuss q-path integrals constructed with the help of contracted ``classical'' algebras.

q-alg↗
arXiv1997

Tetrahedron Reflection Equation

Reflection equation for the scattering of lines moving in half-plane is obtained. The corresponding geometric picture is related with configurations of half-planes touching the boundary plane in 2+1 dimensions. This equation can be obtained as an additional to the tetrahedron equation consistency condition for a modified Zamolodchikov algebra.

hep-th↗
arXiv1996

Irreducible representations of deformed oscillator algebra and q-special functions

Different generators of a deformed oscillator algebra give rise to one-parameter families of $q$-exponential functions and $q$-Hermite polynomials related by generating functions. Connections of the Stieltjes and Hamburger classical moment problems with the corresponding resolution of unity for the $q$-coherent states and with 'coordinate' operators - Jacobi matrices, are also pointed out.

q-alg↗
arXiv1996

New rational solutions of Yang-Baxter equation and deformed Yangians

In this paper a class of new quantum groups is presented: deformed Yangians. They arise from rational solutions of the classical Yang-Baxter equation of the form $c_2 /u + const$ . The universal quantum $R$-matrix for a deformed Yangian is described. Its image in finite-dimensional representaions of the Yangian gives new matrix rational solutions of the Yang-Baxter equation (YBE).

q-alg↗
arXiv1995

Quantum group covariant systems

The meaning of quantum group transformation properties is discussed in some detail by comparing the (co)actions of the quantum group with those of the corresponding Lie group, both of which have the same algebraic (matrix) form of the transformation. Various algebras are considered which are covariant with respect to the quantum (super) groups $SU_q(2),\; SU_q(1, 1),\; SU_q(1|1),\; SU_q(n), \\ SU_q(m|n),\; OSp_q(1|2)$ as well as deformed Minkowski space-time algebras.

q-alg↗
arXiv1995

Gauss Decomposition for Quantum Groups and Duality

The Gauss decomposition of quantum groups and supergroups are considered. The main attention is paid to the R-matrix formulation of the Gauss decomposition and its properties as well as its relation to the contraction procedure. Duality aspects of the Gauss decomposition are also touched. For clarity of exposition a few simple examples are considered in some details.

q-alg↗