Local field theory on κ-Minkowski space, star products and noncommutative translations

We consider local field theory on κ-deformed Minkowski space which is an example of solvable Lie-algebraic noncommutative structure. Using integration formula over κ-Minkowski space and κ-deformed Fourier transform, we consider for deformed local fields the reality conditions as well as deformation of action functionals in standard Minkowski space. We present explicit formulas for two equivalent star products describing CBH quantization of field theory on κ-Minkowski space. We express also via star product technique the noncommutative translations in κ-Minkowski space by commutative translations in standard Minkowski space. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000. Presented by J. Lukierski.     

Representations of the nonstandard (twisted) deformationU′ q(so n ) forq a root of unity

Irreducible representations of the algebrasU′ _q(so _n ) forq a root of unityq ^p=1 are given. The main class of these representations act onp ^N-dimensional linear space (whereN is a number of positive roots of the Lie algebra so _n ) and are given byr = dim so _n complex parameters. Some classes of degenerate irreducible representations are also described. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000. The research described in this publication was made possible in part by Grant UP1-2115 of the U.S. Civilian Research and Development Foundation for the Independent States of the Former Soviet Union (CRDF).     

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.     

q-quantum plane,ψ(q)-umbral calculus, and all that

It is known that one can formulateq-extended finite operator calculus with help of “quantumq-plane”q-commuting variablesA, B : AB − qBA ≡ [A, B]q=0. We shall recall this simple fact in its natural entourage which is the so-calledψ(q)-extension of Rota’s finite operator calculus. We aim to convince the audience that this is a natural and elementary method for formulation and treatment ofq-extended and possiblyR-extended orψ(q)-extended models for quantum-likeψ(q)-deformed oscillators. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001.     

Bicrossproduct structure and graded contractions of deformed algebras

It is shown that all members in the family of deformed Hopf algebras corresponding to the graded contractions of the inhomogeneous algebrasiso(p,q),p +q=N, have a bicrossproduct structure. Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997. This research has been partially supported by a research grant from the Spanish CICYT.     

Cartan-Weyl Basis For Quantum Affine SuperalgebraU q (\widehat{osp}(1|2))(1|2))

A Cartan-Weyl basis for the quantum affine superalgebraU _q (osp(1|2)) is constructed in an explicit form. Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997. Supported by KNB grant No. 2P 30208706. Supported by Russian Foundation for Fundamental Research, grant No. 96-01-01421.     

Embedding euclidean lie algebras into quantum structures

We show that it is possible to express the basis elements of the Lie algebra of the Euclidean group,E(2), as simple irrational functions of certainq deformed expressions involving the generators of the quantum algebraU _q (so(2, 1)). We consider implications of these results for the representation theory of the Lie algebra ofE(2). We briefly discess analogous results forU _q (so(2, 2)). Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997.     

Principal chiral models with non-constant metric

Field equations for generalized principal chiral models with non-constant metric and their possible Lax formulation are considered. Ansatz for Lax operators is taken linear in currents. Results of a complete investigation of models allowing Lax formulation with linear ansatz for Lax operators on solvable 2- and 3-dimensional groups are given; all such models appear to be almost linear. Also models on simple groupSU(2) with diagonal metric are considered; it turns out that Lax formulation exists in this case for constant metrics only. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. This work was partially supported by grant No. 1929/2001 of Fund for Development of Higher Education.     

q-Lorentz group and braided coaddition onq-Minkowski space

We present a new version ofq-Minkowski space, which has both a coaddition law and anSL _q (2, ) decomposition. The additive structure forms a braided group rather than a quantum one. In the process, we obtain aq-Lorentz group which coacts covariantly on thisq-Minkowski space.     

On braided poisson and quantum inhomogeneous groups

The well known incompatibility between inhomogeneous quantum groups and the standardq-deformation is shown to disappear (at least in certain cases) when admitting the quantum group to be braided. Braided quantumISO(p, N - p) containingSO _q (p, N - p) with |q|=1 are constructed forN=2p, 2p + 1, 2p + 2. Their Poisson analogues (obtained first) are presented as an introduction to the quantum case. Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997.     

Wave equations onq-Minkowski space

We give a systematic account of a component approach to the algebra of forms onq-Minkowski space, introducing the corresponding exterior derivative, Hodge star operator, coderivative, Laplace-Beltrami operator and Lie-derivative. Using this (braided) differential geometry, we then give a detailed exposition of theq-d'Alembert andq-Maxwell equation and discuss some of their non-trivial properties, such as for instance, plane wave solutions. For theq-Maxwell field, we also give aq-spinor analysis of theq-field strength tensor.     

On factorizing twists for the Yangian and the quantum affine algebra

We calculate explicit expressions for factorizing Drinfel’d twists in evaluation representations of the YangianY(sl₂) and of the quantum affine algebraUq . From the twists we derive a closed and representation-independent form of theR-matrices in these representations. Employing a carefully chosen basis it is possible to recover the results for the Yangian (rational case) as theq → 1 limit of the expressions for the quantum affine algebra (trigonometric case). Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001.     

On the relation of Manin’s quantum plane and quantum Clifford algebras

In a recent work we have shown that quantum Clifford algebras — i.e. Clifford algebras of an arbitrary bilinear form — are closely related to the deformed structures asq-spin groups, Hecke algebras,q-Young operators and deformed tensor products. The question to relate Manin’s approach to quantum Clifford algebras is addressed here. Explicit computations using the CLIFFORD Maple package are exhibited. The meaning of non-commutative geometry is reexamined and interpreted in Clifford algebraic terms. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

q-path integrals

We construct path integral representations for the evolution operator ofq-oscillators using Bargmann-Fock representations with commuting and non-commuting variables, the differential calculi beingq-deformed for both types of variables. The cases of real and root of unity values ofq-parameter are considered. Forq ²=–1 case we obtain a new form of Grassmann-like path integral.Presented at the 4th Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.On leave of absence from Nuclear Physics Institute, Moscow State University, 119899, Moscow, Russia.     

Braided matrix structure ofq-Minkowski space andq-Poincaré group

We clarify the relation between the approach toq-Minkowski space of Carow-Watamura et al. with an approach based on the idea of 2×2 braided Hermitean matrices. The latter are objects like super-matrices but with Bose-Fermi statistics replaced by braid statistics. We also obtain new R-matrix formulae for theq-Poincaré group in this framework.     

q-conformally covariant q-Minkowski space-time and invariant equations

We present explicitly the covariant action of the q-conformal algebra on the q-Minkowski space we proposed earlier. We also present some q-conformally invariant equations, namely a hierarchy of q-Maxwell equations, and also a q-d'Alembert equation, proposed earlier by us, in a form different from the original.     

Observables in 3d gravity and geodesic algebras

The algebra of quantum geodesics obtained by quantizing the coordinates of the Teichmller spaces is the quantumso _q(m) algebra by Nelson and Regge. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

q-Fuzzy Spheres and Quantum Differentials on Bq[SU2] and Uq(su2)

We provide a new unified construction of the two-parameter Podleś two-spheres as characterised by a projector e with trace _q (e) = 1 + λ. In our formulation the limit in which q → 1 with λ fixed is the fuzzy sphere, while the limit λ → 0 with q fixed is the standard q-deformed sphere. We show further that the non-standard Podleś spheres arise geometrically as ‘constant time slices’ of the unit hyperboloid in q-Minkowski space viewed as the braided group B _q [SU ₂]. Their localisations are then isomorphic to quotients of U _q (su ₂) at fixed values of the q-Casimir precisely q-deforming the fuzzy case. We also use transmutation and twisting theory to introduce a C_q[G_\mathbb C]{C_q[G_\mathbb {C}]} -covariant differential calculus on general B _q [G] and U _q (g), with Ω(B _q [SU ₂]) and Ω(U _q (su ₂) given in detail. To complete the picture, we show how the covariant calculus on the 3D bicrossproduct spacetime arises from Ω(C _q [SU ₂]) prior to twisting.     

Differential calculus on quantum Euclidean spheres

We study covariant differential calculus on the quantum Euclidean spheres S _q ^N−1 which are quantum homogeneous spaces with coactions of the quantum groups O _q (N). First order differential calculi on the quantum Euclidean spheres satisfying a dimension constraint are found and classified: ForN≥6, there exist exactly two such calculi one of which is closely related to the classical differential calculus in the commutative case. Higher order differential forms and symmetry are discussed. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.