The Inhomogeneous Quantum Invariance Group of The Two Parameter Deformed Boson Algebra

We consider two parameter deformed boson algebra and investigate the inhomogeneous invariance quantum group of this system. We find the R-matrix which collects all information about the non-commuting structure of the quantum group. We extend our study to the d-dimensional case.     

A Quantum Analogue of the First Fundamental Theorem of Classical Invariant Theory

We establish a noncommutative analogue of the first fundamental theorem of classical invariant theory. For each quantum group associated with a classical Lie algebra, we construct a noncommutative associative algebra whose underlying vector space forms a module for the quantum group and whose algebraic structure is preserved by the quantum group action. The subspace of invariants is shown to form a subalgebra, which is finitely generated. We determine generators of this subalgebra of invariants and determine their commutation relations. In each case considered, the noncommutative modules we construct are flat deformations of their classical commutative analogues. Our results are therefore noncommutative generalisations of the first fundamental theorem of classical invariant theory, which follows from our results by taking the limit as q → 1. Our method similarly leads to a definition of quantum spheres, which is a noncommutative generalisation of the classical case with orthogonal quantum group symmetry.     

The multi-dimensional <Emphasis Type="Italic">q</Emphasis>-deformed bosonic Newton oscillator and its inhomogeneous quantum invariance group

We obtain the inhomogeneous invariance quantum group for the multi-dimensional q-deformed bosonic Newton oscillator algebra. The homogenous part of this quantum group is given by the multiparameter quantum group $ GL_{X;q_{ij} } $ GL_{X;q_{ij} } of Schirrmacher where q _ij’s take some special values. We find the R-matrix which gives the non-commuting structure of the quantum group for the two dimensional case.     

Associated quantum vector bundles and symplectic structure on a quantum space

We define a quantum generalization of the algebra of functions over an associated vector bundle of a principal bundle. Here the role of a quantum principal bundle is played by a Hopf-Galois extension. Smash products of an algebra times a Hopf algebra H are particular instances of these extensions, and in these cases we are able to define a differential calculus over their associated vector bundles without requiring the use of a (bicovariant) differential structure over H. Moreover, if H is coquasitriangular, it coacts naturally on the associated bundle, and the differential structure is covariant.We apply this construction to the case of the finite quotient of the SL _q(2) function Hopf algebra at a root of unity (q ³ = 1) as the structure group, and a reduced 2-dimensional quantum plane as both the base manifold and fibre, getting an algebra which generalizes the notion of classical phase space for this quantum space. We also build explicitly a differential complex for this phase space algebra, and find that levels 0 and 2 support a (co)representation of the quantum symplectic group. On this phase space we define vector fields, and with the help of the Sp _q structure we introduce a symplectic form relating 1-forms to vector fields. This leads naturally to the introduction of Poisson brackets, a necessary step to do classical mechanics on a quantum space, the quantum plane.     

Quantum group gauge theory on quantum spaces

We construct quantum group-valued canonical connections on quantum homogeneous spaces, including aq-deformed Dirac monopole on the quantum sphere of Podles with quantum differential structure coming from the 3D calculus of Woronowicz onSU _q (2). The construction is presented within the setting of a general theory of quantum principal bundles with quantum group (Hopf algebra) fibre, associated quantum vector bundles and connection one-forms. Both the base space (spacetime) and the total space are non-commutative algebras (quantum spaces).Supported by St. John's College, Cambridge and KBN grant 202189101     

Quantum spin chains with quantum group symmetry

We consider actions of quantum groups on lattice spin systems. We show that if an action of a quantum group respects the local structure of a lattice system, it has to be an ordinary group. Even allowing weakly delocalized (quasi-local) tails of the action, we find that there are no actions of a properly quantum group commuting with lattice translations. The non-locality arises from the ordering of factors in the quantum groupC ^*-algebra, and can be made one-sided, thus allowing semi-local actions on a half chain. Under such actions, localized quantum group invariant elements remain localized. Hence the notion of interactions invariant under the quantum group and also under translations, recently studied by many authors, makes sense even though there is no global action of the quantum group. We consider a class of such quantum group invariant interactions with the property that there is a unique translation invariant ground state. Under weak locality assumptions, its GNS representation carries no unitary representation of the quantum group.Supported in part by NSF Grant # PHY90-19433 A02Copyright © 1995 by the authors. Faithful reproduction of this article by any means is permitted for non-commercial purposes.     

UniversalR-matrix for quantized (super)algebras

For quantum deformations of finite-dimensional contragredient Lie (super)algebras we give an explicit formula for the universalR-matrix. This formula generalizes the analogous formulae for quantized semisimple Lie algebras obtained by M. Rosso, A. N. Kirillov, and N. Reshetikhin, Ya. S. Soibelman, and S. Z. Levendorskii. Our approach is based on careful analysis of quantized rank 1 and 2 (super)algebras, a combinatorial structure of the root systems and algebraic properties ofq-exponential functions. We don't use quantum Weyl group.     

Quantum Lie Algebras, Their Existence, Uniqueness and q-Antisymmetry

Quantum Lie algebras are generalizations of Lie algebras which have the quantum parameter h built into their structure. They have been defined concretely as certain submodules of the quantized enveloping algebras . On them the quantum Lie product is given by the quantum adjoint action. Here we define for any finite-dimensional simple complex Lie algebra an abstract quantum Lie algebra independent of any concrete realization. Its h-dependent structure constants are given in terms of inverse quantum Clebsch-Gordan coefficients. We then show that all concrete quantum Lie algebras are isomorphic to an abstract quantum Lie algebra . In this way we prove two important properties of quantum Lie algebras: 1) all quantum Lie algebras associated to the same are isomorphic, 2) the quantum Lie product of any is q-antisymmetric. We also describe a construction of which establishes their existence. Received: 23 May 1996 / Accepted: 17 October 1996     

Quantum mechanics on Riemannian manifold in Schwinger's quantization approach II

The extended Schwinger quantization procedure is used for constructing quantum mechanics on a manifold with a group structure. The considered manifold M is a homogeneous Riemannian space with the given action of an isometry transformation group. Using the identification of M with the quotient space G/H, where H is the isotropy group of an arbitrary fixed point of M, we show that quantum mechanics on G/H possesses a gauge structure, described by a gauge potential that is the connection 1-form of the principal fiber bundle G(G/H, H). The coordinate representation of quantum mechanics and the procedure for selecting the physical sector of the states are developed. Received: 27 June 2000 / Revised version: 10 May 2001 / Published online: 19 July 2001     

Integrable operator representations of ℝ q 2 ,X q,γ andSL q (2, ℝ)

We introduce and study action of quantum groups on skew polynomial rings and related rings of quotients. This leads to a q-deformation of the Gel'fand-Kirillov conjecture which we partially prove. We propose a construction of automorphisms of certain non-commutative rings of quotients coming from complex powers of quantum group generators; this is applied to explicit calculation of singular vectors in Verma modules over . We finally give a definition of aq-connection with coefficients in a ring of skew polynomials and study the structure of quantum group modules twisted by aq-connection.Supported by the Japan Society for the Promotion of Science Post-Doctoral Fellowship for Foreign Researchers in Japan.     

Exchange Dynamical Quantum Groups

For any simple Lie algebra ? and any complex number q which is not zero or a nontrivial root of unity, %but may be equal to 1 we construct a dynamical quantum group (Hopf algebroid), whose representation theory is essentially the same as the representation theory of the quantum group U _q (?). This dynamical quantum group is obtained from the fusion and exchange relations between intertwining operators in representation theory of U _q (?), and is an algebraic structure standing behind these relations. Received: 24 March 1998 / Accepted: 14 February 1999     

Invariance quantum group of the fermionic oscillator

The fermionic oscillator defined by the algebraic relations cc ^* +c ^* c = 1 and c ² = 0 admits the homogeneous group O(2) as its invariance group. We show that the structure of the inhomogeneus invariance group of this oscillator is a quantum group. Received: 15 July 2002 / Revised version: 14 October 2002 / Published online: 19 February 2003     

Quantum deformation of space-time symmetries and interactions

We discuss quantum deformations of Lie algebra as described by the noncoassociative modification of its coalgebra structure. We consider for simplicity the quantum D = 1 Galilei algebra with four generators: energy H, boost B, momentum P and central generator M (mass generator). We describe the nonprimitive coproducts for H and B and show that their noncocommutative and noncoassociative structure is determined by the two-body interaction terms. Further we consider the case of physical Galilei symmetry in three dimensions. Finally we discuss the noninteraction theorem for manifestly covariant two-body systems in the framework of quantum deformations of D = 4 Poincaré algebra and a possible way out.     

Symmetries of quantum spaces. Subgroups and quotient spaces of quantumSU(2) andSO(3) groups

We prove that each action of a compact matrix quantum group on a compact quantum space can be decomposed into irreducible representations of the group. We give the formula for the corresponding multiplicities in the case of the quotient quantum spaces. We describe the subgroups and the quotient spaces of quantumSU(2) andSO(3) groups.     

Relating the approaches to quantised algebras and quantum groups

This paper constructs two representations of the quantum groupU _q g' by exploiting its quotient structure and the quantum double construction. Here the quantum group is taken as the dual to the quantised algebraU _q g, a one parameter deformation of the universal enveloping algebra of the Lie algebra g, as in Drinfel'd [6] and Jimbo [10]. From the two representations, the Hopf structure of the quantised algebraU _q g is reexpressed in a matrix format. This is the very structure given by Faddeev et al. [7], in their approach to defining quantum groups and quantised algebras via the quantisation of the function space of the associated Lie group to g.Supported by a SERC studentship     

弱激光改善血液携氧功能机制分析及临床研究

本文用群论、量子力学、量子统计力学和非线性科学理论,结合血液中与携氧有关成分(卟啉)的结构功能对激光改善血液的携氧、输氧机制作了分析.量子统计和群论分析结果表明F-F’>0(F、F’分别为有、无激光作用时系统的自由能),这些结果能较好解释我们用激光血管内照射的临床治疗及血氧检测结果.     

Quantum double and differential calculi

We show that bicovariant bimodules as defined by Woronowicz are in one-to-one correspondence with the Drinfeld quantum double representations. We then prove that a differential calculus associated to a bicovariant bimodule of dimension n is connected to the existence of a particular (n+1)-dimensional representation of the double. An example of bicovariant differential calculus on the nonquasitriangular quantum group E _q (2) is developed. The construction is studied in terms of Hochschild cohomology and a correspondence between differential calculi and 1-cocycles is proved. Some differences of calculi on quantum and finite groups with respect to Lie groups are stressed.     

Spin effects in diffractive Q $$\bar Q$$ production and compass experimentproduction and compass experiment

It is known that the inhomogeneous quantum group IGL _q,r(2) can be constructed as a quotient of the multiparameter q-deformation of GL(3). We show that a similar result holds for the inhomogeneous Jordanian deformation and exhibits its Hopf structure.     

2T Physics, Scale Invariance and Topological Vector Fields

We construct, in classical two-time physics, the necessary structure for the most general configuration space formulation of quantum mechanics containing gravity in d+2 dimensions. This structure is composed of a symmetric Riemannian metric tensor and of a vector field that defines a section of a flat U(1) bundle over space-time. This construction is possible because of the existence of a finite local scale invariance of the Hamiltonian and because two-time physics contains, at the classical level, a local generalization of the discrete duality symmetry between position and momentum that underlies the structure of quantum mechanics.