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By building a multi-fermion realization of the quantum algebra SU q (2), we generalized the Lipkin-Meshkov-Glick (LMG) model to test the validity of many-body approximation methods. Our general LMG model incorporates the quantum group symmetry of SU q (2) and the many-body force besides the two-body one in many-fermions systems, which give a probe of the dynamic process that the quantum group symmetry should describe. The exact solvability is still guaranteed by the finite dimensional representation of SU q (2). The deformation parameter q is introduced in the generalized model to characterize the difference between this model and the original one. Especially, it leads to the crossing of different energy levels.

In this paper the representation theory of q-boson algebra in the non-generic case thatq is a root of unity is studied from a purely mathematical point of view. Emphasis is placed on its cyclic representations. Various cyclic representations of the q-boson algebra are thereby constructed in explicit form. Using them, we obtain the cyclic representations of quantum superalgebrasUqosp (1, 2),Uqosp(1, 4) and quantum algebra (C2)q through the q-deformed-boson realization method.

This paper is devoted to building the representation theory of the general deformation for su(2) and then using it to generalize the quasi-exactly-solvable quantal (QESQ) problems. For the finite dimensional and infinite dimensional representations, two classes of generalized QESQ models are respectively constructed in terms of the differential realization of the general deformation of su(2). When the deformation is the q-deformation in the quantum group theory, the QESQ model is discussed in detail by associating it with the nonlinear precession of high spins in an external field.

Endowing a simple C algebra generated by two commuting elements and a unit with a noncocommutative Hopf algebra structure, a new quantum double and the corresponding universal R matrix for the Yang–Baxter equation is constructed. The finite‐dimensional representations of the quantum double is also studied and a concrete R matrix is presented as an example.

We investigate the St\"{u}ckelberg oscillations of a spin-1 Bose-Einstein condensate subject to a spatially inhomogeneous transverse magnetic field and a periodic longitudinal field. We show that the time-domain St\"{u}ckelberg oscillations result in interference fringes in the density profiles of all spin components due to the spatial inhomogeneity of the transverse field. This phenomenon represents the Landau-Zener-St\"{u}ckelberg interference in the space-domain. Since the magnetic dipole-dipole interaction between spin-1 atoms induces an inhomogeneous effective magnetic field, interference fringes also appear if a dipolar spinor condensate is driven periodically. We also point out some potential applications of this spatial Landau-Zener-St\"{u}kelberg interference., Comment: 8 pages, 7 figures

This paper reviews quantum spin squeezing, which characterizes the sensitivity of a state with respect to an SU(2) rotation, and is significant for both entanglement detection and high-precision metrology. We first present various definitions of spin squeezing parameters, explain their origin and properties for typical states, and then discuss spin-squeezed states produced with the Ising and the nonlinear twisting Hamiltonians. Afterwards, we explain correlations and entanglement in spin-squeezed states, as well as the relations between spin squeezing and quantum Fisher information, where the latter plays a central role in quantum metrology. We also review the applications of spin squeezing for detecting quantum chaos and quantum phase transitions, as well as the influence of decoherence on spin-squeezed states. Finally, several experiments are discussed including: producing spin squeezed states via particle collisions in Bose-Einstein condensates, mapping photon squeezing onto atomic ensembles, and quantum non-demolition measurements., 99 pages, 25 figures

Taking into account the deformation of the Fermi surface, we investigate the spin-singlet and -triplet BCS pairings in a mixture of fermionic polar molecules with two different hyperfine states. In particular, we explore the relation between the critical temperatures and the Fock-exchange interaction. We also show that, by tuning short-range interaction between interspecies molecules, the singlet- and triplet-paired superfluids may coexist.

We consider a system where a waveguide is coupled to a cavity embedded with a two-level system (TLS), and study the effects when a two-photon quantum state is injected into the waveguide. The wave function of two outgoing photons is exactly solved using the Lehmann-Symanzik-Zimmermann (LSZ) reduction formalism. Our results explicitly exhibit the photon blockade effects in the strong atom-cavity coupling regime. The quantum statistical characters of the outgoing photons, including the photon bunching and anti-bunching behaviors, are also investigated in both the strong and weak coupling regimes. These results agree with the observations of recent experiments., 4 pages, 3 figures

It is typically assumed, without justification, that a weak coupling between a system and a bath is a necessary condition for the equivalence of a canonical ensemble and a microcanonical ensemble. For instance, in a canonical ensemble, temperature emerges if the system and the bath are uncoupled or weakly coupled. We investigate the validity region of this weak coupling approximation, using a coupled composite-spin system. Our results show that the spin coupling strength can be as large as the level spacing of the system, indicating that the weak coupling approximation has a much wider region of validity than usually expected., 5 pages, 5 figures, 4 tables, in RevTex 4.1 format

We study the measurement-induced enhancement of the spontaneous decay (called quantum anti-Zeno effect) for a two-level subsystem, where measurements are treated as couplings between the excited state and an auxiliary state rather than the von Neumann's wave function reduction. The photon radiated in a fast decay of the atom, from the auxiliary state to the excited state, triggers a quasi-measurement, as opposed to a projection measurement. Our use of the term "quasi-measurement" refers to a "coupling-based measurement". Such frequent quasi-measurements result in an exponential decay of the survival probability of atomic initial state with a photon emission following each quasi-measurement. Our calculations show that the effective decay rate is of the same form as the one based on projection measurements. What is more important, the survival probability of the atomic initial state which is obtained by tracing over all the photon states is equivalent to the survival probability of the atomic initial state with a photon emission following each quasi-measurement to the order under consideration. That is because the contributions from those states with photon number less than the number of quasi-measurements originate from higher-order processes., 7 pages, 3 figures

We study the multipartite correlations of the multi-atom dark states, which are characterized by the atomic squeezing beyond the pairwise entanglement. It is shown that, in the photon storage process with atomic ensemble via electromagnetically induced transparency (EIT) mechanism, the atomic squeezing and the pairwise entanglement can be created by adiabatically manipulating the Rabi frequency of the classical light field on the atomic ensemble. We also consider the sudden death for the atomic squeezing and the pairwise entanglement under various decoherence channels. An optimal time for generating the greatest atomic squeezing and pairwise entanglement is obtained by studying in details the competition between the adiabatic creation of quantum correlation in the atomic ensemble and the decoherence that we describe with three typical decoherence channels., Comment: 11 pages, 13 figures

We study a washboard potential in a light-driven ensemble of three-level atoms induced by the externally appliedmagnetic fields under electromagnetically induced transparency (EIT). We find that a dark-state polariton, which is formed by an atomic ensemble dressed by the photon can display a typical quantum interference effect (i.e., the quasiparticle Bloch oscillation). The dark-state polariton is subjected to a washboard configuration potential with a spatially uniform force and a periodic potential provided by the linearly and periodically external magnetic fields, respectively. Due to the slower group velocity of the photons in the EIT medium, the period of Bloch oscillation can achieve a millisecond order of magnitude. Also, in our protocol, the Bloch oscillation of the dark-state polariton is feasibly modulated through the applied control and magnetic fields.

We study the role of the dimer structure of light-harvesting complex II (LH2) in excitation transfer from the LH2 (without a reaction center (RC)) to the LH1 (surrounding the RC), or from the LH2 to another LH2. The excited and un-excited states of a bacteriochlorophyll (BChl) are modeled by a quasi-spin. In the framework of quantum open system theory, we represent the excitation transfer as the total leakage of the LH2 system and then calculate the transfer efficiency and average transfer time. For different initial states with various quantum superposition properties, we study how the dimerization of the B850 BChl ring can enhance the transfer efficiency and shorten the average transfer time., Comment: 11 pages, 6 figures

A resonant two level atom doped in one dimensional waveguide behaves as a mirror, but this single-atom "mirror" can only reflect single photon perfectly at a specific frequency. For a one dimensional coupled-resonator waveguide, we propose to extend the perfect reflection region from a specific frequency to a wide band by placing many atoms individually in the resonators in a finite coordinate region of the waveguide. Such a doped resonator array promises us to control the propagation of a practical photon wave packet with certain momentum distribution instead of a single photon, which is ideally represented by a plane wave with specific momentum. The studies based on the discrete-coordinate scattering theory display that such hybrid structure indeed provides a near-perfect reflection for single photon in a wide band. We also calculated photon group velocity distribution, which shows that the perfect reflection with wide band exactly corresponds to the stopping light region., 8 pages, 10 figures

We study supersymmetric (SUSY) responses to a photoassociation process in a mixture of Bose molecules $b$ and Fermi atoms $f$ which turn to mutual superpartners for a set of proper parameters. We consider the molecule $b$ to be a bound state of the atom $f$ and another Fermi atom $F$ with different species. The $b$-$f$ mixture and a free $F$ atom gas are loaded in an optical lattice. The SUSY nature of the mixture can be signaled in the response to a photon induced atom-molecule transition: While two new types of fermionic excitations, an individual $b$ particle-$f$ hole pair continuum and the Goldstino-like collective mode, are concomitant for a generic $b$-$f$ mixture, the former is completely suppressed in the SUSY $b$-$f$ mixture and the zero-momentum mode of the latter approaches to an exact eigenstate. This SUSY response can be detected by means of the spectroscopy method, e.g., the photoassociation spectrum which displays the molecular formation rate of $% Ff\to b$., Comment: 5 pages, 3 figures

We study coherent energy transfer of a single excitation and quantum entanglement in a dimer, which consists of a donor and an acceptor modeled by two two-level systems. Between the donor and the acceptor, there exists a dipole-dipole interaction, which provides the physical mechanism for coherent energy transfer and entanglement generation. The donor and the acceptor couple to two independent heat baths with diagonal couplings that do not dissipate the energy of the non-coupling dimer. Special attention is paid to the effect on single-excitation energy transfer and entanglement generation of the energy detuning between the donor and the acceptor and the temperatures of the two heat baths. It is found that, the probability for single-excitation energy transfer largely depends on the energy detuning in the low temperature limit. Concretely, the positive and negative energy detunings can increase and decrease the probability at the steady state, respectively. In the high temperature limit, however, the effect of the energy detuning on the probability is neglectably small. We also find that the probability is neglectably dependent on the bath temperature difference of the two heat baths. In addition, it is found that quantum entanglement can be generated in the process of coherent energy transfer. As the bath temperature increases, the generated steady state entanglement decreases. For a given bath temperature, the steady-state entanglement decreases with the increasing of the absolute value of the energy detuning., Comment: 13 pages, 11 figures

Through a general q-boson realization of quantum algebra sl q(2) and its universal R matrix an operator R matrix with many parameters is obtained in terms of q-boson operators. Building finite-dimensional representations of q-boson algebra, we construct various colored R matrices associated with nongeneric representations of sl q(2) with dimension-independent parameters. The “nonstandard” R matrices obtained by Lee-Couture and Murakami are their special examples. We also study the factorizable structure of some Rmatrices for the indecomposable representations used in its construction.

In this paper it is proved that the nontrivial finite‐dimensional irreducible representations of the quantum matrix algebra Mq(2) [or the coordinate ring of the quantum group GLq(2)] exist only when q is a root of unity, and thereby construct such representations for Mq(2). Some finite‐dimensional indecomposable representations of Mq(2) are also explicitly presented. Finally, the q‐boson realization of Mq(2) is discussed, and it is shown that it works in this case as well as in the quantum algebra case.

We generalise the conception of q-analogue coherent state to the cases of charged coherent state and charged, hypercharged coherent state. Their explicit expressions in the ordinary Fock space are derived.

Both the highest weight representations and the cyclic representation of the quantum superalgebra Uqosp(1, 2)=A are explicitly constructed in this letter. By making use of the obtained finite dimensional representations with highest weights, the new R-matrices (solutions) of the Yang-Baxter equation are obtained through the universal R-matrix. In the main discussion of this paper emphasis is placed on the case that q is root of unity.

For pt.I see ibid., vol.24, p.3265, (1991). Applying the method of the q-deformed boson realization for the quantum group to the case where q is a root of unity the authors establish a set of standard basis for the representation space of the quantum universal enveloping algebra Uq(Cl) associated with a typical subalgebra chain of Uq(Cl). On this basis they systematically obtain irreducible and indecomposable representations of Uq(Cl) and its subalgebras. They discuss Uq(C2) in detail.

A q-deformed oscillator system with quantum group SLq(l) symmetry is constructed in terms of the q-deformed boson operators. The energy level degeneracy associated with the irreducible representation of SLq(l) is analysed, especially for the case that q is a root of unity.

The authors introduce the concept of cyclic boson algebra and study its representations. Using this algebra to realize slq(3), they construct the cyclic representation of the quantum universal enveloping algebra Uq(sl(3)) identical to slq(3) on the q-Fock space. Restricting this representation to the subalgebra slq(2) (slq(3)), they naturally obtain the irreducible cyclic representations of slq(2).

Using the Born-Oppenheimer approximation, we derive an effective Hamiltonian for an optomechanical system that leads to a nonlinear Kerr effect in the system's vacuum. The oscillating mirror at one edge of the optomechanical system induces a squeezing effect in the intensity spectrum of the cavity field. A near-resonant laser field is applied at the other edge to drive the cavity field, in order to enhance the Kerr effect. We also propose a quantum-nondemolition-measurement setup to monitor a system with two cavities separated by a common oscillating mirror, based on our effective Hamiltonian approach., Comment: 4 pages, 2 figures

We propose a quantum storage scheme independent of the current time-control schemes, and study a "quantum data bus" (transmission line resonator) in a hybrid system consisting of a circuit QED system integrated with a cold molecular ensemble. Here, an effective interaction between charge qubit and molecule is mediated by the off-resonate field in the data bus. Correspondingly, the charge state can be mapped into the collective quasi-spin state of the molecular ensemble via the standard dark state based adiabatic manipulation., Comment: 6 pages, 6 figures

In order to witness multipartite correlations beyond pairwise entanglement, spin-squeezing parameters are analytically calculated for a spin ensemble in a collective initial state under three different decoherence channels. It is shown that, in analogy to pairwise entanglement, the spin squeezing described by different parameters can suddenly become zero at different vanishing times. This finding shows the general occurrence of sudden vanishing phenomena of quantum correlations in many-body systems, which here is referred to as spin-squeezing sudden death (SSSD). It is shown that the SSSD usually occurs due to decoherence and that SSSD never occurs for some initial states in the amplitude-damping channel. We also analytically obtain the vanishing times of spin squeezing., 16 pages, 4 figures, 2 tables, final extended version

We theoretically study how to control transport, bound states, and resonant states of a single photon in a one-dimensional coupled-cavity array. We find that the transport of a single photon in the cavity array can be controlled by tuning the frequency of either one or two cavities. If one of the cavities in the array has a tunable frequency, and its frequency is tuned to be larger (or smaller) than those of other cavities, then there is a photon bound state above (or below) the energy band of the coupled cavity array. However, if two cavities in the array have tunable frequencies, then there exist both bound states and resonant states. When the frequencies of the two cavities are chosen to be much larger than those of other cavities and the hopping couplings between any two nearest-neighbor cavities are weak, a single photon with a resonant wave vector can be trapped in the region between the two frequency-tunable cavities. In this case, a quantum supercavity can be formed by these two frequency-tunable cavities. We also study how to apply this photon transport control to an array of coupled superconducting transmission line resonators., 10 pages, 11 figures, references updated

We study the optic control for birefringence of a polarized light by an atomic ensemble with a tripod configuration, which is mediated by the electromagnetically induced transparency with a spatially inhomogeneous laser. The atom ensemble splits the linearly polarized light ray into two orthogonally-polarized components, whose polarizations depend on quantum superposition of the initial states of the atom ensemble. Accompanied with this splitting, the atom ensemble behaves as a birefringent lens, which allows one polarized light ray passing through straightly while focus another orthogonal to this polarization with finite aberration of focus., 4 pages, 3 figures

We propose a special cavity design that is constructed by terminating a one-dimensional waveguide with a perfect mirror at one end and doping a two-level atom at the other. We show that this atom plays the intrinsic role of a semi-transparent mirror for single photon transports such that quasi-normal modes (QNM's) emerge spontaneously in the cavity system. This atomic mirror has its reflection coefficient tunable through its level spacing and its coupling to the cavity field, for which the cavity system can be regarded as a two-end resonator with a continuously tunable leakage. The overall investigation predicts the existence of quasi-bound states in the waveguide continuum. Solid state implementations based on a dc-SQUID circuit and a defected line resonator embedded in a photonic crystal are illustrated to show the experimental accessibility of the generic model., Comment: 4 pages,5 figures, Comments welcome

The long time accumulation of the \textit{random} actions of a single particle "reservoir" on its coupled system can transfer some temperature information of its initial state to the coupled system. This dynamic process can be referred to as a quantum thermalization in the sense that the coupled system can reach a stable thermal equilibrium with a temperature equal to that of the reservoir. We illustrate this idea based on the usual micromaser model, in which a series of initially prepared two-level atoms randomly pass through an electromagnetic cavity. It is found that, when the randomly injected atoms are initially prepared in a thermal equilibrium state with a given temperature, the cavity field will reach a thermal equilibrium state with the same temperature as that of the injected atoms. As in two limit cases, the cavity field can be cooled and "coherently heated" as a maser process, respectively, when the injected atoms are initially prepared in ground and excited states. Especially, when the atoms in equilibrium are driven to possess some coherence, the cavity field may reach a higher temperature in comparison with the injected atoms. We also point out a possible experimental test for our theoretical prediction based on a superconducting circuit QED system., 9 pages,4 figures

We propose and study an approach to realize quantum switch for single-photon transport in a coupled superconducting transmission line resonator (TLR) array with one controllable hopping interaction. We find that the single-photon with arbitrary wavevector can transport in a controllable way in this system. We also study how to realize controllable hopping interaction between two TLRs via a superconducting quantum interference device (SQUID). When the frequency of the SQUID is largely detuned from those of the two TLRs, the variables of the SQUID can be adiabatically eliminated and thus a controllable interaction between two TLRs can be obtained., 4 pages,3 figures

We study the topological properties of quantum states for the spinless particle hopping in a Mobius ladder. This system can be regarded as a molecular device possibly engineered from the Mobius annulenes, which enjoys a pseudospin orbital interaction described by a non-Abelian gauge structure. It results from the nontrivial topology of configuration space and results in various observable effects, such as optical spectral splitting. The transmission spectrum through the Mobius molecular device is calculated to demonstrate a topological effect as destructive interferences in the conduction band. The induced interaction also leads to an entanglement between the transverse and longitudinal modes for any locally factorized state.

We propose to the create vortices in spin-1 condensates via magnetic dipole-dipole interaction. Starting with a polarized condensate prepared under large axial magnetic field, we show that by gradually inverting the field, population transfer among different spin states can be realized in a controlled manner. Under optimal condition, we generate a doubly quantized vortex state containing nearly all atoms in the condensate. The resulting vortex state is a direct manifestation of the dipole-dipole interaction and spin textures in spinor condensates. We also point out that the whole process can be qualitatively described by a simple rapid adiabatic passage model., Comment: 4 pages, 4 figures

We study the nonlinear dynamics of collective excitation in a $N$-site $XXZ$ quantum spin chain, which is manipulated by an oblique magnetic field. We show that, when the tilted field is applied along the magic angle $\theta_{0} =\pm\arccos \sqrt{1/3}$, the anisotropic Heisenberg spin chain becomes isotropic and thus an free propagating spin wave is stimulated. And in the regime of the tilted angle larger and smaller then the magic angle, two types of nonlinear excitations appear, which are bright soliton and dark soliton., Comment: 7 pages 4 figures

The observation of quantized nanomechanical oscillations by detecting femtometer-scale displacements is a significant challenge for experimentalists. We propose that a phonon blockade can serve as a signature of quantum behavior in nanomechanical resonators. In analogy to the photon blockade and Coulomb blockade for electrons, the main idea for phonon blockade is that the second phonon cannot be excited when there is one phonon in the nonlinear oscillator. To realize phonon blockade, a superconducting quantum two-level system is coupled to the nanomechanical resonator and is used to induce the phonon self-interaction. Using Monte Carlo simulations, the dynamics of the induced nonlinear oscillator is studied via the Cahill-Glauber s-parametrized quasiprobability distributions. We show how the oscillation of the resonator can occur in the quantum regime and demonstrate how the phonon blockade can be observed with the currently accessible experimental parameters.

We study how the non-adiabatic effect causes the observable fluctuation in the "geometric phase" for a two-level system, which is defined as the experimentally measurable quantity in the adiabatic limit. From the Rabi's exact solution to this model, we give a reasonable explanation to the experimental discovery of phase fluctuation in the superconducting circuit system [P. J. Leek, \textit{et al}., Science \textbf{318}, 1889 (2007)], which seemed to be regarded as the conventional experimental error., Comment: 4 pages, 3 figures

We study the dynamic sensitivity of an atomic ensemble dressed by a single-mode cavity field (called a photon-dressed atomic ensemble), which is described by the Dicke model near the quantum critical point. It is shown that when an extra atom in a pure initial state passes through the cavity, the photon-dressed atomic ensemble will experience a quantum phase transition, showing an explicit sudden change in its dynamics characterized by the Loschmidt echo of this quantum critical system. With such dynamic sensitivity, the Dicke model can resemble to the cloud chamber for detecting a flying particle by the enhanced trajectory due to the classical phase transition., Comment: 7 pages, 4 figures

The method of q-deformed boson realization is extended to the case that q is a root of unity. By using the method, many irreducible and indecomposable representations of each subalgebra in a subalgebra chain of the quantum universal enveloping algebra (quantum algebra) (Cl)q are obtained on the q-deformed Fock space and its quotient space. As an example, the representations of (C3)q are constructed in explicit forms. Besides, by embedding (A1)q = (C1)q, into the subalgebra chain, a new realisation of (A1)q and its representations are obtained.

In this Letter a series of new solutions for the quantum Yang-Baxter equation is obtained by constructing a new boson realization of λ-dependent representations of the quantum algebra U q sl(3) with q a root of unity.

A general form of solutions of braid relations associated with Vj(X)V1/2 for Su(2) is calculated. A Yang-Baxterization prescription is presented to generate the corresponding solutions of Yang-Baxter equations. The authors show that this general form of solutions gives rise to a new construction of quantum group structure even though the q-commutation relations are kept standard.

When q 1s a root of unity, the representations of the quantum universal enveloping algebra slq(2) with multiplicity two are constructed from the q-deformed boson realization with an arbitrary parameter which is in a very general form and is first presented in this Letter. The new solutions to the Yang-Baxter equation are obtained from these representations through the universal R-matrix.

The Wei-Norman algebraic method is used to solve the evolution of electron in two-dimensional cases in this letter. We show that the coherent states of electron in two dimensions can be generated by the interaction of electron with a certain type of external electromagnetic field.

In this paper a new q‐deformed boson realization of quantum algebra slq(n+1) is presented for the first time, and its representations are obtained in the nongeneric case that q is a root of unity. A new type of slq(2) R‐matrices are systematically constructed through the universal R‐matrix.

By generalizing De Concini and Kac's cyclic representation theory of quantum groups at roots of unity, the cyclic representations of the quantum superalgebra Uq osp(2, 1) are constructed in three classes: irreducible representations with single multiplicities, irreducible representations with the multiplicities larger than one, and indecomposable representations.

Verma representation theory for classical Lie algebra is extended to study the representation of quantum universal enveloping algebra (quantum algebra) for the nongeneric case that q is a root of unity. On certain subspaces and quotient spaces of the Verma space, finite- and infinite-dimensional irreducible or indecomposable representations of slq(3)=Uq(sl(3)) are obtained in explicit matrix forms.

The properties of q-deformed boson operators with non-generic q (q is a root of unity) are analysed by using the representation theory method and their finite-dimensional representations are thereby obtained. Based on this discussion, reducibilities and decompositions of q-deformed boson-realized representations of quantum universal enveloping algebra UqSL(l) are studied for non-generic cases. The explicit matrix elements of some indecomposable representations are obtained on the q-deformed Fock spaces. Necessary details are provided for UqSL(2) and UqSL(3). In particular, the Lusztig operator extension of UqSL(2) is discussed in an explicit form.

In this paper the high-order adiabatic approximation (HOAA) method is formulated in a new form so that the calculation is greatly simplified. Using this improved HOAA method, we study the Berry phase effects of the Alkali atom in a slowly-changing strong magnetic field and also the non-adiabatic transitions between the instantaneous angular momentum states. The possible observability is also pointed out.

The induced topological action relevant to the non-Abelian Berry's phase factor (NABPF) is derived by combining the high-order adiabatic approximation (HOAA) method with Kuratsuji and Iida's adiabatic path integral formulation. The non-adiabatic corrections for the transition amplitude are thereby obtained and the corresponding graphic representations are defined for convenience in the calculation. The gauge invariant properties and the dynamical implications of the non-Abelian induced topological action are also analyzed in detail. Finally, some explicit calculations for a model of physics are carried out as examples.

By establishing a new boson realization of quantum universal enveloping algebra slq(2) and its representations in the non-generic case that q is a root of unity, the authors systematically construct non-generic R-matrices of slq(2) through the universal R-matrix. These new R-matrices are not covered by the standard R-matrices constructed in terms of quantum groups and the non-standard ones obtained by using the extended Kauffman's diagram technique.

In this paper, a realization of the q‐deformed boson operators on the Fock space from a generally algebraic point of view is given. The representations of the quantum group (Cn)q are thereby constructed in terms of this realization. Some infinite‐ and finite‐dimensional representations of the q‐analog of the Heisenberg–Weyl algebra are obtained on certain quotient spaces. Finally, the q‐deformed differential realization of quantum group given by Alvarez‐Gaume, Gomez, and Sierra (Preprint CERN‐Th 5369/89) is derived from the boson realization.