Cyclic Representation of Quantum Group GL(2)q at qp=1 from the Regular Representation of M(2)q1

Regular representation of the matrix element algebra M(2)_q, of the quantum group GL(2)_q is studied, then the cyclic representation of GL(2)_q, is obtained as a quotient module of the regular representation of M(2)_q, with respect to some submodules at q^p = 1.     

Two Types of Infinite-Dimensional Representations of the Quantum Algebra OSPq(l, 4)

In this letter, we discuss the quantum superalgebra OSP_q(l,4). With the help of the SU_q(2)×U(1) basis, we construct two types of representations for the quantum superalgebra, called the-1/2-representation and the 1/2-representation. Both of them are infinitedimensional.     

Boson Representations of Subalgebra Chain for Quantum Algebra (Cl)q with q a Root of Unity

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

Multiboson Realizations of Lie and Quant urn Algebras

A unified method to construct the (multi-)boson realizations of the Lie and quantum algebras is proposed based on a universal deformation of boson (Heisenberg-Weyl) algebra and its multiboson realization. Some explicit examples, the Lie algebras sl(2) and su (Ill), q-boson algebra, quantum algebras sl(2)_q and su (l,l)_q, along with the q-ladder algebra are studied in detail. In particular, the square boson realizations of su (1,l) and su (l,l)_q are naturally obtained as special cases.     

Steady-State Analysis of a Single-Mode Laser Driven by Colored Pump Noise with Cross-Correlation Between Real and Imaginary Parts of Quantum Noise

Applying the method of the unified colored noise approximation and phase lock, we study in this paper the stationary intensity distribution of the single-mode laser driven by colored pump noise with cross-correlation between the real and imaginary parts of the quantum noise. We present a thorough discussion of how the cross-correlation λq between the real and imaginary parts of the quantum noise and the self-correlation time τ of the pump noise determine the behaviors of the stationary distribution Qst(I), the mean (I）, and the variance λ2(0) of the laser intensity. It is shown that cross-correlation intensity λq of the complex quantum noise can induce a first-order-like transition. When the pump noise is colored noise (τ≠0), improving the pump parameters monotonously will make the curves of Qst(I) exhibit reentrant phase transition. The fluctuations of laser intensity are strongly influenced by λq and τ when the laser is operated near or below threshold. Especially when τ≠0, the heights of the peaks of the curves of λ2(0)-α0 and α3(0)-α0, (here a0 is the net gain coefficient) go up as λq increases. However the entire curves λ2(0)-α0 and λ3(0)-α0 are abruptly suppressed when λq = 1, in similarity to phase transition of stationary intensity distribution.     

Inhomogeneous q-Differential Realization,Inhomogeneous q-Boson Realization of Quantum Group SU(n)q and Its Indecomposable Representations

An inhomogeneous q-differential realization (q-IHDR) and its corresponding inhomogeneous q-boson realization (q-IHBR) of quantum group SU(n)_q are studied. By making use of the q-IHBR a kind of infinite dimensional indecomposable and irreducible representations of SU(n)_q is studied on the q-Fcck space. The finite dimensional irreducible representations of SU(n), are obtained on the finite dimensional subspaces of q-Fock space. As an example, these representations of SU(2)_q are studied in detail and Jimbo standard irreducible representations are obtained.     

The Two-Variable Coherent States of the Quantum Algebra suq(l,1)

By means of the k-order q-boson realizations of su_q(1,1), a new kind of q-coherent states, the two-variable coherent states, are introduced for the quantum algebra su_q(l,1).They are relative to a definite irreducible representation of su_q(1,1) and shown to satisfy a completeness relation in a subspace of q-boson's Fock space. With the help of these coherent states the z-representation, representation in functional space, for su_q(1,1) is also discussed.     

The spin-1/2 Heisenberg Chains in Constant Magnetic Field and Coherent states1

In the ferromagnetic Heisenberg chains of XXX and XXZ types with the hidden symmetries of Lie bi-algebra su(2) and quantum bi-algebra su_q(2), we show that at thermodynamic limit the algebra contractions give the boson algebra h(4) and the q-deformed boion algebra h_q(2) as the hidden symmetries respectively. The chains in constant magnetic field are studied and the ground states and lowest excited states are given explicitly with energy spectra. The phonon (or angular momentum) excitations are shown to be bosonic for the isotropic case and q-bosonic for the anisotropic case, and the ground states and lowest excited states of the systems of the chains in field are given explicitly. We give the phonon coherent states in the isotropic Heisenberg chain and the q-coherent states of the anisotropic chain at the thermodynamic limit. The q-coherent states are shown to be a squeezed states of phonon excitations.     

Quantum Affine Superalgebra Uq(gl(1|1)) and Its

Drinfeld's realization of quantum affine superalgebra U_q(gl(1|1)) is given based on the super version of RS construction method and Gauss decomposition.     

A Categorification of Quantum \widehat{sl_2}

In this paper, we categorify the algebraU_q(\widehat{sl_2}) with the same approach as in [A. Lauda, Adv. Math. (2010), arXiv:math.QA/0803.3652;
M. Khovanov, Comm. Algebra11 (2001) 5033]. The algebra \dot{U}=\dot{U}_q(\widehat{sl_2}) is obtained fromU_q(\widehat{sl_2}) by adjoining a collection of orthogonal idempotents 1_λ,λ \in P, in which P is the weight lattice ofU_q(\widehat{sl_2}). Under such construction the algebraU is decomposed into a direct sum \bigoplus_{\lambda\in P} 1_{\lambda'}{U} 1_λ. We set the collection of λ\in P as the objects of the categoryU, 1-morphisms fromλ toλ' are given by 1_λ' }k U1_λ, and 2-morphisms are constructed by some semilinear form defined onU. Hence we get a 2-category {\cal U} from the algebraU_q{(\widehat{sl_2})}.     

The Dyson Realization of the Quantum Group SU(1,1)q and Its Coherent state

The Dyson realization of the quantum group SU(1,1)_q is constructed and the transformation matrix, which turns the Dyson realization into the Holstein Primakoff realization, is given. The coherent state for the Dyson realization is discussed and is found to be the qdeformed Weyl Heisenberg coherent state.     

Universal R-matrix of the quantum superalgebra osp(2 | 1)

A quantum analogue of the simplest superalgebra osp(2 | 1) and its finite-dimensional, irreducible representations are found. The corresponding constant solution to the Yang-Baxter equation is constructed and is used to formulate the Hopf superalgebra of functions on the quantum supergroup OSp(2 | 1).     

The Boundary States of the q-Deformed Supersymmetric t-J Model with a Boundary

The q-deformed supersymmetric t-J model on a semi-infinite lattice is diagonalized by using the level-one vertex operators of the quantum affine superalgebra U_q[\widehat{sl(2|1)}]. We give the bosonization of the boundary states.     

叠加q-相干态的量子统计性质

构造了叠加ｑ－相干态, 研究了它的压缩和反聚束效应等量子统计性质, 用数值计算的方法讨论了叠加系数、变形参数对统计性质的影响。奇偶ｑ－相干态作为特例包含在本文的一般结论之中。     

One-Parameter Indecomposable and Irreducible Representations of gl(2 | 1) Superalgebra

Using inhomogeneous boson–fermion realization, one-parameter indecomposable and irreducible representations of the gl(2 | 1) superalgebra are studied on subspace and quotient spaces of the universal enveloping algebra of Heisenberg–weyl superalgebra. All the finite-dimensional irreducible representations of one-parameter of the gl(2 | 1) superalgebra are naturally obtained as special cases. The parameter has relation to the Hubbard interaction parameter U in the Hubbard model for correlated electrons.     

The q-Boson Realizationo f Parametrized Cyclic Representations of Quantum Superalgebra

By constructing a parametrized cyclic (PC) representation of the q-deformed Heisenberg-Weyl algebras the PC.represen tations of quan tum superalgebra U_qosp(l, 2n) are studied in terms of its q-boson realization. A new type of PC representations of the rank-1 quantum superalgebra U_qosp(l, 2) is obtained by this way.     

kth-order antibunching effect for a new kind of excited even and odd q-coherent states

A new kind of excited even q-coherent states （aq^-1）^m｜α〉q^e and excited odd q-coherent states （aq^-1）^m｜α〉q^o is constructed by acting with inverse boson operators on the even and odd q-coherent states. The m dependence of the kth-order antibunching effect is numerically studied for k = 2, 3, 4, 5. It is shown that the kth-order antibunching effect enhances as m increases. The larger k, the quicker the antibunching effect enhances.     

The algebraic Bethe ansatz for rational braid-monoid lattice models

In this paper we study isotropic integrable systems based on the braid-monoid algebra. These systems constitute a large family of rational multistate vertex models and are realized in terms of the B_n, C_nand D_n Lie algebra and by the superalgebra Osp(n||2m). We present a unified formulation of the quantum inverse scattering method for many of these lattice models. The appropriate fundamental commutation rules are found, allowing us to construct the eigenvectors and the eigenvaluesof the transfer matrix associated to the B_n, C_n, D_n, Osp(2nt-1||2), Osp(2||2nt-2), Osp(2nt-2||2) and Osp(1||2n) models. The corresponding Bethe ansatz equations can be formulated in terms of the root structure of the underlying algebra.     

Superposition of Generalized q-Coherent States of the Non-harmonicOscillator and Their Nonclassical Properties

The superposition states, which are composed of the generalized q-coherent states of the non-harmonic oscillator, are presented. The quantum statistical properties of these states are studied. Then the influence of the superposition phase δ and the deformation parameter q on these nonclassical properties are discussed. It is shown that the superposition states exhibit higher-order (order of odd number) squeezing effect and antibunching effect respectively. When the superposition phase δ is taken values approach to 0 or π/2, these results approach the properties of the even or odd generalized q-coherent states of the non-harmonic oscillator. Especially, the effects are influenced by the parameter q. When the values of q is taken departure from 1 greatly, the nonclassical properties are more evidently.     

Multifractality of a Cartesian Product of Two Fractals

We discuss fractal measures on a bifractal, which is defined as a Cartesian product of two fractals and represents a class of self-affine fractals. It is shown that the global D_q, f_q and α_q are the sums of the corresponding exponents for the subfractals. The relation between the global f (α) spectrum and those for subfractals is also derived.