q-Deformations of 3-Lie Algebras

q-Deformations of 3-Lie algebras and representations of q-3-Lie algebras are discussed. A q-3-Lie algebra （A, [, ,]q, [, ,]＇q, Ja）, where q ∈ F and q ≠ 0, is a vector space A over a field F with 3-cry linear multiplications [,, ]q and [,, ]＇q from A 3 to A, and a map Jq ： A 5→ F satisfying the q-Jacobi identity Jq（x1, x2, x3, x4, x5）[x1, x2, [x3, x4, x5]q]＇q = Jq（x4, x5, x1, x2, x3）[x4, x5, [x1, x2, x3]q]＇q for all x1 ∈ A. If the multiplications satisfy that [,,]q = [,,]＇q and [,,]q is skew-symmetry, then （A, [,,]q, Jq） is called a type （I）-q-3- Lie algebra. From q-Lie algebras, group algebras and commutative associative algebras, q-3-Lie algebras and type （I）-q-3-Lie algebras are constructed. Also, the non-trivial one- dimensional central extension of q-3-Lie algebras is investigated, and new q-3-Lie algebras （DerqC[x,x-1], [, ,]q, [, ,]＇q, Jq）, and （Derδq C[x,x-1], [, ,]q, [,,]＇q, Jq） are obtained.

q-Deformation of W (2 , 2) Lie Algebra Associated with Quantum Groups

The q-deformation of W（2, 2） Lie algebra is well defined based on a realization of this Lie algebra by using the famous bosonic and fermionic oscillators in physics. Furthermore, the quantum group structures on the q-deformation of W（2, 2） Lie algebra are completely determined. Finally, the 1-dimensional central extension of the q-deformed W（2, 2） Lie algebra is studied, which turns out to be coincided with the conventional W（2, 2） Lie algebra in the q → 1 limit.

q-Deformed Gelfand-Dickey系列规范变换的行列式表示

Gelfand-Dickey系列是十分重要的可积系统.本文用规范变换来讨论q-Deformed Gelfand-Dickey系列的解,得到了由两种基本规范变换迭代(n+k)次产生的规范变换算子Tn+k的行列式表达,并由此给出规范变换后的T函数Tq(n+k)的形式.

Symbolic Computation and q-Deformed Function Solutions of （2＋1）-Dimensional Breaking Soliton Equation

In this paper, by using symbolic and algebra computation, Chen and Wang＇s multiple R/ccati equations rational expansion method was further extended. Many double soliton-like and other novel combined forms of exact solutions of the （2＋1）-dimensional Breaking soliton equation are derived by using the extended multiple Riccatl equations expansion method.

Kac Determinant Formula for the q-deformed Virasoro Algebra of Hom-type

We investigate the highest weight representations of the q-deformed Virasoro algebra of Hom-type. In order to determine its unitarity and irreducible highest weight representations, we present its Kac determinant formula when q is nonzero and non-root of unity.

Exact solution of Schrdinger equation with q-deformed quantum potentials using Nikiforov–Uvarov method

In this paper, we present the exact solution of the one-dimensional Schrrdinger equation for the q-deformed quantum potentials via the Nikiforov-Uvarov method. The eigenvalues and eigenfunctions of these potentials are obtained via this method. The energy equations and the corresponding wave functions for some special cases of these potentials are briefly discussed. The PT-symmetry and Hermiticity for these potentials are also discussed.

Comment on “Symbolic Computation and q-Deformed Function Solutions of the (2+1)-Dimensional Breaking Soliton Equation”

In a recent article [Commun. Theor. Phys. （Beijing, China） 47 （2007） 270], Cao et al. gave some nontrivial solutions of a Riccati equation by using symbolic and algebra computation. They took these solutions, which are in the form of q-deformed hyperbolic and triangular functions as new solutions. In this comment, we will show that these solutions are just the special cases of some known solutions of the Riccati equation and thus they are not new solutions.

Approximate analytical solution of the Dirac equation with q-deformed hyperbolic Poschl–Teller potential and trigonometric Scarf II non-central potential

An approximate solution of the Dirac equation for a spin-1/2 particle under the influence of q-deformed hyperbolic P ¨oschl–Teller potential combined with trigonometric Scarf II non-central potential is studied analytically. It is assumed that the scalar potential equals the vector potential in order to obtain analytical solutions. Both radial and angular parts of the Dirac equation are solved using the Nikiforov–Uvarov method. A relativistic energy spectrum and the relation between quantum numbers can be obtained using this method. Several quantum wave functions corresponding to several states are also presented in terms of the Jacobi Polynomials.

Q-Deformed Bosons at Quasi-classical Limit of SO(3)

When a boson interacts with another to form a composite system with SO（3） dynamic symmetry, it is shown that there exists the q-deformed 5osonic excitation satisfying the q-deformed Heisenberg commutation relation in the quasi-classical limit that the angular momentum j for SO（3） is large, but not infinite. In second quantization this quasi-excitation is associated with the boson realization of SO（3） Lie algebra. Physically, the phenomena of q-deformed excitation can happen in many models of quantum dynamics, such as super emission from a system of many identical two-level atoms, the spin wave in Heisenberg chain, the high spin precession and the coherent output of Bose-Einstein atoms in a trap. Especially, in these models, the deformation parameter q is well defined intrinsically by a conservative quantity, such as the total atomic number and the angular momentum.

An extensive analytical and numerical study of the generalized q-deformed Sinh-Gordon equation

In this paper,the q-deformed Sinh-Gordon equation is solved analytically using a new general form based on the extended tanh approach.The numerical solutions of the equation is obtained using a b-spline finite element method.Also,we present numerous figures to demonstrate the various solitons propagation patterns.This type of equation has not been previously dealt with in such ways,whether analytical or numerical.This study is very useful in studying several physical systems that have lost their symmetry.

q-Deforrned Entangled States Representations and Some Applications

The q-deformed entangled states are introduced by using deformation quantization methods and new normal ordering of the vacuum projection operator for q-deformed boson oscillator. In similar way, by virtue of the technique of integration within an ordered product （IWOP） of operators, the new completeness and orthogonMity relations composed of the bra and ket, which are not mutually Hermitian conjugates are obtained. Furthermore, the property of squeezing operator represented by the q-deformed entangled states is exhibited. Lastly, the nonclassical properties of the q-deformed two-mode squeezed vacuum state are studied.

Non-Relativistic Bound State Solutions of Modified Quadratic Yukawa plus <i>q</i>-Deformed Eckart Potential

We have obtained approximate bound state solutions of Schr&#246;dinger wave equation with modified quadratic Yukawa plus q-deformed Eckart potential Using Parametric Nikiforov-Uvarov (NU) method. However, we obtained numerical energy eigenvalues and un-normalized wave function using confluent hypergeometric function (Jacobi polynomial). With some modifications, our potential reduces to a well-known potential such as Poschl-Teller and exponential inversely quadratic potential. Numerical bound state energies were carried out using a well-designed Matlab algorithm while the plots were obtained using origin software. The result obtained is in agreement with that of the existing literature.

Explicit Forms of q-Deformed Lévy-Meixner Polynomials and Their Generating Functions

The classical Levy-Meixner polynomials are distinguished through the special forms of their generating functions. In fact, they are completely determined by 4 parameters： c1, c2,γ and β. In this paper, for-1 〈q〈 1, we obtain a unified explicit form of q-deformed Levy-Meixner polynomials and their generating functions in term of c1, c2, γand β, which is shown to be a reasonable interpolation between classical case （q=1） and fermionic case （q=-1）.In particular, when q=0 it＇s also compatible with the free case.

Thermostatistic Properties of the Relativistic q-Deformed Ideal Fermi Gas at High Temperature

The q-deformed Fermi-Dirac distribution is used to study the high-temperature（T TF） ∨behavior of a relativistic q-deformed ideal Fermi gas. The effects of the q-deformation and relativity on the properties of the system are discussed,and then,the example of the neutrinos near the surface of the earth is used to calculate approximately. It shows that the q-deformation increases the total energy but decreases the chemical potential and heat capacity,whereas mc 2 ε F0increases the chemical potential and total energy but decreases the heat capacity（ ε F0is the Fermi energy of ultrarelativistic undeformed Fermi gas） . The larger the deformation parameter q and the value of mc 2 ε F0are,the more remarkable the effects of them on the thermostatistic properties will be. However,the effects of both q-deformation and relativity become weak with increasing temperature. When the temperature T →∞,the thermostatistic properties of the system are reduced to those of ordinary Boltzmann gases and independent of q and relativity effect completely,which implies that the q-deformation is a pure quantum effect.

Low Dimensional Cohomology of Hom-Lie Algebras and q-deformed W(2, 2) Algebra

This paper aims to study low dimensional cohomology of Hom-Lie algebras and the qdeformed W（2, 2） algebra. We show that the q-deformed W（2, 2） algebra is a Hom-Lie algebra. Also,we establish a one-to-one correspondence between the equivalence classes of one-dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the second cohomology group of the q-deformed W（2, 2） algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras.As application, we compute all αk-derivations and in particular the first cohomology group of the q-deformed W（2, 2） algebra.

Investigation of Bose-Einstein Condensates in q-Deformed Potentials with First Order Perturbation Theory

The Gross-Pitaevskii equation, which is the governor equation of Bose-Einstein condensates, is solved by first order perturbation expansion under various q-deformed potentials. Stationary probability distributions reveal one and two soliton behavior depending on the type of the q-deformed potential. Additionally a spatial shift of the probability distribution is found for the dark soliton solution, when the q parameter is changed.

Addition Formulae of Discrete KP,q-KP and Two-Component BKP Systems

In this paper,we construct the addition formulae for several integrable hierarchies,including the discrete KP,the q-deformed KP,the two-component BKP and the D type Drinfeld–Sokolov hierarchies.With the help of the Hirota bilinear equations and τ functions of different kinds of KP hierarchies,we prove that these addition formulae are equivalent to these hierarchies.These studies show that the addition formula in the research of the integrable systems has good universality.

Biderivations of Hom-Lie Algebras and Superalgebras

On Hom-Lie algebras and superalgebras,we introduce the notions of biderivations and linear commuting maps,and compute them for some typical Hom-Lie algebras and superalgebras,including the q-deformed W(2,2)algebra,the q-deformed Witt algebra and superalgebra.