Fusion rules of U q(sl(2, ℂ)), q m = 1

The decomposition of multiple tensor products of finite-dimensional irreducible representations of U _q(sl(2, )), for q's which are roots of 1, is studied in detail.     

Realisations of quantum GL p,q(2) and Jordanian GLh,h′(2)

The quantum group GL _p,q(2) is known to be related to the Jordanian GL_h,h(2) via a contraction procedure. It can also be realised using the generators of the Hopf algebra G _r,s. We contract the G _r,s quantum group to obtain its Jordanian analogue G _m,k, which provides a realisation of GL_h,h(2) in a manner similar to the q-deformed case.     

A borel-weil-bott approach to representations of sl q (2, ℂ)

We use a quite concrete and simple realization of sl _q (2, ) involving finite difference operators. We interpret them as derivations (in the noncommutative sense) on a suitable graded algebra, which gives rise to the noncommutative scheme ¹ II ^1* as the counterpart of the standard ¹ = Sl(2, )/B.     

What if Ω ≠ 2q?

We discuss various possible scenarios where 2q, where stands for the density parameter andq for the deceleration parameter. We further estimate the corrections necessary when a variable cosmological constant is considered in the theory.     

重子中七夸克成分qqq(qˉq)2的低位能谱

在非相对论夸克模型内,利用手征超精细相互作用,把原来对五夸克系统qqqqˉq能谱的预言,扩展到了七夸克系统qqq(qˉq)2的基态和轨道第一激发态能谱的计算,所使用的参数也是来自于五夸克系统.预言了最低七夸克态的能量大约是2.1 GeV,自旋-宇称是JP=1也存在一些三夸克系统和五夸克系统,所以能量在2 GeV以上的重子激发态可能是这3种夸克成分的混合.+或者JP=1?.由于在这个能区22     

Operator representations of U q (sl2(ℝ))

Integrable (well-behaved) operator representations of the *-algebra U _q (sl₂()), |q|= 1, q ² 1, |q|=1, q² 1, in Hilbert space are defined and classified up to unitary equivalence.     

Structures and bonding of C2X2Y q (X=Si,Ge,Sn,Pb; Y=C,Si,Ge,Sn,Pb; q = +1,0,−1): π-type templates for planar tetracoordinate heavier group 14 atoms

Contrasting the big family of the planar tetracoordinate carbon (ptC), species featuring the planar tetracoordinate heavier group element M (ptM) have been largely limited. Effective structural frameworks to accommodate such ptM centres are thus highly desired. In the present article, we report an extensive computational study on 60 pentatomic systems C₂X₂Y^q (X=Si,Ge,Sn,Pb; Y=C,Si,Ge,Sn,Pb; q = +1,0,?1) covering both the low and high spin states. Up to 34 systems were shown to have the very low-lying singlet planar tetracoordinate heavier group 14 (ptM with M=Si,Ge,Sn,Pb) structures bearing the 19 (q = +1), 20 (q = 0) and 21 (q = ?1) valence electrons (ve). Structural and bonding analysis confirmed the effectiveness of the inherent π-type ligand skeleton XCCX or XCCY that each have several sets of π-bonding orbitals to stabilise the ptM centre. The structural and bonding motifs of these ptMs differ greatly from the classic ptMs, which have the σ-type ligand skeleton, smaller number of valence electrons (≤18ve), and the centre → ligand π-delocalisation.     

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.     

Trapezoidal (p,q)-Integral Inequalities Related to (η1,η2)-convex Functions with Applications

In this paper, the authors introduce the (p,q)-trapezoidal integral inequalities, which are the (p,q)-analogues of the recently introduced q-trapezoidal integral inequalities. We derive a new (p,q)-integral identity for twice (p,q)-differentiable function. Utilizing this as an auxiliary result, we establish several new (p,q)-trapezoidal type integral inequalities for the function whose absolute value of twice (p,q)-derivative is (η₁,η₂)-convex functions. Some special means of real numbers are also given. At the end, we give brief conclusion. It is expected that this method which is very useful, accurate, and versatile will open a new venue for the real-world phenomena of special relativity and quantum theory.

     

Representation of Spin Group Spin（p, q）

The representation η（P, q） of spin group Spin（p, q） in any dimensional space is given by induction, and the relation between two representations, which are obtained in two kinds of inductions from Spin（p, q） to Spin（p ＋ 1, q ＋ 1） are studied.     

su q (2)-invariant Schrödinger equations

We set upsu _q (2)-invariant Schrödinger equations within the usual framework of quantum mechanics. We show that the stationary equations reduce to radial ordinary differential equations by using the q-Spherical Harmonics. We apply these results to deformations of the free particle and of the hydrogen atom, for which we explicitly solve the equations and compare the deformed and undeformed theories.Presented at the 4th Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.     

Baxterization of ther-matrix for the adjoint representation of U q [D(2, 1;α)]

The Baxterization procedure is applied to the Braid group representation arising from the adjoint representation of the quantum supergroup U _q [D(2, 1;)]. This yields a two-parameter family of solutions to the (parameter dependent) Yang-Baxter equation of interest in supersymmetric lattice models.     

Unitary representations of suq(2) on the plane for q ∈ R+ or generic q ∈ S1

Some time ago, Rideau and Winternitz introduced a realization of the quantum algebra su _q(2) on a real two-dimensional sphere, or a real plane, and constructed a basis for its representations in terms of q-special functions, which can be expressed in terms of q-Vilenkin functions. In their study, the values of q were implicitly restricted to q R⁺. In the present paper, we extend their work to the case of generic values of q S ¹ (i.e., q values different from a root of unity). In addition, we unitarize the representations for both types of q values, q R⁺ and generic q S ¹, by determining some appropriate scalar products.     

动力学椭圆代数A_(q,p,)()的Drinfeld流

从推广的Yang -Baxter关系“RLL =LLR ”出发 ,利用高秩高斯分解 ,得到了动力学椭圆代数Aq,p ,^π(gln〈)及其对应的Drinfeld流 .其中R ,R 是A(1)n -1面模型对应的谱参数有一关于代数中心平移的动力学R矩阵     

Clebsch–Gordan and Racah–Wigner Coefficients for a Continuous Series of Representations of ?q (??(2, ℝ))

The decomposition of tensor products of representations into irreducibles is studied for a continuous family of integrable operator representations of U _q (sl(2,ℝ). It is described by an explicit integral transformation involving a distributional kernel that can be seen as an analogue of the Clebsch–Gordan coefficients. Moreover, we also study the relation between two canonical decompositions of triple tensor products into irreducibles. It can be represented by an integral transformation with a kernel that generalizes the Racah–Wigner coefficients. This kernel is explicitly calculated. Received: 9 August 2000 / Accepted: 2 July 2001     

q畸变的KdV方程

本文由"量子"Virasoro代数出发,推导出了q畸变的KdV方程,当q→1时,就退化为通常的结果.