Generalization of Drinfeld Quantum Affine Algebras

Drinfeld gave a current realization of the quantum affine algebras as a Hopf algebra with a simple comultiplication for the quantum current operators. In this Letter, we will present a generalization of such a realization of quantum Hopf algebras. As a special case, we will choose the structure functions for this algebra to be elliptic functions to derive certain elliptic quantum groups as a Hopf algebra, which degenerates into quantum affine algebras if we take certain degeneration of the structure functions.     

On the Graded Quantum Yang-Baxter and Reflection Equations

We clarify the extra signs appearing in the graded quantum Yang-Baxter reflection equations, when they are written in a matrix form. We find the boundary K-matrix for the Perk-Schultz six-vertex model, thus give a general solution to the graded reflection equation associated with it.     

An $${n = \left( {1,1} \right)}$$ Super-Toda Model Based on OSp $$1|{\kern 1pt} 4$$

We show that a Hamiltonian reduction of affine Lie superalgebras having bosonic simple roots (such as OSp ) does produce supersymmetric Toda models, with superconformal symmetry being nonlinearly realized for those fields of the Toda system which are related to the bosonic simple roots of the superalgebra. A fermionic b-c system of conformal spin is a natural ingredient of such models.     

Boundary Quantum Group Generators of Type A

We construct boundary quantum group generators which, through linear intertwining relations, determine nondiagonal solutions of the boundary Yang–Baxter equation for vector representations of A _n-1 ⁽¹⁾ and A ₂ ⁽²⁾ .     

The photomagneton\hat B^{\left( 3 \right)} and electrodynamic conservation lawsand electrodynamic conservation laws

It is shown that the basic electrodynamical conservation laws are unaffected by the presence in free space of the photomagneton of light, $\hat B^{\left( 3 \right)} = B^{\left( 0 \right)} \hat J/\rlap{--} h$ , the fundamental photon property responsible for magnetization by light. The expectation value $B^{\left( 3 \right)} = \left\langle {\hat B^{\left( 3 \right)} } \right\rangle $ does not affect the Poynting vector, so that it does not contribute to electromagnetic flux density. The electromagnetic energy density can be expressed in terms ofB ⁽³⁾ through the equation $$\rlap{--} h\omega = \frac{1}{{\mu _0 }}\smallint B^{\left( 3 \right)} \cdot B^{\left( 3 \right) * } dV.$$ When light magnetizes matter, the unitB ⁽³⁾ of magnetic flux density per photon is transferred from light to matter. This is equivalent to an elastic transfer of angular momentum. Experimental indications for the existence ofB ⁽³⁾ are discussed.     

A Remark on the FRTS Realization and Drinfeld Realization of Quantum Affine Superalgebra U q (o $${\hat s} $$ sp(1,2))

In this Letter, we present the hidden symmetry behind the Faddeev–Reshetikhin–Takhtajan–Semenov–Tian–Shansky (FRTS) realization of quantum affine superalgebras U(o sp(1,2)) and add the q-Serre relation to the Drinfeld realization of U(o sp(1,2)), derived from the FRTS realization.     

A Quantum Homogeneous Space of Nilpotent Matrices

A quantum deformation of the adjoint action of the special linear group on the variety of nilpotent matrices is introduced. New non-embedded quantum quasi- homogeneous spaces are obtained related to certain maximal coadjoint orbits, and known quantum homogeneous spaces are revisited.     

The Relation Between Two Types of Characteristic Equations for Quantum Matrices

A definition (modification) of the power of quantum matrices using the -matrix has recently been proven useful to obtain generalizations of many well known theorems from linear algebra to the quantum case, among which are the Cayley–Hamilton theorem and the Newton identities. A separate effort has provided another generalization of the Cayley–Hamilton theorem for GL _q (n), which uses usual matrix powers but diagonal matrices as coefficients.We show that the latter generalization can be derived in the aforementioned more general framework and it is the expression of the modified quantum power in terms of the usual ones that accounts for the appearance of diagonal matrices.     

An admissible level $$\widehat{\mathfrak {osp}} \left( 1 \big \vert 2 \right) $$-model: modular transformations and the Verlinde formula

The modular properties of the simple vertex operator superalgebra associated with the affine Kac–Moody superalgebra \(\widehat{{\mathfrak {osp}}} (1|2)\) at level \(-\frac{5}{4}\) are investigated. After classifying the relaxed highest-weight modules over this vertex operator superalgebra, the characters and supercharacters of the simple weight modules are computed and their modular transforms are determined. This leads to a complete list of the Grothendieck fusion rules by way of a continuous superalgebraic analog of the Verlinde formula. All Grothendieck fusion coefficients are observed to be non-negative integers. These results indicate that the extension to general admissible levels will follow using the same methodology once the classification of relaxed highest-weight modules is completed.     

Output characteristics of mixed holograms in the ($$\left( {\overline {11} 0} \right)$$)-Cut Bi12TiO20 crystal

An analytical solution to the system of linear differential equations that describe the reconstruction of object wave using a transmission holographic grating formed in a cubic photorefractive piezoelectric (\(\left( {\overline {11} 0} \right)\))- cut crystal with the 23 symmetry is derived. The solution is used to interpret the experimental results on the dependence of the diffraction efficiency of mixed holograms on the thickness of the Bi₁₂TiO₂₀ crystal at a fixed orientation angle and three azimuths of linear polarization of the readout reference beam. The best agreement of the theoretical and experimental results on mixed holograms is reached when the inverse piezoelectric effect and photoelasticity are taken into account in addition to the conventional electrooptic effect.     

Study of Inelastic $$\boldsymbol{A}$$ ( $$\boldsymbol{p,p^{\prime}}$$ ) $$\boldsymbol{X}$$ Reaction with $${}^{\mathbf{9}}$$ Be and $${}^{\mathbf{90}}$$ Zr Nuclei at 1 GeV

Physics of Atomic Nuclei - The secondary proton polarization and differential cross sections of the ( $$p,p^{\prime}$$ ) inelastic reaction on nuclei $${}^{9}$$ Be and $${}^{90}$$ Zr at the initial...     

Lasing of optically pumped XeF and analysis of the spectrum of the laser transitionB^2 \sum _{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}^ + - X^2 \sum _{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}^ +

The operating principle of a photodissociation XeF laser (λ?350 nm) is discussed. A laser operating on the gas mixture XeF₂:N₂(SF₆):Ar pumped by radiation from an open high-current discharge is described. The maximum lasing duration is 5 μsec, the maximum output energy is 0.15 J. The vibrational structure of the XeF-molecule transition B²Σ_1/2 ⁺ ? X²Σ_1/2 ⁺, observed in absorption, luminescence, and stimulated emission, is investlgated. The vibrational quantum and the anharmonicity in the upper and lower states are measured: ω_e ^' = 308.7 cm^-1, ω_e ^'x_e ^' = 1.44 cm^-1, ω_e ^'' = 225.7 cm^-1, ω_e ^''x_e ^'' = 11.0 cm^-1. The difference between the equilibrium internuclear distances of the XeF molecule is determined for the excited (B) and ground (X) states, namely r_e ^' - r_e ^'' = 0.33 +- 0.01 Å.     

Properties of 2 × 2 h-Deformed Quantum (Super)Matrices

We investigate the h-deformed quantum (super)group of 2 × 2 matrices and use a kind of contraction procedure to prove that the n-th power of this deformed quantum (super)matrix is quantum (super)matrix with the deformation parameter nh.     

Some Tensor Product Representations of Uq(s $${\hat l}{\kern 1pt} $$ 2) at Roots of Unity

When q is a root of unity, a triangular decomposition of U _q(s ₂) is given and irreducibility conditions concerning some tensor product representations of U _q(s ₂) are presented. Their connection with physics is also pointed out.     

Theoretical study of collision and collision-free radiative lifetimes of {{\text{S}}_2}\left( {{{\text{B}}^3}\Sigma_{\text{u}}^{-} \left( {{\upsilon^{\prime}} = 0 - 9} \right)} \right)

We calculate multireference configuration-interaction wavefunctions and the potential-energy curves for the $ {B^3}\Sigma_u^{-} $ and $ {X^3}\Sigma_g^{-} $ states of the collision-free S₂ molecule and the T-shape collision complex S₂?CHe using cc-pVQZ basis sets. We obtain the transition dipole moments of the $ {{\text{S}}_2}\left( {{B^3}\Sigma_u^{-} \to {X^3}\Sigma_g^{-} } \right) $ and the Franck?CCondon factors between the vibrational levels of this two states. We evaluate the radiative lifetimes of $ {{\text{S}}_2}\left( {{B^3}\Sigma_u^{-} \left( {{\upsilon^{\prime}} = 0 - 9} \right)} \right) $ levels of the collision complex and the collision-free molecule and compare them with the experiments. The collision provides little change in the radiative lifetimes of $ {{\text{S}}_2}\left( {{B^3}\Sigma_u^{-} \left( {{\upsilon^{\prime}} = 0 - 9} \right)} \right) $ according to the previous calculations. We obtain excellent agreement between the theoretical results and the experiments. The data calculated are very useful in the study of the microwave-driven high-pressure sulfur lamp and an S₂ laser pumped by a transverse fast discharge.     

Traces of Intertwiners for Quantum Groups and Difference Equations

In this Letter we study twisted traces of products of intertwining operators for quantum affine algebras. They are interesting special functions, depending on two weights ,, three scalar parameters q,,k, and spectral parameters z ₁,...,z _N , which may be regarded as q-analogs of conformal blocks of the Wess–Zumino–Witten model on an elliptic curve. It is expected that in the rank 1 case they essentially coincide with the elliptic hypergeometric functions defined by Felder and Varchenko. Our main result is that after a suitable renormalization the traces satisfy four systems of difference equations – the Macdonald–Ruijsenaars equation, the q-Knizhnik–Zamolodchikov–Bernard equation, and their dual versions. We also show that in the case when the twisting automorphism is trivial, the trace functions are symmetric under the permutation , k . Thus, our results generalize those of Etingof and Schiffmann, dealing with the case q=1, and Etingof, Varchenko, and Schiffmann, dealing with the finite-dimensional case.     

反射后激光(光子)螺旋性改变的量子论解释和信息论意义

基于近年来王顺金提出的处理螺旋光纤中激光(光子)传播时的螺旋性变化的理论方法和所得结果,分析了激光(光子)在反射后螺旋性(左旋与右旋)互换这一实验事实,发现激光(光子)反射的瞬息过程会产生一个极高的激光束回转频率,它与激光园偏振(光子自旋)发生科式惯性力耦合,进而改变激光(光子)的螺旋性.还发现当激光束回转频率达到极大时,将发生激光(光子)螺旋性(左旋与右旋)互换.讨论了激光反射后螺旋性互换的信息论含义及其应用.     

Generalization of the $$\mathcal{U} $$ q (gl(N)) Algebra and Staggered Models

We develop a technique for the construction of integrable models with a ₂ grading of both the auxiliary (chain) and quantum (time) spaces. These models have a staggered disposition of the anisotropy parameter. The corresponding Yang–Baxter equations are written down and their solution for the gl(N) case is found. We analyze in details the N = 2 case and find the corresponding quantum group behind this solution. It can be regarded as the quantum group , with a matrix deformation parameter q such that (q )² = q ². The symmetry behind these models can also be interpreted as the tensor product of the (–1)-Weyl algebra by an extension of _q (gl(N)) with a Cartan generator related to deformation parameter –1.