A Characterization of Vertex Operator Algebra {L(\frac{1}{2},0)\otimes L(\frac{1}{2},0)}

We study a simple, rational and C ₂-cofinite vertex operator algebra whose weight 1 subspace is zero, the dimension of weight 2 subspace is greater than or equal to 2 and with c = c? = 1. Under some additional conditions it is shown that such a vertex operator algebra is isomorphic to ${L(\frac{1}{2},0)\otimes L(\frac{1}{2},0)}We study a simple, rational and C ₂-cofinite vertex operator algebra whose weight 1 subspace is zero, the dimension of weight 2 subspace is greater than or equal to 2 and with c = c̃ = 1. Under some additional conditions it is shown that such a vertex operator algebra is isomorphic to L(\frac12,0)?L(\frac12,0){L(\frac{1}{2},0)\otimes L(\frac{1}{2},0)}.     

Chiral properties of baryon interpolating fields

We study the chiral transformation properties of all possible local (non-derivative) interpolating field operators for baryons consisting of three quarks with two flavors, assuming good isospin symmetry. We derive and use the relations/identities among the baryon operators with identical quantum numbers that follow from the combined color, Dirac and isospin Fierz transformations. These relations reduce the number of independent baryon operators with any given spin and isospin. The Fierz identities also effectively restrict the allowed baryon chiral multiplets. It turns out that the non-derivative baryons’ chiral multiplets have the same dimensionality as their Lorentz representations. For the two independent nucleon operators the only permissible chiral multiplet is the fundamental one, . For the Δ, admissible Lorentz representations are and . In the case of the chiral multiplet, the Δ field has one chiral partner; otherwise it has none. We also consider the Abelian (U _A(1)) chiral transformation properties of the fields and show that each baryon comes in two varieties: (1) with Abelian axial charge +3; and (2) with Abelian axial charge −1. In case of the nucleon these are the two Ioffe fields; in case of the Δ, the multiplet has an Abelian axial charge −1 and the multiplet has an Abelian axial charge +3.     

The Cubic Dirac Equation: Small Initial Data in {{H^{\frac{1}{2}}} (\mathbb{R}^{2}})

Global well-posedness and scattering for the cubic Dirac equation with small initial data in the critical space \({{H^{\frac{1}{2}}} (\mathbb{R}^{2}}\)) is established. The proof is based on a sharp endpoint Strichartz estimate for the Klein–Gordon equation in dimension n = 2, which is captured by constructing an adapted systems of coordinate frames.     

On the Yangian Y $$(\mathfrak{g}\mathfrak{l}_{m|n} )$$ and its quantum Berezinian

Jonathan Brundan and Alexander Kleshchev recently introduced a new family of presentations for the Yangian Y of the general linear Lie algebra . In this article, we extend some of their ideas to consider the Yangian Y of the Lie superalgebra . In particular, we give a new proof of the result by Nazarov that the quantum Berezinian is central. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

Generalized parton distributions in impact parameter space

The kinematical factor in the positivity bound (36) is incorrect. The bound correctly reads Our corrected result agrees with inequality (25) in [1], taking into account the different normalization conventions here and there.Published online: 9 October 2003Erratum published online: 10 October 2003     

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.     

Modules over\mathfrak{U}_q (\mathfrak{s}\mathfrak{l}_2 )

The restricted quantum universal enveloping algebra decomposes in a canonical way into a direct sum of indecomposable left (or right) ideals. They are useful for determining the direct summands which occur in the tensor product of two simple . The indecomposable finite-dimensional are classified and located in the Auslander-Reiten quiver.     

Spinor Representations of $${U}_q (\hat {\mathfrak{o}}(N))$$ )

This Letter concerns an extension of the quantum spinor construction of . We define quantum affine Clifford algebras based on the tensor category and the solutions of q-KZ equations, and construct quantum spinor representations of .     

The <Emphasis Type="Italic">N</Emphasis> = 1 Triplet Vertex Operator Superalgebras

We introduce a newfamily of C ₂-cofinite N = 1 vertex operator superalgebras , m ≥ 1, which are natural super analogs of the triplet vertex algebra family , p ≥ 2, important in logarithmic conformal field theory. We classify irreducible -modules and discuss logarithmic modules. We also compute bosonic and fermionic formulas of irreducible characters. Finally, we contemplate possible connections between the category of -modules and the category of modules for the quantum group , , by focusing primarily on properties of characters and the Zhu’s algebra . This paper is a continuation of our paper Adv. Math. 217, no.6, 2664–2699 (2008). The second author was partially supported by NSF grant DMS-0802962.     

Simulating spin-\frac{3}{2} particles at colliders

Support for interactions of spin- $\frac{3}{2}$ particles is implemented in the FeynRules and ALOHA packages and tested with the MadGraph 5 and CalcHEP event generators in the context of three phenomenological applications. In the first, we implement a spin- $\frac{3}{2}$ Majorana gravitino field, as in local supersymmetric models, and study gravitino and gluino pair-production. In the second, a spin- $\frac{3}{2}$ Dirac top-quark excitation, inspired from compositeness models, is implemented. We then investigate both top-quark excitation and top-quark pair-production. In the third, a general effective operator for a spin- $\frac{3}{2}$ Dirac quark excitation is implemented, followed by a calculation of the angular distribution of the s-channel production mechanism.     

The Lie algebra $$\mathfrak{s}\mathfrak{l}_2 (\mathbb{F})$$ and quasi-deformations

The Lie algebra is “deformed” using twisted derivations satisfying a twisted Leibniz rule. Some particular algebras appearing in this deformation scheme are discussed. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005. Supported by the Liegrits network Supported by the Crafoord foundation     

The Automorphism Group of a Simple Tracially AI Algebra

The structure of the automorphism group of a simple TAI algebra is studied. In particular, we show that is isomorphic (as a topological group) to an inverse limit of discrete abelian groups for a unital, simple, AH algebra with bounded dimension growth. Consequently, is totally disconnected. Another consequence of our results is the following: Suppose A is the transformation group C*-algebra of a minimal Furstenberg transformation with a unique invariant probability measure. Then the automorphism group of A is an extension of a simple topological group by the discrete group .     

$${f(\mathcal R)}$$ quantum cosmology

We have quantized a flat cosmological model in the context of the metric models, using the causal Bohmian quantum theory. The equations are solved and then we have obtained how the quantum corrections influence the classical equations.     

Coordinates of the quantum plane as q-tensor operators in (\mathfrak{s}\mathfrak{u}(2)*\mathfrak{s}\mathfrak{u}(2))

The relation between the set of transformations of the quantum plane and the quantum universal enveloping algebra U _q (u(2)) is investigated by constructing representations of the factor algebra U _q (u(2))* . The noncommuting coordinates of , on which U _q (2) * U _q (2) acts, are realized as q-spinors with respect to each U _q (u(2)) algebra. The representation matrices of U _q (2) are constructed as polynomials in these spinor components. This construction allows a derivation of the commutation relations of the noncommuting coordinates of directly from properties of U _q (u(2)). The generalization of these results to U _q (u(n)) and is also discussed.     

The Integrals of Motion for the Deformed W-Algebra $${W_{q,t}(\widehat{gl_N})}$$. II. Proof of the Commutation Relations

We explicitly construct two classes of infinitely many commutative operators in terms of the deformed W-algebra , and give proofs of the commutation relations of these operators. We call one of them local integrals of motion and the other nonlocal, since they can be regarded as elliptic deformations of local and nonlocal integrals of motion for the Virasoro algebra and the W ₃ algebra [1,2]. Dedicated to Professor Tetsuji Miwa on the occasion of the 60th birthday     

On a bosonic-parafermionic realization ofU_q (\widehat{sl(2)})

We realize the current algebra at an arbitrary level in terms of one deformed free bosonic field and a pair of deformed parafermionic fields. It is shown that the operator product expansions of these parafermionic fields involve an infinite number of simple poles and simple zeros, which then condensate to form a branch cut in the classical limitq1. Our realization coincides with those of Frenkel-Jing and Bernard when the levelk takes the values 1 and 2, respectively.     

Modules-at-Infinity for Quantum Vertex Algebras

This is a sequel to [Li4] and [Li5] in a series to study vertex algebra-like structures arising from various algebras such as quantum affine algebras and Yangians. In this paper, we study two versions of the double Yangian , denoted by DY _q (sl ₂) and with q a nonzero complex number. For each nonzero complex number q, we construct a quantum vertex algebra V _q and prove that every DY _q (sl ₂)-module is naturally a V _q -module. We also show that -modules are what we call V _q -modules-at-infinity. To achieve this goal, we study what we call -local subsets and quasi-local subsets of for any vector space W, and we prove that any -local subset generates a (weak) quantum vertex algebra and that any quasi-local subset generates a vertex algebra with W as a (left) quasi module-at-infinity. Using this result we associate the Lie algebra of pseudo-differential operators on the circle with vertex algebras in terms of quasi modules-at-infinity.     

Cohomology of $${\mathfrak{osp}}(1|2)$$ Acting on Linear Differential Operators on the Supercircle S1|1

We compute the first cohomology spaces of the Lie superalgebra with coefficients in the superspace of linear differential operators acting on weighted densities on the supercircle S ^1|1. The structure of these spaces was conjectured in (Gargoubi et al. in Lett Math Phys 79:5165, 2007). In fact, we prove here that the situation is a little bit more complicated.      

Two Phases of the Non-Commutative Quantum Mechanics with the Generalized Uncertainty Relations

We consider the quantum mechanics on the noncommutative plane with the generalized uncertainty relations \({\Delta } x_{1} {\Delta } x_{2} \ge \frac {\theta }{2}, {\Delta } p_{1} {\Delta } p_{2} \ge \frac {\bar {\theta }}{2}, {\Delta } x_{i} {\Delta } p_{i} \ge \frac {\hbar }{2}, {\Delta } x_{1} {\Delta } p_{2} \ge \frac {\eta }{2}\). We show that the model has two essentially different phases which is determined by \(\kappa = 1 + \frac {1}{\hbar ^{2} } (\eta ^{2} - \theta \bar {\theta })\). We construct a operator \(\hat {\pi }_{i}\) commuting with \(\hat {x}_{j} \) and discuss the harmonic oscillator model in two dimensional non-commutative space for three case κ > 0, κ = 0, κ < 0. Finally, we discuss the thermodynamics of a particle whose hamiltonian is related to the harmonic oscillator model in two dimensional non-commutative space.