Deformed oscillators algebra formulation of the nonlinear Schrödinger hierarchy and of its symmetry

We present a self-contained formulation of the Nonlinear Schrödinger hierarchy and its Yangian symmetry in terms of deformed oscilator algebra (Z.F. algebra). The link between Yangian Y(gl _N) and finite W(gl _pN, N.gl _p) algebras is also illustrated in this framework.     

Yangian Realization for Dirac Oscillator

We investigate the realizations of Yangian algebra for a Dirac oscillator. Applying the representation theory of Y(sl(2)) to Dirac oscillator, shift operaors for different energy levelsfor this system are obtained.     

Yangian Realization for Dirac Oscillator

We investigate the realizations of Yangian algebra for a Dirac oscillator. Applying the representation theory of Y(sl(2)) to Dirac oscillator, shift operators for different energy levels for this system are obtained.     

Quantum Berezinian and the classical capelli identity

We study a superanalogue of the Yangian of the Lie algebra gl _m . We apply our constructions to invariant theory.     

On the integrals of motion for an exactly solvable model of interacting fermions

Based on the Lax formalism, integrals of motion are constructed for the Sutherland hyperbolic systems of particles with internal degrees of freedom (su(n) spins) situated in an external field with the Morse potential characterized by the parameter τ². It is shown that the corresponding infinite-dimensional algebra determining the hidden symmetry of the systems is not of the Yangian type.     

New Spin Generalisation for Long Range Interaction Models

We study new interactions between degrees of freedom for Calogero, Sutherland and confined Calogero spin models. These interactions are encoded by the generators of the Lie algebra so(N) or sp(N). We find the symmetry algebras of these new models: the half-loop algebra based on so(N) or sp(N) for the Calogero models and the Yangian of so(N) or sp(N) for the two types of other models. Surprisingly, these symmetry occur only for a specific value of the coupling constant.Dedicated to my PhD supervisor and friend D. Arnaudon.     

Some remarks on Yang-Baxter algebras

We point out the existence of an alternative algebraic structure in Yang-Baxter algebra with trigonometric R-matrix, which appears to be the generalization of the Yangian in Yang-Baxter algebras with rational R-matrix and which is described most naturally by q-commutators. Some properties are presented, in particular in the case of the well-known symmetric six-vertex model. Received: 13 February 1998 / Revised: 16 March 1998 / Accepted: 17 April 1998     

Yangian and Topological Basis Expressions of Two Types of XXZ Spin Chains

We construct a set of orthogonal topological basis states, based on which Temperley-Lieb algebra (TLA) generators U_ij can be reduced into four subspaces. According to the relations between the XXZ model and TLA generators U_ij, two types of eigenstates of the XXZ model can be expressed by a topological basis. We also analyze the Yangian realizations of XXZ model systems and illustrate them with figures.

     

Another Free Boson Representation of Yangian Double DYħ(slN) with Arbitrary Level

We derive a free boson representation of the Yangian double DY_ħ(sl_N) with arbitrary level k by the observation that there is a correspondence between the q-affine algebra and Yangian double associated with the same Cartan matrix. Vertex operator and screening currents cannot be obtained in the same way. From this point of view, the Yangian double with centre cannot be regarded as the degenerated case of the quantum affine algebra.     

Spin models and superconformal Yang–Mills theory

We apply novel techniques in planar superconformal Yang–Mills theory which stress the role of the Yangian algebra. We compute the first two Casimirs of the Yangian, which are identified with the first two local Abelian Hamiltonians with periodic boundary conditions, and show that they annihilate the chiral primary states. We streamline the derivation of the R-matrix in a conventional spin model, and extend this computation to the gauge theory. We comment on higher-loop corrections and higher-loop integrability.     

The quantum super-Yangian and Casimir operators of Uq(gl(M |N))

TheZ ₂ graded Yangian Yq(gl(M |N)) associated with the Perk-SchultzR matrix is introduced. Its structural properties, the central algebra in particular, are studied. AZ ₂-graded associative algebra epimorphism Yq(gl(M |N)) Uq (gl(M |N)) is obtained in explicit form. Images of central elements of the quantum super-Yangian under this epimorphism yield the Casimir operators of the quantum supergroup Uq(gl(M |N)) constructed in an earlier publication.     

Schur duality in the toroidal setting

The classical Frobenius-Schur duality gives a correspondence between finite dimensional representations of the symmetric and the linear groups. The goal of the present paper is to extend this construction to the quantum toroidal setup with only elementary (algebraic) methods. This work can be seen as a continuation of [J, D1 and C2] (see also [C-P and G-R-V]) where the cases of the quantum groups U _q (sl(n)), Y(sl(n)) (the Yangian) and U _q (sl(n)) are given. In the toroidal setting the two algebras involved are deformations of Cherednik's double affine Hecke algebra introduced in [C1] and of the quantum toroidal group as given in [G-K-V]. Indeed, one should keep in mind the geometrical construction in [G-R-V] and [G-K-V] in terms of equivariant K-theory of some flag manifolds. A similar K-theoretic construction of Cherednik's algebra has motivated the present work. At last, we would like to lay emphasis on the fact that, contrary to [J, D1 and C2], the representations involved in our duality are infinite dimensional. Of course, in the classical case, i.e.,q=1, a similar duality holds between the toroidal Lie algebra and the toroidal version of the symmetric group. The authors would like to thank V. Ginzburg for a useful remark on a preceding version of this paper. Communicated by M. Jimbo     

On affine yangians

A Yangian , a deformation of the universal enveloping algebra of the two-dimensional loop algebra sl(2) C [t ^–1,t;u], is constructed. This deformation is an analogue of a Yangian which was constructed by V. Drinfeld for any simple Lie algebra. The PBW theorem for is proved and some representations are constructed. Like usual Yangians, possesses a one-dimensional group of auto- morphisms and at zero level - a two-dimensional group of automorphisms. This observation allows one to conjecture that the representation theory of should give rise to new solutions of QYBE.Yangians of other affine algebras can be constructed similarly and they enjoy similar properties.     

The gl (M|N) super Yangian and its finite-dimensional representations

Methods are developed for systematically constructing the finite-dimensional irreducible representations of the super Yangian Y (gl(M|N)) associated with the Lie superalgebra gl(M|N). It is also shown that every finite-dimensional irreducible representation of Y (gl(M|N)) is of highest weight type, and is uniquely characterized by a highest weight. The necessary and sufficient conditions for an irrep to be finite-dimensional are given.     

Yangian symmetry in integrable quantum gravity

Dimensional reduction of various gravity and supergravity models leads to effectively two-dimensional field theories described by gravity coupled G/H coset space σ-models. The transition matrices of the associated linear system provide a complete set of conserved charges. Their Poisson algebra is a semi-classical Yangian double modified by a twist which is a remnant of the underlying coset structure. The classical Geroch group is generated by the Lie-Poisson action of these charges. Canonical quantization of the structure leads to a twisted Yangian double with fixed central extension at a critical level.     

Application of Y（sl（2）） Algebra for Entanglement of Two-Qubit System

We construct the transition operators in terms of the generators of the general Yangian and the reduced Yangian. By acting these operators on a two-qubit pure state, we find that the entanglement degrees of the states are all decreased from the certain values to zero for the reduced Yangian algebra, which makes the state disentangled. This result sheds new light on the physical meaning of Y （sl（2） ） in quantum information.     

Gauss Decomposition of the Yangian $$Y({\mathfrak{gl}}_{m|n})$$

We describe a Gauss decomposition for the Yangian of the general linear Lie superalgebra. This gives a connection between this Yangian and the Yangian of the classical Lie superalgebra Y(A(m − 1, n − 1)) (for m ≠ n) defined and studied in papers by Stukopin, and suggests natural definitions for the Yangians and Y(A(n, n)). We also show that the coefficients of the quantum Berezinian generate the centre of the Yangian . This was conjectured by Nazarov in 1991.     

{\hbar}-adic Quantum Vertex Algebras and Their Modules

This is a paper in a series to study vertex algebra-like structures arising from various algebras including quantum affine algebras and Yangians. In this paper, we study notions of (^h/_2p){\hbar}-adic nonlocal vertex algebra and (^h/_2p){\hbar}-adic (weak) quantum vertex algebra, slightly generalizing Etingof-Kazhdan’s notion of quantum vertex operator algebra. For any topologically free \mathbb C[[(^h/_2p)]]{{\mathbb C}\lbrack\lbrack{\hbar}\rbrack\rbrack}-module W, we study (^h/_2p){\hbar}-adically compatible subsets and (^h/_2p){\hbar}-adically S{\mathcal{S}}-local subsets of (End W)[[x, x ⁻¹]]. We prove that any (^h/_2p){\hbar}-adically compatible subset generates an (^h/_2p){\hbar}-adic nonlocal vertex algebra with W as a module and that any (^h/_2p){\hbar}-adically S{\mathcal{S}}-local subset generates an (^h/_2p){\hbar}-adic weak quantum vertex algebra with W as a module. A general construction theorem of (^h/_2p){\hbar}-adic nonlocal vertex algebras and (^h/_2p){\hbar}-adic quantum vertex algebras is obtained. As an application we associate the centrally extended double Yangian of \mathfrak s\mathfrak l₂{{\mathfrak s}{\mathfrak l}_{2}} to (^h/_2p){\hbar}-adic quantum vertex algebras.     

Multiparameter deformed and non-standard Y(glM) Yangian symmetry in a novel class of spin Calogero-Sutherland models

It is well known through a recent work of Bernard, Gaudin, Haldane and Pasquier (BGHP) that the usual spin Calogero-Sutherland (CS) model, containing particles with M internal degrees of freedom, respects the Y(gl_M) Yangian symmetry. By following and suitably modifying the approach of BGHP, in this article we construct a novel class of spin CS models which exhibit multiparameter deformed or ‘non-standard’ variants of Y(gl_M) Yangian symmetry. An interesting feature of such CS Hamiltonians is that they contain many-body spin-dependent interactions, which can be calculated directly from the associated rational solutions of the Yang-Baxter equation. Moreover, these spin-dependent interactions often lead to ‘anyon-like’ representations of the permutation algebra on the combined internal space of all particles. We also find the general forms of conserved quantities as well as Lax pairs for the above-mentioned class of spin CS models, and describe the method of constructing their exact wave functions.     

An ?-Deformed Virasoro Algebra as Hidden Symmetry of the Restricted sine-Gordon Model

As the Yangian double with center, which is deformed from affine algebra by the additive loop parameter ?, we get the commutation relation and the bosonization of quantum ?-deformed Virasoro algebra. The corresponding Miura transformation, the associated screening operators and the BRST charge have been studied. Moreover, we also construct the bosonization for type Ⅰ and type Ⅱ intertwiner vertex operators. Finally, we show that the commutation relations of these vertex operators in the case of p^′ = γ, p = γ - 1 and ? = π actually give the exact scattering matrix of the restricted sine-Gordon model.