Yangian symmetries and integrability of the Dirac equation with spin symmetry

We show that a Yangian symmetry, namely, Y(su(2))$Y\left(su\left(2\right)\right)$, exists in the Dirac equation with spin symmetry when the potential term takes a Coulomb form. We construct the generators of Y(su(2))$Y\left(su\left(2\right)\right)$ explicitly and get the energy spectrum of this model from the representation theory for Y(su(2))$Y\left(su\left(2\right)\right)$. We also show that this model is integrable, from RTT relations.     

Yangian double

Studying the algebraic structure of the doubleDY(g) of the Yangian Y(g), we present the triangular decomposition ofDY(g) and a factorization for the canonical pairing of the Yangian with its dual inside Y⁰(g). As a consequence, we describe a structure of the universalR-matrixR forDY(g) which is complete forDY(s1₂). We demonstrate how this formula works in evaluation representations of Y(sl₂). We interpret the one-dimensional factor arising in concrete representations ofR as a bilinear form on highest-weight polynomials of irreducible representations of Y(g) and express this form in terms of -functions.Partially supported by ISF grant MBI000 and Russian Foundation for Fundamental Researches     

Yangian symmetric correlators

Similarity transformations and eigenvalue relations of monodromy operators composed of Jordan–Schwinger type L matrices are considered and used to define Yangian symmetric correlators of n-dimensional theories. Explicit expressions are obtained and relations are formulated. In this way basic notions of the Quantum inverse scattering method provide a convenient formulation for high symmetry and integrability not only in lower dimensions.     

Higher symmetries in a class of cosmological models

A particular class of solutions of Einstein's field equations, in which the source is a perfect fluid and the geometry admits a two-parameter Abelian isometry group with spacelike orbits, is examined for the possible admission of higher symmetries. It is shown that in certain cases the geometry is spatially homogeneous, being either of the orthogonal Bianchi type I or of the tilted Bianchi-Behr type VIU_h withh = –4, and that no other cases of higher symmetry (in the sense of isometry groups) are possible. The global behaviour of the Bianchi-Behr type VIU_h, models is studied, and conformal diagrams are given. This investigation is extended to the case when additional homothetic vectors are admitted, and a parallel study of the global behaviour, complete with conformal diagrams, is provided. A brief argument also shows that, within the class considered, whenever the fluid flow-lines possess identical thermal histories, the geometry is necessarily spatially homogeneous.     

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.     

Drinfeld Functor and Finite-Dimensional Representations of Yangian

We extend the results of Drinfeld on the Drinfeld functor to the case . We present the character of finite-dimensional representations of the Yangian in terms of the Kazhdan–Lusztig polynomials as a consequence. Received: 4 September 1998 / Accepted: 12 February 1999     

Parabolic Presentations of the Yangian

We introduce some new presentations for the Yangian associated to the Lie algebra These presentations are parametrized by tuples of positive integers summing to n. At one extreme, for the tuple (n), the presentation is the usual RTT presentation of Y_n. At the other extreme, for the tuple (1ⁿ), the presentation is closely related to Drinfelds presentation. In general, the presentations are useful for understanding the structure of the standard parabolic subalgebras of Y_n.Research partially supported by the NSF (grant no. DMS-0139019).Acknowledgement The second author would like to thank Arun Ram for stimulating conversations.     

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.     

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.     

Yangian construction of the Virasoro algebra

We show that a Yangian construction based on the algebra of an infinite number of harmonic oscillators (i.e. a vibrating string) terminates after one step, yielding the Virasoro algera.     

Yangian superalgebras in conformal field theory

Quantum Yangian symmetry in several sigma models with supergroup or supercoset as target is established. Starting with a two-dimensional conformal field theory that has current symmetry of a Lie superalgebra with vanishing Killing form we construct non-local charges and compute their properties. Yangian axioms are satisfied, except that the Serre relations only hold for a subsector of the space of fields. Yangian symmetry implies that correlation functions of fields in this sector satisfy Ward identities. We then show that this symmetry is preserved by certain perturbations of the conformal field theory.     

Traces of Intertwining Operators for the Yangian Double

The traces over infinite-dimensional representations of the central extended Yangian double for the product of operators which intertwine these representations are calculated. For the special combinations of the intertwining operators, the traces are identified with form factors of local operators in the SU(2)-invariant Thirring model. This identification is based on the identities which are deformed analogs of the Gauss–Manin connection identities for the hyperelliptic curves.     

Resonance spins and Regge residues

The diffractive component of the slopes of elastic differential cross sections at intermediate energies is examined in the context of a strong coupling multiperipheral cluster resonance model. The crucial ingredient is the incorporation of resonance spins. Qualitative agreement with πp, Kp, and pp scattering is obtained. We also conjecture a mechanism whereby s-channel helicity can be approximately conserved in elastic but not in certain quasi-elastic diffractive processes. The problem of the extension to NAL-ISR energies is briefly discussed.     

Competition between Ising spins and classical n-vector spins on the honeycomb and the diced lattice

The competition between ordering and disordering is investigated for mixed spin models of Ising spins and classical n-vector spins on the honeycomb and the diced lattice. The critical indices of the specific heat and the spontaneous magnetization for the mixed spin models turn out to be the same as those for the two-dimensional Ising model.     

On factorizing twists for the Yangian and the quantum affine algebra

We calculate explicit expressions for factorizing Drinfel’d twists in evaluation representations of the YangianY(sl₂) and of the quantum affine algebraUq . From the twists we derive a closed and representation-independent form of theR-matrices in these representations. Employing a carefully chosen basis it is possible to recover the results for the Yangian (rational case) as theq → 1 limit of the expressions for the quantum affine algebra (trigonometric case). Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001.