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.     

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.     

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.     

The supersymmetric one-dimensional string

We consider a supersymmetric discretized string. The full string theory is defined as the sum over the triangulations of the surface, which is imbedded in the superspace. In the continuum limit such a string theory is described by an appropriate Wess-Zumino model. We present an explicit computation of the properties of the string in the ID case: we find that supersymmetry is spontaneously broken.     

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.