The dressing techniques for intermediate hierarchies

We consider the generalized AKNS systems, introduced and discussed recently in [1]. We have shown that the dressing techniques both in matrix pseudo-differential operators and formal series with respect to the spectral parameter can be developed for these hierarchies.On leave of absence from the ITP, Kiev 252143, Ukraine. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 3, pp. 422–436, June, 1995.     

On the bosonization ofL-operators for the quantum affine algebraU q (sl 2)

Some relations between different objects associated with quantum affine algebras are reviewed. It is shown that the Frenkel-Jing bosonization of a new realization of the quantum affine algebra as well as bosonization of L-operators for this algebra can be obtained from Zamolodchikov-Faddeev algebras defined by the quantum R-matrix satisfying unitarity and crossing-symmetry conditions.On leave of absence from the ITP, Kiev 252143, Ukraine. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 1, pp. 64–77, July, 1995.     

Scalar products in models with the GL(3) trigonometric R-matrix: General case

We study quantum integrable models with the GL(3) trigonometric R-matrix solvable by the nested algebraic Bethe ansatz and obtain an explicit representation for a scalar product of generic Bethe vectors in terms of a sum over partitions of Bethe parameters. This representation generalizes the known formula for scalar products in models with the GL(3)-invariant R-matrix.     

Bethe Vectors for Orthogonal Integrable Models

Theoretical and Mathematical Physics - We consider quantum integrable models associated with the $$\mathfrak{so}_3$$ algebra and describe Bethe vectors of these models in terms of the current...     

Nonlocal integrable partners to generalized MKdV and two-dimensional Toda lattice equation in the formalism of a dressing method with quantized spectral parameter

Two new hierarchies, MILW₂ and a two-dimensional nonlocal Toda lattice are constructed. The characteristic property of the first one is the connection with the ILW₂ hierarchy by means ofgl(2) Miura transformation. On the other hand, MILW₂ equations turn out to be symmetry equations for a two-dimensional nonlocal Toda lattice. A new version of the dressing technique with quantized spectral parameter is proposed.     

Weight Function for the Quantum Affine Algebra Uq( $$\widehat{\mathfrak{s}\mathfrak{l}}_3$$ )

We give a precise expression for the universal weight function of the quantum affine algebra U _q ( ). The calculations use the technique of projecting products of Drinfeld currents on the intersections of Borel subalgebras. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 1, pp. 3–34, October, 2005.     

Quantum relativistic Toda chain at root of unity: invariant approach

Recent results, concerning the quantum relativistic Toda chain at root of unity, are interpreted in this note from the point of view of invariant auxiliary linear problem. Such form of the linear problem arises in the three-dimensional integrable systems, and several results from the three dimensional models are applied to the relativistic Toda chain. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. This work was supported in part by the grant INTAS OPEN 00-00055. S.P.’s work was supported by the grants RFBR 00-02-16477 and grant for support of scientific schools RFBR 00-15-96557, S.S’s work was supported by the grant RFBR 01-01-00201.     

Weight Function and Nested Bethe Ansatz

The projections of the product of the Drinfeld’s currents onto intersections of the different Borel subalgebras in the current realization of the quantum affine algebra are calculated. It was shown that this projections has the structure of the off-shell nested Bethe ansatz vectors. Communicated by Vincent Rivasseau Dedicated to the memory of Daniel Arnaudon Submitted: January 21, 2006; Accepted: March 13, 2006