Diagonal reduction algebras of \gl type

Several general properties, concerning reduction algebras - rings of definition and algorithmic efficiency of the set of ordering relations - are discussed. For the reduction algebras, related to the diagonal embedding of the Lie algebra $gl_n$ into $gl_n \oplus gl_n$, we establish a stabilization phenomenon and list the complete sets of defining relations.Comment: 24 pages, no figure

Yangians and Mickelsson Algebras I

We study the composition of the functor from the category of modules over the Lie algebra gl_m to the category of modules over the degenerate affine Hecke algebra of GL_N introduced by I. Cherednik, with the functor from the latter category to the category of modules over the Yangian Y(gl_n) due to V. Drinfeld. We propose a representation theoretic explanation of a link between the intertwining operators on the tensor products of Y(gl_n)-modules, and the `extremal cocycle' on the Weyl group of gl_m defined by D. Zhelobenko. We also establish a connection between the composition of two functors, and the `centralizer construction' of the Yangian Y(gl_n) discovered by G. Olshanski.Comment: publication details added. arXiv admin note: substantial text overlap with arXiv:math/060627

Twisted Yangians and Mickelsson Algebras II

We introduce a skew analogue of the composition of the Cherednik and Drinfeld functors for twisted Yangians. Our definition is based on the skew Howe duality, and originates from the centralizer construction of twisted Yangians due to Olshanski. Using our functor, we establish a correspondence between intertwining operators on the tensor products of certain modules over twisted Yangians, and the extremal cocycle on the hyperoctahedral group.Comment: final versio

Generating Series for Nested Bethe Vectors

We reformulate nested relations between off-shell $U_q(\widehat{\mathfrak{gl}}_N)$ Bethe vectors as a certain equation on generating series of strings of the composed $U_q(\widehat{\mathfrak{gl}}_N)$ currents. Using inversion of the generating series we find a new type of hierarchical relations between universal off-shell Bethe vectors, useful for a derivation of Bethe equation. As an example of application, we use these relations for a derivation of analytical Bethe ansatz equations [Arnaudon D. et al., Ann. Henri Poincar\'e 7 (2006), 1217-1268, math-ph/0512037] for the parameters of universal Bethe vectors of the algebra $U_q(\widehat{\mathfrak{gl}}_2)$.Comment: This is a contribution to the Special Issue on Kac-Moody Algebras and Applications, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Integrable structure of W_3 Conformal Field Theory, Quantum Boussinesq Theory and Boundary Affine Toda Theory

In this paper we study the Yang-Baxter integrable structure of Conformal Field Theories with extended conformal symmetry generated by the W_3 algebra. We explicitly construct various T- and Q-operators which act in the irreducible highest weight modules of the W_3 algebra. These operators can be viewed as continuous field theory analogues of the commuting transfer matrices and Q-matrices of the integrable lattice systems associated with the quantum algebra U_q(\hat{sl}(3)). We formulate several conjectures detailing certain analytic characteristics of the Q-operators and propose exact asymptotic expansions of the T- and Q-operators at large values of the spectral parameter. We show, in particular, that the asymptotic expansion of the T-operators generates an infinite set of local integrals of motion of the W_3 CFT which in the classical limit reproduces an infinite set of conserved Hamiltonians associated with the classical Boussinesq equation. We further study the vacuum eigenvalues of the Q-operators (corresponding to the highest weight vector of the W_3 module) and show that they are simply related to the expectation values of the boundary exponential fields in the non-equilibrium boundary affine Toda field theory with zero bulk mass.Comment: LaTeX, 87 pages, 1 figure. Misprints correcte

Universal R-matrix for quantum affine algebras Uq(A2(2)) and Uq(osp(1|2)) with Drinfeld comultiplication

AbstractWe derive an integral formula for the universal R-matrix for the twisted quantum affine algebra Uq(A2(2)) and quantum affine superalgebra Uq(osp(1|2)) with Drinfeld comultiplication

Rational and polynomial representations of Yangians

We define natural classes of rational and polynomial representations of the Yangian of the general linear Lie algebra. We also present the classification and explicit realizations of all irreducible rational representations of the Yangian.Comment: 23 pages, affiliations update

Cherednik algebras and Zhelobenko operators

We study canonical intertwining operators between induced modules of the trigonometric Cherednik algebra. We demonstrate that these operators correspond to the Zhelobenko operators for the affine Lie algebra of type A. To establish the correspondence, we use the functor of Arakawa, Suzuki and Tsuchiya which maps certain modules of the affine Lie algebra to modules of the Cherednik algebra