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 ( 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.
For quantum deformations of finite-dimensional contragredient Lie (super)algebras we give an explicit formula for the universal .R-matrix. This formula generalizes the analogous formulae for quantized semisimple Lie algebras obtained by M. Rosso, A. N. Kirillov, and N. Reshetikhin, Ya. S. Soibelman, and S. Z. Levendorskii. Our approach is based on careful analysis of quantized rank 1 and 2 (super)algebras, a combinatorial structure of the root systems and algebraic properties of ^-exponential functions. We don't use quantum Weyl group.
A universal weight function for a quantum affine algebra is a family of functions with values in a quotient of its Borel subalgebra, satisfying certain coalgebraic properties. In representations of the quantum affine algebra it gives off-shell Bethe vectors and is used in the construction of solutions of the qKZ equations. We construct a universal weight function for each untwisted quantum affine algebra, using projections onto the intersection of Borel subalgebras of different types, and study its functional properties.
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