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.

We introduce conformal multi-matrix models (CMM) as an alternative to conventional multi-matrix model description of two-dimensional gravity interacting with c < 1 matter. We define CMM as solutions to (discrete) extended Virasoro constraints. We argue that the so defined alternatives of multi-matrix models represent the same universality classes in continuum limit, while at the discrete level they provide explicit solutions to the multi-component KP hierarchy and by definition satisfy the discrete W -constraints. We prove that discrete CMM coincide with the (p, q)-series of 2d gravity models in a well-defined continuum limit, thus demonstrating that they provide a proper generalization of Hermitian one-matrix model.

We study SU (3)-invariant integrable models solvable by nested algebraic Bethe ansatz. We obtain a determinant representation for the particular case of scalar products of Bethe vectors. This representation can be used for the calculation of form factors and correlation functions of XXX SU (3)-invariant Heisenberg chain.

We study GL(3)-invariant integrable models solvable by the nested algebraic Bethe ansatz. Different formulas are given for the Bethe vectors and the actions of the generators of the Yangian 𝒴(gl3) on the Bethe vectors are considered. These actions are relevant for the calculation of correlation functions and form factors of local operators of the underlying quantum models.

We compute an universal weight function (off-shell Bethe vectors) in any representation with a weight singular vector of the quantum affine algebra U q ( gl N ) applying the method of projections of Drinfeld currents developed in [EKP].

The Baxter-Bazhanov-Stroganov model (also known as the τ (2) model) has attracted much interest because it provides a tool for solving the integrable chiral Z N -Potts model. It can be formulated as a face spin model or via cyclic L-operators. Using the latter formulation and the Sklyanin-Kharchev-Lebedev approach, we give the explicit derivation of the eigenvectors of the component B n (λ) of the monodromy matrix for the fully inhomogeneous chain of finite length. For the periodic chain we obtain the Baxter T-Q-equations via separation of variables. The functional relations for the transfer matrices of the τ (2) model guarantee non-trivial solutions to the Baxter equations. For the N = 2 case, which is free fermion point of a generalized Ising model, the Baxter equations are solved explicitly.

We find Bethe vectors for quantum integrable models associated with the supersymmetric Yangians Y (gl(m|n) in terms of the current generators of the Yangian double DY (gl(m|n)). We use the method of projections onto intersections of different type Borel subalgebras in this infinite dimensional algebra to construct the Bethe vectors. Calculating these projection the supersymmetric Bethe vectors can be expressed through matrix elements of the universal monodromy matrix elements. Using two different but isomorphic current realizations of the Yangian double DY (gl(m|n)) we obtain two different presentations for the Bethe vectors. These Bethe vectors are also shown to obey some recursion relations which prove their equivalence.

We study SU (3)-invariant integrable models solvable by a nested algebraic Bethe ansatz. We obtain determinant representations for form factors of diagonal entries of the monodromy matrix. This representation can be used for the calculation of form factors and correlation functions of the XXX SU (3)-invariant Heisenberg chain.