Iorgov N. Quantum integrable systems with quantum algebraic symmetries

The thesis is devoted to elaboration of symmetry methods, in particular the theory of quantum algebras and their representations, for construction of the eigenstates of quantum integrable systems. For the Baxter-Bazhanov-Stroganov quantum chain, the eigenstates were derived by the separation of variables method. An integral representation for the wave functions of the quantum Toda chain with integrable boundary interaction is found. The radial components of the Casimir elements of the quantum algebra U_q(gl(n)) corresponding the quantum analogue of GL(n)/SO(n) are derived. It is shown that they coincide with the Hamiltonians of quantum n-particle Ruijsenaars model.