Skrypnyk T. The method of non-skew-symmetric r-matrices and new integrable classical and quantum system

The thesis is devoted to elaboration of the methods of the theory of finite-dimensional classical and quantum integrable systems, of the theory of soliton equations and of the theory of infinite-dimensional Lie algebras of their hidden symmetries. The key object in the proposed investigations is classical r-matrix r(u,v) with spectral parameters. In the thesis we have constructed new examples of non-skew-symmetric classical r-matrices. With its help we have explicitly constructed infinite-dimensional quasi-graded Lie algebras with the Kostant-Adler-Symmes decomposition, that serve as the algebras of the hidden symmetries of a wide class of the classical integrable system admitting Lax representation. We have constructed the explicit formulae for all possible meromorphic Lax matrices of the finite-dimensional integrable systems and U-V pairs of soliton equations that are connected with the given classical r-matrix. We have shown that reduction both in finite-dimensional integrable systems and hierarchies of soliton equations is connected with special points of the classical r-matrices were they become degenerated. We have constructed new examples of finite-dimensional integrable systems and soliton equations associated with the classical non-skew-symmetric r-matrices. In particular, we have constructed integrable modifications of the Toda chains and integrable modifications of the abelian and non-abelian two-dimensional Toda field theories. In the thesis we have shown that quantum integrable systems can be associated also with general non-skew-symmetric classical r-matrices with spectral parameters. We have constructed a lot of new classes of such the systems. Among them there are integrable spin chains with the long-ranged interaction among the spins of the chain --- generalized Gaudin systems and generalized Gaudin systems in an external magnetic field; integrable spin-boson models, namely, generalized n-level, many-mode Jaynes-Cummings-Dicke models; integrable boson models, namely, many-boson gene-ralizations of Bose-Hubbard dimmers. In the case of the Lie algebra gl(2) we have developed the algebraic ansatz method based on the general non-skew-symmetric classical r-matrices with spectral parameters. In the case of the Lie algebra gl(n), n>2 and general non-skew-symmetric r-matrices we have generalized hierarchical (nested) Bethe ansatz that is based both on the standard chain of subalgebras gl(n)> gl(n-1)> ...> gl(1) and non-standard chains of subalgebras compatible with different restrictions of the type gl(n)> gl(n-k)+ gl(k). With the help of this method we have found the spectrа of the wide class of quantum-integrable models. In particular, we have found the spectra of Zp-graded models of the Gaudin-type, of Jaynes-Cummings-Dicke models and of many-boson generalizations of Bose-Hubbard dimmer model.