NEW FUNDAMENTAL SYMMETRIES OF INTEGRABLE SYSTEMS AND PARTIAL BETHE-ANSATZ

We introduce a new concept of quasi-Yang-Baxter algebras. These algebr as, being simple but non-trivial deformations of ordinary algebras of monodromy matrices, realize a new type of quantum dynamical symmetry a nd find unexpected and remarkable applications in quantum inverse scat tering method (QISM). We show that applying to quasi-Yang-Baxter algeb ras the standard procedure of QISM one obtains new wide classes of qua ntum models which, being integrable (i.e., having enough number of com muting integrals of motion), are only quasi-exactly solvable (i.e., ad mit an algebraic Bethe ansatz solution for arbitrarily large but limit ed parts of the spectrum). These quasi-exactly solvable models natural ly arise as deformations of known exactly solvable ones. A general the ory of such deformations is proposed. The correspondence ''Yangian-qua si-Yangian'' and ''XXX spin models-quasi-XXX spin models'' is discusse d in detail. We also construct the classical conterparts of quasi-Yang -Baxter algebras and show that they naturally generate new classes of classical integrable models. We conjecture that these models are quasi -exactly solvable, i.e., admit only partial construction of action-ang le variables in the framework of the classical inverse scattering meth od. The mathematical formalism elaborated in this paper naturally lead s to the notions of ''quasi-commutators'' and ''quasi-Poisson brackets '' which can be considered as special deformations of the ordinary com mutators and Poisson brackets and may play a fundamental role in the t heory of integrable and quasi-exactly solvable systems. (C) 1998 Acade mic Press.