Infinite-dimensional algebras in exactly solvable models of strongly correlated electrons and statistical mechanics

Several aspects of infinite-dimensional Lie algebras and quantum groups are considered in relation to exactly solvable models of statistical mechanics:;I. It is shown that the Lie algebra of the automorphic, meromorphic sl(2, C)-valued functions on a torus is a geometric realization of a certain infinite-dimensional finitely generated Lie algebra. In the trigonometric limit, when the modular parameter of the torus goes to zero, the former Lie algebra goes over into the sl(2, C)-valued loop algebra, while the latter one--into the Lie algebra (;II. The quantum bialgebra related to the Baxter's eight-vertex R-matrix is found as a quantum deformation of the Lie algebra of sl(2)-valued automorphic functions on a complex torus.;III. A new hidden symmetry of the one-dimensional Hubbard model is discovered. It is shown that the one-dimensional Hubbard model on the infinite chain has an infinite-dimensional algebra of symmetries. This algebra is a direct sum of two sl(2)-Yangians. This ;IV. The representations of the degenerate affine Hecke algebra in which the analogues of the Dunkl operators are given by finite-difference operators are introduced. The non-selfadjoint lattice analogues of the spin Calogero-Sutherland Hamiltonians are analyzed by Bethe-Ansatz. The gl(m)-Yangian representations arising from the finite-difference representations of the degenerate affine Hecke algebra are shown to be related to the Yangian representation of the 1-d Hubbard Model.;V. An sl(N) analog of Onsager's Algebra is defined through a finite set of relations that generalize the Dolan Grady defining relations for the original Onsager's Algebra. This infinite-dimensional Lie Algebra is shown to be isomorphic to a fixed point subalgebra of sl(N) Loop Algebra with respect to a certain involution. As the consequence of the generalized Dolan Grady relations a Hamiltonian linear in the generators of sl(N) Onsager's Algebra possesses an infinite number of mutually commuting integrals of motion.