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 (A_sp {1}{(1)})^' /(centre). 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 Y(sl(2)) oplus Y(sl(2)) symmetry is an extension of the well-known sl(2)oplus sl(2). The deformation parameters of the Yangians are proportional to the coupling constant of the Hubbard model Hamiltonian. 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.