Sl(N) Onsager's algebra and integrability

Abstract

We define an $ sl(N) $ analog of Onsager's Algebra 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 is shown to posses an infinite number of mutually commuting integrals of motion.