sl(N) Onsager's Algebra and Integrability

TL;DR

This paper introduces a generalization of Onsager's Algebra for sl(N), demonstrating its structure and showing that Hamiltonians based on it have infinite conserved quantities, indicating integrability.

Contribution

It defines an sl(N) analog of Onsager's Algebra, establishes its isomorphism to a fixed point subalgebra of sl(N) Loop Algebra, and proves the integrability of associated Hamiltonians.

Findings

Defined an sl(N) Onsager's Algebra with generalized relations.

Proved the algebra's isomorphism to a fixed point subalgebra of sl(N) Loop Algebra.

Showed Hamiltonians have infinite mutually commuting integrals of motion.

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.