The Onsager Algebra

TL;DR

This thesis explores four different realizations of the Onsager algebra, providing new insights into its structure, ideal classification, and subalgebra relationships.

Contribution

It introduces novel perspectives on the Onsager algebra by presenting four realizations and classifying its ideals within these frameworks.

Findings

Classified all closed ideals of the Onsager algebra as an equivariant map algebra.

Explicitly described all ideals of the Onsager algebra as a subalgebra of the tetrahedron algebra.

Connected the Onsager algebra to the tetrahedron algebra, enriching its structural understanding.

Abstract

In this thesis, four realizations of the Onsager algebra are explored. We begin with its original definition as introduced by Lars Onsager. We then examine how the Onsager algebra can be presented as a Lie algebra with two generators and two relations. The third realization of the Onsager algebra consists of viewing it as an equivariant map algebra which then gives us the tools to classify its closed ideals. Finally, we examine the Onsager algebra as a subalgebra of the tetrahedron algebra. Using this fourth realization, we explicitly describe all its ideals.