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

The Onsager Algebra