Tridiagonal pairs, the Onsager algebra, and the three-point sl(2) loop algebra

In this thesis we explain how the Onsager Lie algebra O , the three-point sl2 loop algebra, and tridiagonal pairs of linear transformations are related. Here is a summary of our results. O has a well known presentation involving two generators, said to be standard, and two relations. We show that the standard generators of O act on each finite-dimensional irreducible O -module as a tridiagonal pair. We also classify up to isomorphism which tridiagonal pairs arise in this way. To illuminate how the three-point sl2 loop algebra is related to O we give a new presentation of the three-point sl2 loop algebra via generators and relations. To obtain this presentation we define a Lie algebra ⊠ by generators and relations and eventually show that ⊠ is isomorphic to the three-point sl2 loop algebra. ⊠ has essentially six generators and it is natural to identify them with the six edges of a tetrahedron. We show that each pair of opposite edges in ⊠ are the standard generators for a subalgebra of ⊠ that is isomorphic to O . Let us call these Onsager subalgebras. We show that the vector space ⊠ is the direct sum of its three Onsager subalgebras. We also obtain a bijection between the isomorphism classes of finite-dimensional irreducible ⊠ -modules and the isomorphism classes of finite-dimensional irreducible O -modules where the action of each standard generator of O has trace 0. In order to describe how ⊠ is related to tridiagonal pairs we introduce a certain relationship between two given tridiagonal pairs which we call adjacency. We then show that the three pairs of opposite edges in ⊠ act on each finite-dimensional irreducible ⊠ -module as three mutually adjacent tridiagonal pairs.