Four bases for the Onsager Lie algebra related by a ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ action

Abstract

The Onsager Lie algebra $O$ is an infinite-dimensional Lie algebra defined by generators $A$, $B$ and relations $[A,[A,[A,B]]]=4[A,B]$ and $[B,[B,[B,A]]]=4[B,A]$. Using an embedding of $O$ into the tetrahedron Lie algebra $⊠$, we obtain four direct sum decompositions of the vector space $O$, each consisting of three summands. As we will show, there is a natural action of ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ on these decompositions. For each decomposition, we provide a basis for each summand. Moreover, we describe the Lie bracket action on these bases and show how they are recursively constructed from the generators $A$, $B$ of $O$. Finally, we discuss the action of ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ on these bases and determine some transition matrices among the bases.

25 pages