Four bases for the Onsager Lie algebra related by a $\mathbb{Z}_2 \times \mathbb{Z}_2$ action
Jae-Ho Lee
TL;DR
This work analyzes the Onsager Lie algebra $O$ through its embedding into the tetrahedron algebra $\boxtimes$, revealing four path-shaped direct-sum decompositions of $O$ into three summands under a natural $\mathbb{Z}_2\times\mathbb{Z}_2$ action. For each decomposition, the authors construct explicit bases for the summands, describe the Lie bracket action in terms of recursive relations that build higher elements from the generators $A$ and $B$, and study the action of the Klein four-group on these bases. They organize the ambient structure using the three-point $\mathfrak{sl}_2$ loop algebra $L(\mathfrak{sl}_2)^+$ and the equitable basis of $\mathfrak{sl}_2$, leveraging the isomorphism $\sigma: \boxtimes \to L(\mathfrak{sl}_2)^+$ to realize $O$ as the intersection of well-chosen $X_{ij}$-like subspaces. The paper culminates with an explicit description of the four bases, the $G$-action on them, and exact transition matrices between adjacent bases, thus enabling concrete computations and deeper understanding of $O$’s internal structure and its relationship to $L(\mathfrak{sl}_2)^+$.
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 $\boxtimes$, 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.