The standard generators of the tetrahedron algebra and their look-alikes

TL;DR

This paper explores the structure of the tetrahedron algebra, introducing elements that resemble standard generators and establishing a basis where each element is similar to specific generators, revealing new insights into its algebraic structure.

Contribution

The paper defines a new class of elements called 'look-alikes' in the tetrahedron algebra and constructs a basis where each element is either a look-alike of certain standard generators.

Findings

Established a basis where each element is $x_{ij}$-like, $x_{jk}$-like, or $x_{ki}$-like.

Characterized the properties of these look-alike elements and their relations.

Provided multiple perspectives on the new basis within the algebra.

Abstract

The tetrahedron algebra $\boxtimes$ is an infinite-dimensional Lie algebra defined by generators $\{x_{ij} \mid i, j \in \{0, 1, 2, 3\}, i \neq j\}$ and some relations, including the Dolan-Grady relations. These twelve generators are called standard. We introduce a type of element in $\boxtimes$ that "looks like" a standard generator. For mutually distinct $h, i, j, k \in \{0, 1, 2, 3\}$, consider the standard generator $x_{ij}$ of $\boxtimes$. An element $\xi \in \boxtimes$ is called $x_{ij}$-like whenever both (i) $\xi$ commutes with $x_{ij}$; (ii) $\xi$ and $x_{hk}$ satisfy a Dolan-Grady relation. Pick mutually distinct $i,j,k \in \{0,1,2,3\}$. In our main result, we find an attractive basis for $\boxtimes$ with the property that every basis element is either $x_{ij}$-like or $x_{jk}$-like or $x_{ki}$-like. We discuss this basis from multiple points of view.