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

Abstract

The tetrahedron algebra $⊠$ is an infinite-dimensional Lie algebra defined by generators $\{x_{ij}\mid i,j\in \{0,1,2,3\},i\ne j\}$ and some relations, including the Dolan-Grady relations. These twelve generators are called standard. We introduce a type of element in $⊠$ 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 $⊠$. An element $\xi \in ⊠$ 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 $⊠$ 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.

30 pages