Linking the special orthogonal algebra \(\mathfrak{so}_4\) and the tetrahedron algebra \(\boxtimes \)
The tetrahedron algebra \(\boxtimes\) was defined by \textit{B. Hartwig} and \textit{P. Terwilliger} [J. Algebra 308, No. 2, 840--863 (2007; Zbl 1163.17026)] as the Lie algebra with generators \(x_{ij}\), \(0\leq i\neq j\leq 3\) (one generator for each edge of a tetrahedron) and relations: \(x_{ij}+x_{ji}=0\), \([x_{ij},x_{jk}]=2(x_{ij}+x_{jk})\), \([x_{hi},[x_{hi},[x_{hi},x_{jk}]]]=4[x_{hi},x_{jk}]\), for distinct \(i,j,k,h\). Each `face' of the tetrahedron has \(3\) edges, whose corresponding generators span a subalgebra isomorphic to \(\mathfrak{sl}_2\). Hartwig and Terwilliger proved, in the reference above, that the tetrahedron algebra is isomorphic to the three-point \(\mathfrak{sl}_2\) loop algebra: \(\mathfrak{sl}_2\otimes \mathbb{F}[t,t^{-1},(1-t)^{-1}]\). It follows that any nonzero homomorphism \(\boxtimes\rightarrow \mathfrak{sl}_2\) is determined by an automorphism of \(\mathfrak{sl}_2\) and a ring homomorphism \(\mathbb{F}[t,t^{-1},(1-t)^{-1}]\rightarrow \mathbb{F}\), which in turn is determined by a scalar \(p\in\mathbb{F}\setminus\{0,1\}\) (the image of \(t\)). In the paper under review, some explicit homomorphisms from \(\boxtimes\) to the special orthogonal Lie algebra \(\mathfrak{so}_4\simeq \mathfrak{sl}_2\times \mathfrak{sl}_2\) are given. This allows us to view the finite-dimensional irreducible modules for \(\mathfrak{so}_4\) as modules for \(\boxtimes\). Different bases for these modules are considered, and these are shown to be related by the so-called Krawtchouk polynomials. Extensions of these results substituting \(\mathfrak{so}_4\) by an arbitrary finite cartesian product of copies of \(\mathfrak{sl}_2\) are considered too.
scientific article; zbMATH DE number 7462795
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