Leonard triples associated with hypercubes and their antipodal quotients

George Brown (University of Wisconsin - Madison, USA)

Let $\mathcal{A}$ be the unital associative algebra over $\mathbb{C}$ with generators $x,y,z$ and relations $xy+yx=2z$, $yz+zy=2x$ and $zx+xz=2y$.  We find the finite-dimensional irreducible $\mathcal{A}$-modules and show that $x,y,z$ act on these modules as bipartite or almost bipartite Leonard triples.  We define an operator $s$ on finite-dimensional \mathfrak{sl}_{2}-modules that gives them an $\mathcal{A}$-structure.

Let $d$ denote a nonnegative integer and let $Q_{d}$ denote the graph of the $d$-dimensional hypercube.  It is known that the Terwilliger algebra of $Q_{d}$ has an \mathfrak{sl}_{2}-module structure.  When $d$ is even, we show that applying $s$ to the Terwilliger algebra of $Q_{d}$ produces the Terwilliger algebra of the alternate $Q$-polynomial structure of $Q_{d}$.  When $d$ is odd, we show that applying $s$ to the Terwilliger algebra of $Q_{d}$ produces the Terwilliger algebra of the antipodal quotient of $Q_{d}$.