Speaker
Alison Gordon Lynch
(University of Wisconsin-Madison)
Description
The quantum algebra $U_q(\mathfrak{sl}_2)$ has connections to $Q$-polyonomial distance-regular graphs, tridiagonal pairs of linear transformations, the $q$-tetrahedron algebra, as well as many other combinatorial and algebraic objects. In 2006, Ito, Terwilliger, and Weng gave a presentation for $U_q(\mathfrak{sl}_2)$ in generators $x, y, y^{-1}, z$, called the {\it equitable presentation}, and showed that $\{x^r y^s z^t : r, t \in \mathbb{N}, s \in \mathbb{Z}\}$ is a basis for $U_q(\mathfrak{sl}_2)$. In 2013, Bockting-Conrad and Terwilliger introduced a subalgebra $\mathcal{A}$ of $U_q(\mathfrak{sl}_2)$ spanned by the elements $\{x^r y^s z^t : r, s, t \in \mathbb{N}, r+s+t \ {\rm even}\}$. In this talk, we give a presentation for the algebra $\mathcal{A}$ and we show that for every $d \ge 1$, there exists a unique irreducible $\mathcal{A}$-module of dimension $d$.
Primary author
Alison Gordon Lynch
(University of Wisconsin-Madison)
