Peter Chenwei Ruan (University of Wisconsin–Madison)「Distance-regular graphs and the positive part $U_q^+$ of the $q$-deformed enveloping algebra for affine $\mathfrak{sl}_2$」
Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with diameter $d \geq 3$. It is known that the adjacency matrix and dual adjacency matrix of $\Gamma$ satisfy a pair of relations called the tridiagonal relations. The two matrices generate an algebra called the Terwilliger algebra $T$. We will consider a famous special case of $Q$-polynomial distance-regular graphs, said to have classical parameters $(d,b,\alpha,\sigma)$ with $b \neq 1$ and $\alpha=b-1$. In this case, the tridiagonal relations become the $q$-Serre relations ($b=q^2$) after an affine transformation. We will be discussing the positive part $U_q^+$ of the $q$-deformed enveloping algebra for affine $\mathfrak{sl}_2$. As we will review, there exists a surjective algebra homomorphism $U_q^+ \to T$. In an effort to study $T$ via this homomorphism, we will bring in three PBW bases for $U_q^+$. Originally these PBW bases were developed by Damiani, Beck, and Terwilliger using different points of view. We will give a uniform approach to these PBW bases.