Dual polar graphs, the quantum algebra U_q(sl_2), and Leonard systems of dual q-Krawtchouk type

Abstract

In this paper we consider how the following three objects are related: (i) the dual polar graphs; (ii) the quantum algebra U_q(sl_2); (iii) the Leonard systems of dual q-Krawtchouk type. For convenience we first describe how (ii) and (iii) are related. For a given Leonard system of dual q-Krawtchouk type, we obtain two U_q(sl_2)-module structures on its underlying vector space. We now describe how (i) and (iii) are related. Let \Gamma denote a dual polar graph. Fix a vertex x of \Gamma and let T = T(x) denote the corresponding subconstituent algebra. By definition T is generated by the adjacency matrix A of \Gamma and a certain diagonal matrix A* = A*(x) called the dual adjacency matrix that corresponds to x. By construction the algebra T is semisimple. We show that for each irreducible T-module W the restrictions of A and A* to W induce a Leonard system of dual q-Krawtchouk type. We now describe how (i) and (ii) are related. We obtain two U_q(sl_2)-module structures on the standard module of \Gamma. We describe how these two U_q(sl_2)-module structures are related. Each of these U_q(sl_2)-module structures induces a $\mathbb{C}$-algebra homomorphism U_q(sl_2) \rightarrow T. We show that in each case T is generated by the image together with the center of T. Using the combinatorics of \Gamma we obtain a generating set L, F, R, K of T along with some attractive relations satisfied by these generators.