  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.
