Summary: In this paper we consider how the following three objects are related:

(i)

the dual polar graphs;

(ii)

the quantum algebra \(U_{q}(\mathfrak{sl}_{2})\);

(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 \(U_{q}({\mathfrak{sl}}_{2})\)-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 \(U_{q}(\mathfrak{sl}_{2})\)-module structures on the standard module of \(\Gamma\). We describe how these two \(U_{q}(\mathfrak{sl}_{2})\)-module structures are related. Each of these \(U_{q}(\mathfrak{sl}_{2})\)-module structures induces a \(\mathbb{C}\)-algebra homomorphism \(U_{q}(\mathfrak{sl}_{2}) \to T\). 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.