  The Clebsch--Gordan coefficients of $U(\mathfrak{sl}_2)$ are expressible in
terms of Hahn polynomials. The phenomenon can be explained by an algebra
homomorphism $\natural$ from the universal Hahn algebra $\mathcal H$ into
$U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$. Let $\Omega$ denote a finite
set of size $D$ and $2^\Omega$ denote the power set of $\Omega$. It is
generally known that $\mathbb C^{2^\Omega}$ supports a
$U(\mathfrak{sl}_2)$-module. Let $k$ denote an integer with $0\leq k\leq D$ and
fix a $k$-element subset $x_0$ of $\Omega$. By identifying $\mathbb
C^{2^\Omega}$ with $\mathbb C^{2^{\Omega\setminus x_0}}\otimes \mathbb
C^{2^{x_0}}$ this induces a $U(\mathfrak{sl}_2)\otimes
U(\mathfrak{sl}_2)$-module structure on $\mathbb C^{2^\Omega}$ denoted by
$\mathbb C^{2^\Omega}(x_0)$. Pulling back via $\natural$ the
$U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$-module $\mathbb
C^{2^\Omega}(x_0)$ forms an $\mathcal H$-module. When $1\leq k\leq D-1$ the
$\mathcal H$-module $\mathbb C^{2^\Omega}(x_0)$ enfolds the Terwilliger algebra
of the Johnson graph $J(D,k)$ with respect to $x_0$. This result connects these
two seemingly irrelevant topics: The Clebsch--Gordan coefficients of
$U(\mathfrak{sl}_2)$ and the Terwilliger algebras of Johnson graphs.
Unfortunately some steps break down in the $q$-analog case. By making detours,
the imperceptible connection between the Clebsch--Gordan coefficients of
$U_q(\mathfrak{sl}_2)$ and the Terwilliger algebras of Grassmann graphs is
successfully disclosed in this paper.
