As another example of the second category in Kostantinos Kanakoglou's answer I think it is fair to mention **quantum-integrable systems**: this topic in physics was pivotal in the *historical development* of the notion of quantum groups by the Leningrad group (Faddeev *et al*), and the Japanese group (Jimbo and Miwa *et al*).

In particular, $U_q(\widehat{\mathfrak{sl}_2})$, the quantum-affine version of $\mathfrak{sl}_2$, naturally arises in the (algebraic Bethe-ansatz) analysis of the partially isotropic (or "XXZ") Heisenberg quantum spin chain, and the closely related six-vertex model in statistical physics. Here the deformation parameter $q$ characterizes the spin chain's partial anisotropy as $\Delta = (q+q^{-1})/2$, with $\Delta=q=1$ the completely isotropic ($\mathfrak{sl}_2$-invariant) point.

More, including many references to published articles, can e.g. be found in the books

Finally, although I am by no means an expert on the topic, let me point out that the second half of the book by Gómez *et al* is devoted to quantum groups in **conformal field theory**, see also