We explore the relationship between algebraic structures and integrable systems, with the Yang–Baxter equation at the core of our research. The quantum group (or quantum coordinate ring) was introduced as a mathematical framework to derive, in a unified way through representation theory, the L-operators that play a crucial role in integrable systems. A quantum group is a Hopf algebra, defined via solutions to the quantum Yang-Baxter equation. In our work, we use solutions to a generalized version—the quantum dynamical Yang–Baxter equation—to construct Hopf algebroids with desirable properties. From the representations of these Hopf algebroids, we can define new L-operators. We have also studied quotient categories by relations arising from solutions to the Yang–Baxter equation on quivers, and have shown that these categories possess good properties from the viewpoint of Garside theory. Conversely, starting from quotient categories—or more precisely, from suitable pairs of a quiver and relations—we have succeeded in constructing solutions to the Yang–Baxter equation on the quiver. Going forward, we aim to deepen our understanding of algebraic structures and to uncover new connections with integrable systems, with the Yang-Baxter equation as our guiding theme.