Abstract: The emergence of the quantum R-matrix in the double-scaled SYK model suggests a hidden quantum group structure. In this talk, I will demonstrate how the quantum group U_q(su(1,1)) underlies the structure of the chord Hilbert space with matter insertions. Specifically, we construct an isometric factorization of one-particle states into two boundary states without matter, allowing a decomposition of the one-particle Hilbert space into positive discrete representations of the quantum group.  We then build the multi-particle Hilbert space by fusing these one-particle states, where the quantum R-matrix naturally arises as the operator exchanging fusion channels, corresponding to the crossing of matter chords in the standard chord rules. Along the way, we derive the chord Yang–Baxter relation, and explore the Schwarzian regime of the quantum group structure.  Finally, I will comment on a similar chord description of relational observables in closed universes based on work in progress, suggesting a broader relevance of the chord-based construction to gravitational systems.