Quantum symmetry and geometry in double-scaled SYK

A bstract The emergence of the quantum R -matrix in the double-scaled SYK model points to an underlying quantum group structure. In this work, we identify the quantum group 𝒰 q (𝔰𝔲(1, 1)) as a subalgebra of the chord algebra. Specifically, we construct the generators of 𝒰 q (𝔰𝔲(1, 1)) from combinations of operators within the chord algebra and show that the one-particle chord Hilbert space decomposes into the positive discrete series representations of 𝒰 q (𝔰𝔲(1, 1)). Using the coproduct structure of the quantum group, we build the multi-particle Hilbert space and establish its equivalence with previous results defined by the chord rules. In particular, we show that the quantum R -matrix acts as a swapping operator that reverses the ordering of open chords in each fusion channel while incorporating the corresponding q -weighted penalty factors. This action enables an explicit derivation of the chord Yang-Baxter relation. We further explore a realization of the quantum group generators on the quantum disk, and present a novel factorization formula for the bulk gravitational wavefunction in the presence of matter. We also discuss the relation between the 𝒰 q (𝔰𝔲(1, 1)) structure uncovered here and the 𝒰 q (𝔰𝔩(2 , ℝ)) algebra previously studied from the boundary perspective. Finally, we study the gravitational wavefunction with matter in the Schwarzian regime.