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We study the double-scaling limit of SYK (DS-SYK) model and elucidate theunderlying quantum group symmetry. The DS-SYK model is characterized by aparameter $q$, and in the $q\rightarrow 1$ and low-energy limit it goes over tothe familiar Schwarzian theory. We relate the chord and transfer-matrix pictureto the motion of a ``boundary particle" on the Euclidean Poincar{\'e} disk,which underlies the single-sided Schwarzian model. $AdS_2$ carries an action of$\mathfrak{s}\mathfrak{l}(2,{\mathbb R}) \simeq \mathfrak{s}\mathfrak{u}(1,1)$,and we argue that the symmetry of the full DS-SYK model is a certain$q$-deformation of the latter, namely $\mathcal{U}_{\sqrtq}(\mathfrak{s}\mathfrak{u}(1,1))$. We do this by obtaining the effectiveHamiltonian of the DS-SYK as a (reduction of) particle moving on a latticedeformation of $AdS_2$, which has this $\mathcal{U}_{\sqrtq}(\mathfrak{s}\mathfrak{u}(1,1))$ algebra as its symmetry. We also exhibit theconnection to non-commutative geometry of $q$-homogeneous spaces, by obtainingthe effective Hamiltonian of the DS-SYK as a (reduction of) particle moving ona non-commutative deformation of $AdS_3$. There are families of possiblydistinct $q$-deformed $AdS_2$ spaces, and we point out which are relevant forthe DS-SYK model.

Quantum groups, non-commutative $AdS_2$, and chords in the double-scaled SYK model