Holography on the quantum disk

Motivated by recent study of DSSYK and the non-commutative nature of its bulk dual, we review and analyze an example of a non-commutative spacetime known as the quantum disk proposed by L. Vaksman. The quantum disk is defined as the space whose isometries are generated by the quantum algebra $U_q\left(\mathfrak{s}{\mathfrak{u}}_{1,1}\right)$ \( {U}_q\left(\mathfrak{s}{\mathfrak{u}}_{1,1}\right) \) . We review how this algebra is defined and its associated group SU_q(1, 1) that it generates, highlighting its non-trivial coproduct that sources bulk non-commutativity. We analyze the structure of holography on the quantum disk and study the imprint of non-commutativity on the putative boundary dual.