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We investigate the underlying quantum group symmetry of 2d Liouville anddilaton gravity models, both consolidating known results and extending them tothe cases with $\mathcal{N} = 1$ supersymmetry. We first calculate the mixedparabolic representation matrix element (or Whittaker function) of$\text{U}_q(\mathfrak{sl}(2, \mathbb{R}))$ and review its applications toLiouville gravity. We then derive the corresponding matrix element for$\text{U}_q(\mathfrak{osp}(1|2, \mathbb{R}))$ and apply it to explainstructural features of $\mathcal{N} = 1$ Liouville supergravity. We show thatthis matrix element has the following properties: (1) its $q\to 1$ limit is theclassical $\text{OSp}^+(1|2, \mathbb{R})$ Whittaker function, (2) it yields thePlancherel measure as the density of black hole states in $\mathcal{N} = 1$Liouville supergravity, and (3) it leads to $3j$-symbols that match with thecoupling of boundary vertex operators to the gravitational states asappropriate for $\mathcal{N} = 1$ Liouville supergravity. This object shouldlikewise be of interest in the context of integrability of supersymmetricrelativistic Toda chains. We furthermore relate Liouville (super)gravity todilaton (super)gravity with a hyperbolic sine (pre)potential. We do so byshowing that the quantization of the target space Poisson structure in the(graded) Poisson sigma model description leads directly to the quantum group$\text{U}_q(\mathfrak{sl}(2, \mathbb{R}))$ or the quantum supergroup$\text{U}_q(\mathfrak{osp}(1|2, \mathbb{R}))$.

From Quantum Groups to Liouville and Dilaton Quantum Gravity