Matrix model approach to minimal Liouville gravity revisited

Using the connection with the Frobenius manifold structure, we study the matrix model description of minimal Liouville gravity (MLG) based on the Douglas string equation. Our goal is to find an exact discrete formulation of the (q,p) MLG model that intrinsically contains information about the conformal selection rules. We discuss how to modify the Frobenius manifold structure appropriately for this purposes. We propose a modification of the construction for Lee-Yang series involving the $A_{p-1}$ algebra instead of the previously used $A_1$ algebra. With the new prescription, we calculate correlators on the sphere up to four points and find full agreement with the continuous approach without using resonance transformations.