The ring of physical states in the $M(2,3)$ minimal Liouville gravity

We consider the $M(2,3)$ minimal Liouville gravity, whose state space in the gravity sector is realized as irreducible modules of the Virasoro algebra. We present a recursive construction for BRST cohomology classes based on using an explicit form of singular vectors in irreducible modules of the Virasoro algebra. We find a certain algebra acting on the BRST cohomology space and use this algebra to find the operator algebra of physical states.