THE COCYCLE CONDITION OF LIOUVILLE THEORY AND ITS APPLICATION TO 2-D INDUCED GRAVITY .1.

The cocycle condition of the action in the Liouville theory has been proposed. Several composition laws could be deduced from the cocycle condition, such as the composition law of the action in 2-d induced gravity in the light-cone gauge, and that of the geometric action in coadjoint Diffs 1 orbits proposed by Alekseev and Shatashivilli