Interactions of Reggeized Gluons in the M\"obius Representation

We investigate consequences of the M\"obius invariance of the BFKL Hamiltonian and of the triple Pomeron vertex. In particular, we show that the triple Pomeron vertex in QCD, when restricted to the large $N_c$ limit and to the space of M\"obius functions, simplifies and reduces to the vertex used in the Balitsky-Kovchegov (BK) equation. As a result, the BK equation for the dipole density appears as a special case of the nonlinear evolution equation which sums the fan diagrams for BFKL Green's functions in the M\"obius representation. We also calculate the corrections $O(1/(N_c^2-1)$ to the triple Pomeron vertex in the space of M\"obius functions, and we present a generalization of the BK-equation in the next-to-leading order approximation in the $1/N_c$ expansion.