The pentagon equation and mapping-class groups of punctured surfaces

In the quantum Teichmuller theory, the mapping-class groups of punctured su rfaces are represented projectively based on Penner coordinates. Algebraica lly, the representation is based on the pentagon equation together with pai r of additional relations. Two more examples of solutions of these equation s are connected with matrix (or operator) generalizations of the Rogers dil ogarithm. The corresponding central charges are rational. It is possible th at this system of equations admits many different solutions.