  We study the Long dimodule category in conection with a nonlinear equation;
we called the D-equation. The category of Long dimodules will play for the
D-equation the same role as the category of Yetter-Drinfel'd (crossed) modules
play for the quantum Yang-Baxter equation. A FRT type theorem is given and new
class of coalgebras are introduced. Several new examples of noncommutative
noncocomutative bialgebras are given.
