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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 theD-equation the same role as the category of Yetter-Drinfel'd (crossed) modulesplay for the quantum Yang-Baxter equation. A FRT type theorem is given and newclass of coalgebras are introduced. Several new examples of noncommutativenoncocomutative bialgebras are given.Comment: 20 pages, latex2e, Algbreas and Repr. theory, in pres

The Long dimodule category and nonlinear equation