The Long Dimodules Category and Nonlinear Equations

Let H be a bialgebra and _H L^H be the category of Long H-dimodules defined, for a commutative and co-commutative H, by F. W. Long and studied in connection with the Brauer group of a so-called H-dimodule algebra. For a commutative and co-commutative H, _H L^H =_H YD^H (the category of Yetter–Drinfel'd modules), but for an arbitrary H, the categories _H L^H and _H YD^H are basically different. Keeping in mind that the category _H YD^H is deeply involved in solving the quantum Yang–Baxter equation, we study the category _H L^H of H-dimodules in connection with what we have called the D-equation: R¹² R²³ = R²³ R¹², where R End_k(M M) for a vector space M over a field k. The main result is a FRT-type theorem: if M is finite-dimensional, then any solution R of the D-equation has the form R = R_{(M, , )}, where (M, , ) is a Long D(R)-dimodule over a bialgebra D(R) and R_{(M, , )} is the special map R_{(M, , )}(m n) : = n₁ m n₀. In the last section, if C is a coalgebra and I is a coideal of C, we introduce the notion of D-map on C, that is a k-bilinear map : C C / I k satisfying a condition which ensures on the one hand that, for any right C-comodule, the special map R is a solution of the D-equation and, on the other, that, in the finite case, any solution of the D-equation has this form.