Militaru's \(D\)-equation in monoidal categories
The authors generalise Militaru's universal construction of a bialgebra \(D(R)\) out of the \(D\)-operator \(R:M\otimes M\to M\otimes M\) [\textit{G. Militaru}, Algebr. Represent. Theory 2, 177--200 (1999; Zbl 0936.16037)]. In a first step Militaru's results will be extended to the more general context of a category \({\mathcal V}\) which is monoidal and has coequalizers. Thereby the authors have found a generalised FRT type universal construction of a coalgebra \(D(R)\) for any morphism \(R:M\otimes M\to M\otimes M\) for any object \(M\) in \({\mathcal V}\) which has a left dual. Moreover, if \(R\) is a so-called \(D\)-morphism then a universal Long system for \(M\) exists by which \(R\) arises naturally from the Long comodule \(M\) over \(D(R)\). The authors then define \(D\)-structures and \(D\)-bialgebras in braided categories. It will be shown that the category \({\mathcal C}omod(A)\) of a \(D\)-bialgebra \(A\) has a natural \(D\)-structure. Conversely, let \(F:{\mathcal C}\to{\mathcal V}_0\) be a monoidal functor of the monoidal category \({\mathcal C}\) with \(D\)-structure into the full subcategory of objects with left dual of a cocomplete braided category \({\mathcal V}\) in which the tensor product commutes with colimits. Then the bialgebra \(A=\text{coend}(F)\) is a \(D\)-bialgebra whose \(D\)-morphism is canonically related to the \(D\)-structure of \({\mathcal C}\). As a consequence, the generalized FRT construction over an object \(M\) with left dual and \(D\)-morphism in such a cocomplete braided category \({\mathcal V}\) yields a \(D\)-morphism on the FRT bialgebra \(A(R)\). Eventually the authors apply these results to Frobenius algebras, from which \(D\)-morphisms arise quite naturally. They construct new \(D\)-bialgebras and new solutions of the \(D\)-equation.
scientific article; zbMATH DE number 1968818