Militaru's \(D\)-equation in monoidal categories
type: Item
instance of: scholarly article
title: Militaru's \(D\)-equation in monoidal categories
zbMATH Open document ID: 1050.18006
DOI: 10.1023/A:1024458931031
author: Julien Bichon
author: Ross H. Street
published in: Applied Categorical Structures
publication date: 2003-08-21T00:00:00Z
review text: 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.
reviewed by: Bernhard Drabant
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57T05
zbMATH DE Number: 1968818
zbMATH Keywords: monoidal categories
zbMATH Keywords: coalgebras
zbMATH Keywords: bialgebras
zbMATH Keywords: \(D\)-equation
MaRDI profile type: Publication
full work available at URL: a:1024458931031
OpenAlex ID: W90792916
description: scientific article; zbMATH DE number 1968818