Hopf algebras and their generalizations from a category theoretical point of view
type: Item
instance of: scholarly article
title: Hopf algebras and their generalizations from a category theoretical point of view
zbMATH Open document ID: 1417.16034
DOI: 10.1007/978-3-319-98137-6
published in: Lecture Notes in Mathematics
publication date: 2018-08-27T00:00:00Z
review text: These lecture notes provide a unified category theoretical introduction to the notion of Hopf algebra and many of its generalizations. The approach has as its point of departure the fact that the category of modules over such algebraic object turns out to be a closed monoidal category and this structure is strictly preserved by some forgetful functor, the target of which is responsible for each of the specific generalizations. One of the main achievements of the book is to derive the axioms defining a given Hopf algebraic structure from the mentioned feature of its category of modules. \par The strategy consists in regarding the category of modules over an algebra, as well as its generalizations, as the Eilenberg-Moore category of some induced monad such that the monoidal structure of the base category is lifted to the Eilenberg-Moore category, the last condition being a reformulation of the condition that the structure is preserved by a forgetful functor. \par Along its seven chapters the book discusses (Hopf) bimonads on monoidal categories, (Hopf) bialgebras over a field, (Hopf) bialgebroids over an arbitrary associative and unital base algebra $B$, weak (Hopf) bialgebras and (Hopf) bimonoids in so-called \textit{duoidal categories}, which are categories endowed with two compatible monoidal structures introduced by \textit{M. Aguiar} and \textit{S. Mahajan} [Monoidal functors, species and Hopf algebras. Providence, RI: American Mathematical Society (AMS) (2010; Zbl 1209.18002)] under the name \textit{2-monoidal category}: such (Hopf) bimonoids include as particular examples Hopf monoids in braided monoidal categories, small groupoids, Hopf algebroids over a commutative base algebra, weak Hopf algebras, Hopf monads and certain coalgebra-enriched categories called \textit{Hopf categories}. \par The book offers a self-contained presentation, starting from the basic notions of categories and functors. The introduction of the different algebraic structures is illustra…
reviewed by: Sonia Natale
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 16T10
Mathematics Subject Classification ID: 18C15
Mathematics Subject Classification ID: 18D15
zbMATH DE Number: 6927920
zbMATH Keywords: Hopf algebra
zbMATH Keywords: monoidal category
zbMATH Keywords: Hopf monad
zbMATH Keywords: Eilenberg-Moore category
zbMATH Keywords: Hopf algebroid
zbMATH Keywords: weak Hopf algebra
zbMATH Keywords: bimonoid
zbMATH Keywords: duoidal category
zbMATH Keywords: Hopf group algebra
author: Gabriella Böhm
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-98137-6
OpenAlex ID: W2898746062
description: scientific article; zbMATH DE number 6927920