The structure of Hopf algebras with a weak projection
type: Item
instance of: scholarly article
title: The structure of Hopf algebras with a weak projection
zbMATH Open document ID: 0973.16023
DOI: 10.1023/A:1009993517021
author: Peter Schauenburg
published in: Algebras and Representation Theory
publication date: 2001-11-25T00:00:00Z
review text: The author studies the structure of Hopf algebras that have a weak projection. Let \(H'\) be a Hopf subalgebra of the bialgebra \(H\) over a field \(k\). A map \(\pi\colon H\to H'\) is a weak (left) projection if \(\pi\) is a left \(H'\)-linear coalgebra map and \(\pi|_{H'}=\text{id}_{H'}\). In this situation, \(Q=H^{\prime+}H\) is a right \(H'\)-comodule coalgebra with the \(H'\)-comodule structure given by \(\rho'(\overline h)=\overline h_{[0]}\otimes\overline h_{[1]}=\overline h_{(2)}\otimes S(\pi(h_{(1)}))\pi(h_{(2)})\), and \(\psi=(\pi\otimes\nu)\Delta\colon H\to H'\times Q\) is an \(H'\)-module coalgebra isomorphism, where \(H'\times Q\) is the cosmash product and \(\nu\colon H\to Q\) is the canonical epimorphism. Let \(H'\) be a bialgebra and \(Q\) a right \(H'\)-module coalgebra with the comodule structure given by \(Q\ni q\mapsto q_{[0]}\otimes q_{[1]}\in Q\otimes H'\). Then one can form a cosmash product \(H'\times Q\). Assume there are maps \(Q\otimes H'\ni q\otimes x\mapsto q\leftharpoonup x\in Q\), \(Q\otimes H'\ni q\otimes x\mapsto q\rightharpoonup x\in H'\), \(Q\otimes Q\ni p\otimes q\mapsto p\cdot q\in Q\) and \(Q\otimes Q\ni p\otimes q\mapsto\tau(p|q)\in H'\), and an element \(1\in Q\). The paper describes equivalent conditions for \(H'\times Q\) to be a bialgebra with unit \(1\times 1\) and multiplication \[ (x\times p)(y\times q)=x(p_{(1)}\rightharpoonup y_{(1)})\tau(p_{(2)}\leftharpoonup y_{(2)}|q_{(1)[0]})\times (p_{(3)}\leftharpoonup y_{(3)}q_{(1)[1]})\cdot q_{(2)}, \] such that the map \(j\colon H'\ni x\mapsto x\times 1\in H'\times Q\) is a bialgebra map satisfying \(\pi j=\text{id}_{H'}\), where \(\pi\colon H'\times Q\ni x\times q\mapsto x\varepsilon(q)\in H'\). It is also proved that every bialgebra structure on \(H'\times Q\) such that \(H'\) is a subbialgebra of \(H'\times Q\) and \(\pi\colon H'\times Q\to H'\) is a left \(H'\)-linear coalgebra map with \(\pi|_{H'}=\text{id}_{H'}\) is of the form described above. Furthermore, let \(H'\time…
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16S40
zbMATH DE Number: 1569447
zbMATH Keywords: module coalgebras
zbMATH Keywords: comodule coalgebras
zbMATH Keywords: coalgebra maps
zbMATH Keywords: Hopf algebras
zbMATH Keywords: weak projections
zbMATH Keywords: bialgebras
zbMATH Keywords: cosmash products
MaRDI profile type: Publication
full work available at URL: a:1009993517021
OpenAlex ID: W204802450
description: scientific article; zbMATH DE number 1569447