Generalized smash products.
type: Item
instance of: scholarly article
title: Generalized smash products.
zbMATH Open document ID: 1078.16049
publication date: 2004-11-26T00:00:00Z
review text: Doi introduced the definition of the generalized smash product \(\#(D,B)\): (1) It is equal to \(\Hom_k(D,B)\) as a vector space; (2) \((f\cdot g)(d)=\sum f(g(d_2)_1\rightharpoonup d_1)g(d_2)_0\) for any \(f,g\in\#(D,B)\), \(d\in D\), where \((B,\rho)\) is a right \(A\)-comodule algebra, and \((D,\rightharpoonup)\) is a left \(A\)-module coalgebra for a bialgebra \(A\). The author considers the ring \(\#(D,B)\). First, it is shown that the category of \(BU\)-modules is equivalent to both the category of \(\#\)-rational left modules and the category of all \((B,D)\)-Hopf modules \(_BM^D\). The case when \(D=B=A\) has been studied by Cai and Chen. Secondly it is shown \(\#(A,A)\) is a dense subring of \(\text{End}_k(Q)\), where \(Q\) is a certain subspace of \(\#(A,A)\) under the condition that the antipode is bijective. Furthermore if \(A\) has nonzero left integral, \(Q\) can be chosen to be \(A\) and \(\#(A,A)\) is both left and right primitive. In fact, Cai and Chen have already shown that if \(A\) has a nonzero left integral then \(A\#A^{*rat}\) is a dense subring of \(\text{End}_k(A)\). Moreover, the author gives that the left singular ideal of the ring \(\#(A,A)\) is zero. As a corollary, it is shown that the ring \(\#(A,A)\) has finite uniform dimension.
reviewed by: Fang Li
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16E10
zbMATH DE Number: 2119277
zbMATH Keywords: generalized smash products
zbMATH Keywords: rational modules
zbMATH Keywords: uniform dimension
zbMATH Keywords: right comodule algebras
zbMATH Keywords: left module coalgebras
zbMATH Keywords: bialgebras
zbMATH Keywords: categories of modules
zbMATH Keywords: Hopf modules
zbMATH Keywords: dense subrings
zbMATH Keywords: antipodes
zbMATH Keywords: left integrals
zbMATH Keywords: right primitive rings
author: Zhixiang Wu
MaRDI profile type: Publication
full work available at URL: s10114-003-0293-z
OpenAlex ID: W2053310823
cites work: Galois actions on rings and finite Galois coverings.
cites work: Hopf Galois extensions, smash products, and Morita equivalence
cites work: Hopf algebras and Galois theory
cites work: Q4315517
cites work: Strongly inner actions, coactions, and duality theorems
cites work: Coactions, smash products, and Hopf modules
cites work: Finiteness conditions for a Hopf algebra with a nonzero integral
cites work: Equivalence theorems and Hopf-Galois extensions
cites work: Homological coalgebra
cites work: A duality theorem for Hopf module algebras
cites work: Rings of quotients and Morita contexts
DOI: 10.1007/S10114-003-0293-Z
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 2119277