Quantum lines over non-cocommutative cosemisimple Hopf algebras.
type: Item
instance of: scholarly article
title: Quantum lines over non-cocommutative cosemisimple Hopf algebras.
zbMATH Open document ID: 1042.16022
author: Akira Masuoka
author: C. Calinescu
author: Sorin Dascalescu
author: Claudia Menini
published in: Journal of Algebra
publication date: 2004-03-14T00:00:00Z
review text: Let \(q\) be a primitive root of 1 of degree \(n\) in a field \(k\) and \(R_q\) the graded \(k\)-algebra \(k[Y]/(y^n)\) endowed with a coalgebra structure such that \(\tfrac{y^j}{(j)_q}\), \(1\leqslant j\leqslant n-1\), forms a divided power sequence. Let \(H\) be one of the following non-commutative cosemisimple Hopf algebras \[ \widehat{\mathcal D}_{2n},\;n\geqslant 3,\quad\widehat{\mathcal T}_{4m},\;m\geqslant 2,\quad\widehat{\mathcal A}_{4m},\;m\geqslant 3,\quad\widehat{\mathcal B}_{4m},\;m\geqslant 2 \] from the paper of \textit{A. Masuoka} [Isr. J. Math. 92, No. 1-3, 361-373 (1995; Zbl 0839.16036)]. There is given a classification of biproducts \(R_q\#H\) as graded and ungraded Hopf algebras. In particular there is given a classification of Hopf algebras \(A\) with coradical \(H\) such that \(\text{gr\,}A\) is isomorphic to \(R_q\#H\). The problem of a selfduality of \(R_q\#H\) is also solved.
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16W35
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 2055694
zbMATH Keywords: Hopf algebras
zbMATH Keywords: graded algebras
reviewed by: Vyacheslav A. Artamonov
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.08.006
OpenAlex ID: W2117159078
cites work: Triangular Hopf algebras with the Chevalley property
cites work: Hopf algebras of order \(p^2\) and braided Hopf algebras of order \(p\)
cites work: Lifting of quantum linear spaces and pointed Hopf algebras of order \(p^3\)
cites work: Finite quantum groups and Cartan matrices
cites work: Q4537941
cites work: Q3153375
cites work: Duals of pointed Hopf algebras
cites work: Constructing pointed Hopf algebras by Ore extensions
cites work: Classifying pointed hopf algebras of dimension 16
cites work: Classification of semisimple Hopf algebras of dimension 16
cites work: Cross products by braided groups and bosonization
cites work: Semisimple Hopf algebras of dimension \(6\), \(8\)
cites work: Cocycle deformations and Galois objects for some cosemisimple Hopf algebras of finite dimension
cites work: Q4650799
cites work: Biinvertible actions of Hopf algebras
cites work: Bialgebras of type one*
cites work: The structure of Hopf algebras with a projection
cites work: Minimal quasitriangular Hopf algebras
cites work: THE NICHOLS ZOELLER THEOREM FOR HOPF ALGEBRAS IN THE CATEGORY OF YETTER DRINFELD MODULES
cites work: A family of braided cosemisimple Hopf algebras of finite dimension
cites work: Q5590089
cites work: Tensor categories with fusion rules of self-duality for finite abelian groups
DOI: 10.1016/J.JALGEBRA.2003.08.006
description: scientific article; zbMATH DE number 2055694