scholarly journals A note on split extensions of bialgebras

2018 ◽  
Vol 30 (5) ◽  
pp. 1089-1095 ◽  
Author(s):  
Xabier García-Martínez ◽  
Tim Van der Linden
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Closed Field ◽  
Split Extension ◽  
Algebraically Closed Field ◽  
Split Extensions

AbstractWe prove a universal characterization of Hopf algebras among cocommutative bialgebras over an algebraically closed field: a cocommutative bialgebra is a Hopf algebra precisely when every split extension over it admits a join decomposition. We also explain why this result cannot be extended to a non-cocommutative setting.

HOPF ALGEBRAS OF DIMENSION 30

2010 ◽  
Vol 09 (01) ◽  
pp. 11-15 ◽  
Author(s):  
DAIJIRO FUKUDA
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Finite Dimensional

This paper contributes to the classification of finite dimensional Hopf algebras. It is shown that every Hopf algebra of dimension 30 over an algebraically closed field of characteristic zero is semisimple and thus isomorphic to a group algebra or the dual of a group algebra.

On Semisimple Hopf Algebras of Dimension pqn

2014 ◽  
Vol 57 (2) ◽  
pp. 264-269
Author(s):  
Li Dai ◽  
Jingcheng Dong
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Prime Numbers ◽  
Closed Field ◽  
Semisimple Hopf Algebra ◽  
Algebraically Closed Field

AbstractLet p, q be prime numbers with p2 < q, n ∊ ℕ, and H a semisimple Hopf algebra of dimension pqn over an algebraically closed field of characteristic 0. This paper proves that H must possess one of the following two structures: (1) H is semisolvable; (2) H is a Radford biproduct R#kG, where kG is the group algebra of group G of order p and R is a semisimple Yetter–Drinfeld Hopf algebra in of dimension qn.

scholarly journals Semisimple Hopf Algebras of Dimension 2q3

2015 ◽  
Vol 65 (1) ◽  
Author(s):  
Jingcheng Dong ◽  
Li Dai
Keyword(s):  
Hopf Algebra ◽  
Prime Number ◽  
Hopf Algebras ◽  
Closed Field ◽  
Semisimple Hopf Algebra ◽  
Algebraically Closed Field

AbstractLet q be a prime number, k an algebraically closed field of characteristic 0, and H a non-trivial semisimple Hopf algebra of dimension 2q

scholarly journals NON-AFFINE HOPF ALGEBRA DOMAINS OF GELFAND–KIRILLOV DIMENSION TWO

2017 ◽  
Vol 59 (3) ◽  
pp. 563-593
Author(s):  
K. R. GOODEARL ◽  
J. J. ZHANG
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Integral Domains ◽  
Algebraically Closed Field ◽  
Kirillov Dimension

AbstractWe classify all non-affine Hopf algebras H over an algebraically closed field k of characteristic zero that are integral domains of Gelfand–Kirillov dimension two and satisfy the condition Ext1H(k, k) ≠ 0. The affine ones were classified by the authors in 2010 (Goodearl and Zhang, J. Algebra324 (2010), 3131–3168).

scholarly journals An Application of Finite Groups to Hopf algebras

2021 ◽  
Vol 14 (3) ◽  
pp. 816-828
Author(s):  
Tahani Al-Mutairi ◽  
Mohammed Mosa Al-shomrani
Keyword(s):  
Finite Group ◽  
Hopf Algebra ◽  
Finite Groups ◽  
Hopf Algebras ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
A Finite Group

Kaplansky’s famous conjectures about generalizing results from groups to Hopf al-gebras inspired many mathematicians to try to find solusions for them. Recently, Cohen and Westreich in [8] and [10] have generalized the concepts of nilpotency and solvability of groups to Hopf algebras under certain conditions and proved interesting results. In this article, we follow their work and give a detailed example by considering a finite group G and an algebraically closed field K. In more details, we construct the group Hopf algebra H = KG and examine its properties to see what of the properties of the original finite group can be carried out in the case of H.

Ideals of Finite-Dimensional Pointed Hopf Algebras of Rank One

2021 ◽  
Vol 28 (02) ◽  
pp. 351-360
Author(s):  
Yu Wang ◽  
Zhihua Wang ◽  
Libin Li
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Adjoint Action ◽  
Prime Ideals ◽  
Closed Field ◽  
Finite Dimensional ◽  
Maximal Ideals ◽  
Primitive Ideals ◽  
Rank One ◽  
Pointed Hopf Algebras

Let [Formula: see text] be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed field of characteristic zero. In this paper we show that any finite-dimensional indecomposable [Formula: see text]-module is generated by one element. In particular, any indecomposable submodule of [Formula: see text] under the adjoint action is generated by a special element of [Formula: see text]. Using this result, we show that the Hopf algebra [Formula: see text] is a principal ideal ring, i.e., any two-sided ideal of [Formula: see text] is generated by one element. As an application, we give explicitly the generators of ideals, primitive ideals, maximal ideals and completely prime ideals of the Taft algebras.

scholarly journals Special rays in the Mori cone of a projective variety

2002 ◽  
Vol 168 ◽  
pp. 127-137 ◽  
Author(s):  
Marco Andreatta ◽  
Gianluca Occhetta
Keyword(s):  
Projective Variety ◽  
Maximal Length ◽  
Closed Field ◽  
Algebraically Closed Field

AbstractLet X be a smooth n-dimensional projective variety over an algebraically closed field k such that KX is not nef. We give a characterization of non nef extremal rays of X of maximal length (i.e of length n – 1); in the case of Char(k) = 0 we also characterize non nef rays of length n – 2.

scholarly journals STRUCTURE OF CORADICAL FILTRATION AND ITS APPLICATION TO HOPF ALGEBRAS OF DIMENSIONpq

2008 ◽  
Vol 50 (2) ◽  
pp. 183-190 ◽  
Author(s):  
DAIJIRO FUKUDA
Keyword(s):  
Hopf Algebras ◽  
Classification Problem ◽  
Closed Field ◽  
Algebraically Closed Field

AbstractThis paper contributes to the classification problem ofpqdimensional Hopf algebrasHover an algebraically closed fieldkof characteristic 0, wherep,qare odd primes. It is shown that such Hopf algebrasHare semisimple for the pairs of odd primes (p,q)=(3,11),(3,13),(3,19),(5,17),(5,19),(5,23),(5,29),(7,17),(7,19),(7,23),(7,29),(11,29),(13,29).

On the classification of finite quasi-quantum groups

2017 ◽  
Vol 29 (10) ◽  
pp. 1730003 ◽  
Author(s):  
Mamta Balodi ◽  
Hua-Lin Huang ◽  
Shiv Datt Kumar
Keyword(s):  
Quantum Groups ◽  
Hopf Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Tools And Methods

We give an overview of the classification results obtained so far for finite quasi-quantum groups over an algebraically closed field of characteristic zero. The main classification results on basic quasi-Hopf algebras are obtained by Etingof, Gelaki, Nikshych, and Ostrik, and on dual quasi-Hopf algebras by Huang, Liu and Ye. The objective of this survey is to help in understanding the tools and methods used for the classification.

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