scholarly journals A semidirect product decomposition for certain Hopf algebras over an algebraically closed field

1976 ◽  
Vol 59 (1) ◽  
pp. 29-29 ◽  
Author(s):  
Richard K. Molnar
Keyword(s):  
Semidirect Product ◽  
Hopf Algebras ◽  
Closed Field ◽  
Product Decomposition ◽  
Algebraically Closed Field

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.

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.

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 Finite Symmetric Integral Tensor Categories with the Chevalley Property with an Appendix by Kevin Coulembier and Pavel Etingof

2019 ◽  
Author(s):  
Pavel Etingof ◽  
Shlomo Gelaki
Keyword(s):  
Hopf Algebras ◽  
Tensor Category ◽  
Group Scheme ◽  
Symmetric Tensor ◽  
Closed Field ◽  
Characteristic P ◽  
Tensor Categories ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Chevalley Property

Abstract We prove that every finite symmetric integral tensor category $\mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $p&gt;2$ admits a symmetric fiber functor to the category of supervector spaces. This proves Ostrik’s conjecture [25, Conjecture 1.3] in this case. Equivalently, we prove that there exists a unique finite supergroup scheme $\mathcal{G}$ over $k$ and a grouplike element $\epsilon \in k\mathcal{G}$ of order $\le 2$, whose action by conjugation on $\mathcal{G}$ coincides with the parity automorphism of $\mathcal{G}$, such that $\mathcal{C}$ is symmetric tensor equivalent to $\textrm{Rep}(\mathcal{G},\epsilon )$. In particular, when $\mathcal{C}$ is unipotent, the functor lands in $\textrm{Vec}$, so $\mathcal{C}$ is symmetric tensor equivalent to $\textrm{Rep}(U)$ for a unique finite unipotent group scheme $U$ over $k$. We apply our result and the results of [17] to classify certain finite dimensional triangular Hopf algebras with the Chevalley property over $k$ (e.g., local), in group scheme-theoretical terms. Finally, we compute the Sweedler cohomology of restricted enveloping algebras over an algebraically closed field $k$ of characteristic $p&gt;0$, classify associators for their duals, and study finite dimensional (not necessarily triangular) local quasi-Hopf algebras and finite (not necessarily symmetric) unipotent tensor categories over an algebraically closed field $k$ of characteristic $p&gt;0$. The appendix by K. Coulembier and P. Etingof gives another proof of the above classification results using the recent paper [4], and more generally, shows that the maximal Tannakian and super-Tannakian subcategory of a symmetric tensor category over a field of characteristic $\ne 2$ is always a Serre subcategory.

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).

H-Simple module algebras for eight-dimensional non-semisimple Hopf algebras

2016 ◽  
Vol 15 (07) ◽  
pp. 1650134
Author(s):  
Fengxia Gao ◽  
Shilin Yang
Keyword(s):  
Hopf Algebras ◽  
Characteristic Zero ◽  
Jacobson Radical ◽  
Simple Module ◽  
Simple Algebra ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Module Algebras

Let [Formula: see text] be an algebraically closed field of characteristic zero. For all eight-dimensional non-semisimple Hopf algebras [Formula: see text] which are either pointed or unimodular, we characterrize all finite-dimensional [Formula: see text]-simple module algebras. As a bonus of our approach, it is shown that for any [Formula: see text]-simple algebra, the nilpotent index of the Jacobson radical is at most three.

Sign in / Sign up

Export Citation Format

Share Document