scholarly journals On the antipode of Hopf algebras with the dual Chevalley property

2021 ◽  
pp. 106871
Author(s):  
Kangqiao Li ◽  
Gongxiang Liu
Keyword(s):  
Hopf Algebras ◽  
Chevalley Property

scholarly journals Examples of finite-dimensional Hopf algebras with the dual Chevalley property

2017 ◽  
Vol 61 ◽  
pp. 445-474 ◽  
Author(s):  
N. Andruskiewitsch ◽  
C. Galindo ◽  
M. Müller
Keyword(s):  
Hopf Algebras ◽  
Finite Dimensional ◽  
Chevalley Property

scholarly journals Triangular Hopf algebras with the Chevalley property

2001 ◽  
Vol 49 (2) ◽  
pp. 277-298 ◽  
Author(s):  
Nicolás Andruskiewitsch ◽  
Pavel Etingof ◽  
Shlomo Gelaki
Keyword(s):  
Hopf Algebras ◽  
Chevalley Property

scholarly journals GAUGE DEFORMATIONS FOR HOPF ALGEBRAS WITH THE DUAL CHEVALLEY PROPERTY

2012 ◽  
Vol 11 (03) ◽  
pp. 1250051
Author(s):  
ALESSANDRO ARDIZZONI ◽  
MARGARET BEATTIE ◽  
CLAUDIA MENINI
Keyword(s):  
Hopf Algebra ◽  
Gauge Transformation ◽  
Hopf Algebras ◽  
Characteristic Zero ◽  
Finite Dimensional ◽  
Chevalley Property ◽  
Algebra Over A Field

Let A be a Hopf algebra over a field K of characteristic zero such that its coradical H is a finite-dimensional sub-Hopf algebra. Our main theorem shows that there is a gauge transformation ζ on A such that Aζ ≅ Q#H where Aζ is the dual quasi-bialgebra obtained from A by twisting its multiplication by ζ, Q is a connected dual quasi-bialgebra in [Formula: see text] and Q#H is a dual quasi-bialgebra called the bosonization of Q by H.

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 $p>2$ 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 $p>0$, 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 $p>0$. 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 Nonsemisimple Hopf algebras of dimension 8𝑝 with the Chevalley property

2019 ◽  
pp. 49-66
Author(s):  
Margaret Beattie ◽  
Gastón García ◽  
Siu-Hung Ng ◽  
Jolie Roat
Keyword(s):  
Hopf Algebras ◽  
Chevalley Property
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