Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type III. Semisimple classes in PSL$_n(q)$

Nicolás Andruskiewitsch, Giovanna Carnovale, Gastón Andrés García

Type: Article

Publication Date: 2017-10-02

Citations: 10

DOI: https://doi.org/10.4171/rmi/961

Abstract

We show that Nichols algebras of most simple Yetter–Drinfeld modules over the projective special linear group over a finite field, corresponding to semisimple orbits, have infinite dimension. We introduce a new criterium to determine when a conjugacy class collapses and prove that for infinitely many pairs (n,q) , any finite-dimensional pointed Hopf algebra H with G(H)\simeq\mathbf {PSL}_{n}(q) or \mathbf {SL}_{n}(q) is isomorphic to a group algebra.

Locations

Revista Matemática Iberoamericana - View
arXiv (Cornell University) - View - PDF
Padua Research Archive (University of Padova) - View - PDF
Research Padua Archive (University of Padua) - View - PDF
DataCite API - View

Similar Works

Action	Title	Year	Authors
+	Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type I. Non-semisimple classes in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi mathvariant="bold">PSL</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>	2014	Nicolás Andruskiewitsch Giovanna Carnovale Gastón Andrés García
+	Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type VII. Semisimple classes in PSL(n,q) and PSp(2n,q)	2023	N. Andruskiewitsch G. Carnovale G. García
+ PDF Chat	Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type II: Unipotent classes in symplectic groups	2015	Nicolás Andruskiewitsch Giovanna Carnovale Gastón Andrés García
+	Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type IV. Unipotent classes in Chevalley and Steinberg groups	2016	Nicolás Andruskiewitsch Giovanna Carnovale Gastón Andrés García
+ PDF Chat	Finite-Dimensional Pointed Hopf Algebras Over Finite Simple Groups of Lie Type IV. Unipotent Classes in Chevalley and Steinberg Groups	2019	Nicolás Andruskiewitsch Giovanna Carnovale Gastón Andrés García
+ PDF Chat	Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type V. Mixed classes in Chevalley and Steinberg groups	2020	Nicolás Andruskiewitsch Giovanna Carnovale Gastón Andrés García
+ PDF Chat	Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type VII. Semisimple classes in PSL(q) and PSp2(q)	2023	Nicolás Andruskiewitsch Giovanna Carnovale Gastón Andrés García
+ PDF Chat	Erratum to: Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type V. Mixed classes in Chevalley and Steinberg groups	2020	Nicolás Andruskiewitsch Giovanna Carnovale Gastón Andrés García
+	Finite dimensional Nichols algebras over certain<i>p</i>-groups	2016	Yi-Bo Yang Shenglin Zhu
+	Examples of finite-dimensional Hopf algebras with the dual Chevalley property	2015	Nicolás Andruskiewitsch César Galíndo Monique Müller
+ PDF Chat	Examples of finite-dimensional Hopf algebras with the dual Chevalley property	2017	N. Andruskiewitsch Carlos Galindo Meinard Müller
+	Examples of finite-dimensional Hopf algebras with the dual Chevalley property	2015	Nicolás Andruskiewitsch César Galíndo Monique Müller
+	On pointed Hopf algebras over nilpotent groups	2021	Nicolás Andruskiewitsch
+ PDF Chat	On pointed Hopf algebras over nilpotent groups	2023	Nicolás Andruskiewitsch
+	Pointed $p^3$-dimensional Hopf algebras in positive characteristic	2016	Van C. Nguyen Xingting Wang
+ PDF Chat	Pointed Hopf algebras of odd dimension and Nichols algebras over solvable groups	2024	Nicolás Andruskiewitsch I. Heckenberger L. Vendramin
+ PDF Chat	Finite quantum groups over abelian groups of prime exponent	2002	Nicolás Andruskiewitsch Henriette Schneider
+	Pointed Hopf algebras over the sporadic simple groups	2010	Nicolás Andruskiewitsch Fernando Fantino M. Graña L. Vendramin
+	Representations of finite dimensional pointed Hopf algebras over S3	2010	Agustín García Iglesias
+	Orbits of greatest dimension in semi-simple linear Lie groups	1968	Евгений Михайлович Андреев �. B. Vinberg A. G. �lashvili

Works That Cite This (10)

Action	Title	Year	Authors
+ PDF Chat	Multinomial expansion and Nichols algebras associated to non-degenerate involutive solutions of the Yang-Baxter equation	2023	Yuxing Shi
+ PDF Chat	On pointed Hopf algebras over nilpotent groups	2023	Nicolás Andruskiewitsch
+ PDF Chat	Finite-dimensional Nichols algebras of simple Yetter–Drinfeld modules (over groups) of prime dimension	2024	I. Heckenberger Ehud Meir L. Vendramin
+ PDF Chat	On finite-dimensional copointed Hopf algebras over dihedral groups	2018	Fernando Fantino Gastón Andrés García Mitja Mastnak
+ PDF Chat	Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type VII. Semisimple classes in PSL(q) and PSp2(q)	2023	Nicolás Andruskiewitsch Giovanna Carnovale Gastón Andrés García
+ PDF Chat	Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type VI. Suzuki and Ree groups	2020	Giovanna Carnovale Mauro Costantini
+	Twist equivalence and Nichols algebras over Coxeter groups	2024	Giovanna Carnovale Gabriel Maret
+ PDF Chat	Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type V. Mixed classes in Chevalley and Steinberg groups	2020	Nicolás Andruskiewitsch Giovanna Carnovale Gastón Andrés García
+	Simple Yetter-Drinfeld modules over groups with prime dimension and a finite-dimensional Nichols algebra	2023	I. Heckenberger Ehud Meir L. Vendramin
+	Finite GK-Dimensional Nichols Algebras Over the Infinite Dihedral Group	2023	Yongliang Zhang

Works Cited by This (20)

Action	Title	Year	Authors
+	Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras	2012	Martin W. Liebeck Gary M. Seitz
+	On pointed Hopf algebras associated with alternating and dihedral groups	2007	Nicolás Andruskiewitsch Fernando Fantino
+	Conjugacy Classes in Semisimple Algebraic Groups	2011	James E. Humphreys
+	The Nichols algebra of a semisimple Yetter-Drinfeld module	2010	Nicol�s Andruskiewitsch I. Heckenberger Hans-J�rgen Schneider
+	Linear Algebraic Groups and Finite Groups of Lie Type	2011	Gunter Malle Donna Testerman
+	The classification of Nichols algebras with finite root system of rank two	2013	I. Heckenberger L. Vendramin
+ PDF Chat	On twisted conjugacy classes of type D in sporadic simple groups	2013	Fernando Fantino L. Vendramin
+	Pointed Hopf algebras over the sporadic simple groups	2010	Nicolás Andruskiewitsch Fernando Fantino M. Graña L. Vendramin
+ PDF Chat	Finite-dimensional pointed Hopf algebras with alternating groups are trivial	2010	Nicolás Andruskiewitsch Fernando Fantino M. Graña L. Vendramin
+	From racks to pointed Hopf algebras	2003	Nicolás Andruskiewitsch Matı́as Graña