Revista Matemática Iberoamericana


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Volume 33, Issue 3, 2017, pp. 995–1024
DOI: 10.4171/RMI/961

Published online: 2017-10-02

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

Nicolás Andruskiewitsch[1], Giovanna Carnovale[2] and Gastón Andrés García[3]

(1) Universidad Nacional de Córdoba, Argentina
(2) Università degli Studi di Padova, Italy
(3) Universidad Nacional de La Plata, Argentina

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.

Keywords: Nichols algebra, Hopf algebra, rack, finite group of Lie type, conjugacy class

Andruskiewitsch Nicolás, Carnovale Giovanna, García Gastón Andrés: Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type III. Semisimple classes in PSL$_n(q)$. Rev. Mat. Iberoamericana 33 (2017), 995-1024. doi: 10.4171/RMI/961