Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type V. Mixed classes in Chevalley and Steinbe
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Nicolás Andruskiewitsch Gastón Andrés García
© The Author(s) 2020
· Giovanna Carnovale
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Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type V. Mixed classes in Chevalley and Steinberg groups Received: 8 August 2019 / Accepted: 16 September 2020 Abstract. We show that all classes that are neither semisimple nor unipotent in finite simple Chevalley or Steinberg groups different from PSLn (q) collapse (i.e. are never the support of a finite-dimensional Nichols algebra). As a consequence, we prove that the only finite-dimensional pointed Hopf algebra whose group of group-like elements is PSp2n (q), − 3 P+ 4n (q), P4n (q), D4 (q), E 7 (q), E 8 (q), F4 (q), or G 2 (q) with q even is the group algebra.
Contents Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type V. Mixed classes in Chevalley and Steinberg groups Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The work of N. A. was partially supported by CONICET, Secyt (UNC) and the Alexander von Humboldt Foundation through the Research Group Linkage Programme. The work of G. A. G. was partially supported by CONICET, Secyt (UNLP) and ANPCyT-Foncyt 2014-0507. The work of G. C. was partially supported by Progetto BIRD179758/17 of the University of Padova. The results were obtained during visits of N. A. and G. A. G. to the University of Padova, and of G. C. to the University of Córdoba, partially supported by: the bilateral agreement between these Universities, the ICTP-INdAM Research in Pairs fellowship programme and the Coimbra group Scholarship Programme for Young Professors and Researchers from Latin American Universities. The authors wish to thank the referee for his/her careful reading and suggestions. N. Andruskiewitsch: CIEM – CONICET, FaMAF, Universidad Nacional de Córdoba, Medina Allende s/n, Ciudad Universitaria, 5000 Córdoba, Argentina e-mail: [email protected] G. Carnovale (B): Dipartimento di Matematica Tullio Levi-Civita, Università degli Studi di Padova, via Trieste 63, 35121 Padua, Italy e-mail: [email protected] G. A. García: CONICET, Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, C. C. 172, 1900 La Plata, Argentina e-mail: [email protected] Mathematics Subject Classification: 16T05 · 20D06
https://doi.org/10.1007/s00229-020-01248-5
N. Andruskiewitsch et al. 2. 2 Notation and preliminaries . . . . . . . . . . . . . . . . . 2.1. 2.1 The groups G . . . . . . . . . . . . . . . . . . . . 2.2. 2.2 F-stable tori . . . . . . . . . . . . . . . . . . . . . 2.3. 2.3 The elements wθ . . . . . . . . . . . . . . . . . . 2.4. 2.4 Semisimple classes . . . . . . . . . . . . . . . . . 2.5. 2.5 Commuting elements in a semisimple class . . . . 2.6. 2.6 Real classes . . . . . . . . . . . . . . . . . . . . . 2.7. 2.7 Non-central elements in the torus . . . . . . . . . . 2.8. 2.8 Racks . . . . . . . . . . . . . . . . . . . . . . . . 2.
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