Revista de la
Unión Matemática Argentina
On Hopf algebras over basic Hopf algebras of dimension 24
Rongchuan Xiong

Volume 65, no. 2 (2023), pp. 469–493    

Published online: December 27, 2023

https://doi.org/10.33044/revuma.3018

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Abstract

We determine finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero, whose Hopf coradical is isomorphic to a non-pointed basic Hopf algebra of dimension 24 and the infinitesimal braidings are indecomposable objects. In particular, we obtain families of new finite-dimensional Hopf algebras without the dual Chevalley property.

References

  1. N. Andruskiewitsch and I. Angiono, On finite dimensional Nichols algebras of diagonal type, Bull. Math. Sci. 7 no. 3 (2017), 353–573.  DOI  MR  Zbl
  2. N. Andruskiewitsch and I. Angiono, On Nichols algebras over basic Hopf algebras, Math. Z. 296 no. 3-4 (2020), 1429–1469.  DOI  MR  Zbl
  3. N. Andruskiewitsch and M. Beattie, Irreducible representations of liftings of quantum planes, in Lie Theory and Its Applications in Physics V, World Scientific, River Edge, NJ, 2004, pp. 414–423.  DOI  MR  Zbl
  4. N. Andruskiewitsch and J. Cuadra, On the structure of (co-Frobenius) Hopf algebras, J. Noncommut. Geom. 7 no. 1 (2013), 83–104.  DOI  MR  Zbl
  5. N. Andruskiewitsch and J. M. J. Giraldi, Nichols algebras that are quantum planes, Linear Multilinear Algebra 66 no. 5 (2018), 961–991.  DOI  MR  Zbl
  6. N. Andruskiewitsch and M. Graña, Braided Hopf algebras over non abelian finite groups, Bol. Acad. Nac. Cienc. (Córdoba) 63 (1999), 45–78.  MR  Zbl
  7. N. Andruskiewitsch, I. Heckenberger, and H.-J. Schneider, The Nichols algebra of a semisimple Yetter-Drinfeld module, Amer. J. Math. 132 no. 6 (2010), 1493–1547.  MR  Zbl
  8. N. Andruskiewitsch and H.-J. Schneider, Lifting of quantum linear spaces and pointed Hopf algebras of order $p^3$, J. Algebra 209 no. 2 (1998), 658–691.  DOI  MR  Zbl
  9. I. Angiono, On Nichols algebras of diagonal type, J. Reine Angew. Math. 683 (2013), 189–251.  DOI  MR  Zbl
  10. I. Angiono, A presentation by generators and relations of Nichols algebras of diagonal type and convex orders on root systems, J. Eur. Math. Soc. (JEMS) 17 no. 10 (2015), 2643–2671.  DOI  MR  Zbl
  11. G. A. García and J. M. J. Giraldi, On Hopf algebras over quantum subgroups, J. Pure Appl. Algebra 223 no. 2 (2019), 738–768.  DOI  MR  Zbl
  12. M. Graña, A freeness theorem for Nichols algebras, J. Algebra 231 no. 1 (2000), 235–257.  DOI  MR  Zbl
  13. M. Graña, On Nichols algebras of low dimension, in New Trends in Hopf Algebra Theory (La Falda, 1999), Contemp. Math. 267, Amer. Math. Soc., Providence, RI, 2000, pp. 111–134.  DOI  MR  Zbl
  14. L. Grunenfelder and M. Mastnak, Pointed Hopf algebras as cocycle deformations, 2010. arXiv:1010.4976 [math.QA].
  15. I. Heckenberger, Classification of arithmetic root systems, Adv. Math. 220 no. 1 (2009), 59–124.  DOI  MR  Zbl
  16. I. Heckenberger and H.-J. Schneider, Yetter-Drinfeld modules over bosonizations of dually paired Hopf algebras, Adv. Math. 244 (2013), 354–394.  DOI  MR  Zbl
  17. N. Hu and R. Xiong, On families of Hopf algebras without the dual Chevalley property, Rev. Un. Mat. Argentina 59 no. 2 (2018), 443–469.  DOI  MR  Zbl
  18. L. Krop and D. E. Radford, Finite-dimensional Hopf algebras of rank one in characteristic zero, J. Algebra 302 no. 1 (2006), 214–230.  DOI  MR  Zbl
  19. S. Majid and R. Oeckl, Twisting of quantum differentials and the Planck scale Hopf algebra, Comm. Math. Phys. 205 no. 3 (1999), 617–655.  DOI  MR  Zbl
  20. O. Márquez, D. Bagio, J. M. J. Giraldi, and G. A. García, Finite-dimensional Nichols algebras over dual Radford algebras, J. Algebra Appl. 20 no. 1 (2021), Paper No. 2140001, 39 pp.  DOI  MR  Zbl
  21. S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Regional Conference Series in Mathematics 82, American Mathematical Society, Providence, RI, 1993.  DOI  MR  Zbl
  22. D. E. Radford, Hopf Algebras, Series on Knots and Everything 49, World Scientific, Hackensack, NJ, 2012.  MR  Zbl
  23. R. Xiong, On Hopf algebras over the unique 12-dimensional Hopf algebra without the dual Chevalley property, Comm. Algebra 47 no. 4 (2019), 1516–1540.  DOI  MR  Zbl
  24. R. Xiong and N. Hu, Classification of finite-dimensional Hopf algebras over dual Radford algebras, Bull. Belg. Math. Soc. Simon Stevin 28 no. 5 (2022), 633–688.  DOI  MR  Zbl