Mathematische Zeitschrift

, Volume 288, Issue 3–4, pp 1395–1420 | Cite as

Discriminants and automorphism groups of Veronese subrings of skew polynomial rings

  • K. Chan
  • A. A. Young
  • J. J. Zhang


We study important invariants and properties of the Veronese subalgebras of q-skew polynomial rings, including their discriminant, center and automorphism group, as well as cancellation property and the Tits alternative.


Skew polynomial ring Veronese subring Discriminant Automorphism group Cancellation problem Tits alternative 

Mathematics Subject Classification

Primary 16W20 



The authors would like to thank the referee for his/her very careful reading and extremely valuable comments. A.A. Young was partly supported by the US National Science Foundation (NSF Postdoctoral Research Fellowship, No. DMS-1203744) and J.J. Zhang by the US National Science Foundation (No. DMS-1402863).


  1. 1.
    Alev, J., Chamarie, M.: Dérivations et automorphismes de quelques algébres quantiques. Commun. Algebra 20(6), 1787–1802 (1992)CrossRefzbMATHGoogle Scholar
  2. 2.
    Alev, J., Dumas, F.: Rigidité des plongements des quotients primitifs minimaux de \(U_q(sl(2))\) dans l’algébre quantique de Weyl-Hayashi. Nagoya Math. J. 143, 119–146 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Andruskiewitsch, N., Dumas, F.: On the automorphisms of \(U^+_q (\mathfrak{g})\). In: Quantum Groups. IRMA Lect. Math. Theor. Phys., vol. 12, pp. 107–133. European Mathematical Society, Zürich (2008)Google Scholar
  4. 4.
    Bavula, V.V., Jordan, D.A.: Isomorphism problems and groups of automorphisms for generalized Weyl algebras. Trans. Am. Math. Soc. 353(2), 769–794 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bell, J., Zhang, J.J.: Zariski cancellation problem for noncommutative algebras. Sel. Math. (N.S.) (To appear) (Preprint) (2016). arXiv:1601.04625
  6. 6.
    Berenstein, A., Zelevinsky, A.: Quantum cluster algebras. Adv. Math. 195, 405–455 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ceken, S., Palmieri, J., Wang, Y.-H., Zhang, J.J.: The discriminant controls automorphism groups of noncommutative algebras. Adv. Math. 269, 551–584 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ceken, S., Palmieri, J., Wang, Y.-H., Zhang, J.J.: The discriminant criterion and automorphism groups of quantized algebras. Adv. Math. 285, 754–801 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ceken, S., Palmieri, J., Wang, Y.-H., Zhang, J.J.: Invariant theory for quantum Weyl algebras under finite group action. In: Proceedings of symposia in pure mathematics, vol. 92, Lie algebras, Lie superalgebras, vertex algebras and related topics, pp. 119–135 (2016)Google Scholar
  10. 10.
    Chan, K., Young, A.A., Zhang, J.J.: Discriminant formulas and applications. Algebra Number Theory 10(3), 557–596 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gómez-Torrecillas, J., El Kaoutit, L.: The group of automorphisms of the coordinate ring of quantum symplectic space. Beiträge Algebra Geom. 43(2), 597–601 (2002)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Goodearl, K.R., Yakimov, M.T.: Unipotent and Nakayama automorphisms of quantum nilpotent algebras. Commut. Algebra Noncommut. Algebraic Geom. 2, 181–212 (2015). (Math. Sci. Res. Inst. Publ., 68)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gupta, N.: On the cancellation problem for the affine space \({\mathbb{A}}^3\) in characteristic \(p\). Invent. Math. 195(1), 279–288 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gupta, N.: On Zariski’s cancellation problem in positive characteristic. Adv. Math. 264, 296–307 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gupta, N.: A survey on Zariski cancellation problem. Indian J. Pure Appl. Math. 46(6), 865–877 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Launois, S., Lenagan, T.H.: Automorphisms of quantum matrices. Glasg. Math. J. 55(A), 89–100 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lü, J.-F., Mao, X.-F., Zhang, J.J.: Nakayama automorphism and applications. Trans. Am. Math. Soc. 369(4), 2425–2460 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Shestakov, I., Umirbaev, U.: The tame and the wild automorphisms of polynomial rings in three variables. J. Am. Math. Soc. 17(1), 197–227 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Suárez-Alvarez, M., Vivas, Q.: Automorphisms and isomorphism of quantum generalized Weyl algebras. J. Algebra 424, 540–552 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Tits, J.: Free subgroups in linear groups. J. Algebra 20, 250–270 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Yakimov, M.: The Launois–Lenagan conjecture. J. Algebra 392, 1–9 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Yakimov, M.: The Andruskiewitsch–Dumas conjecture. Sel. Math. (N.S.) 20(2), 421–464 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA
  2. 2.Department of MathematicsDigiPen Institute of TechnologyRedmondUSA

Personalised recommendations