Skip to main content
SpringerLink
Log in

Derived invariants of the fixed ring of enveloping algebras of semisimple Lie algebras

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

Let \({\mathfrak {g}}\) be a semisimple complex Lie algebra, and let W be a finite subgroup of \({\mathbb {C}}\)-algebra automorphisms of the enveloping algebra \(U({\mathfrak {g}})\). We show that the derived category of \(U({\mathfrak {g}})^W\)-modules determines isomorphism classes of both \({\mathfrak {g}}\) and W. Our proofs are based on the geometry of the Zassenhaus variety of the reduction modulo \(p\gg 0\) of \({\mathfrak {g}}.\) Specifically, we use non-existence of certain étale coverings of its smooth locus.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alev, J., Polo, P.: A rigidity theorem for finite group actions on enveloping algebras of semisimple Lie algebras. Adv. Math. 111(2), 208–226 (1995)

    Article  MathSciNet  Google Scholar 

  2. Belov-Kanel, A., Kontsevich, M.: The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture. Mosc. Math. J. 7(2), 209–218 (2007)

    Article  MathSciNet  Google Scholar 

  3. Bezrukavnikov, R., Mirkovic, I., Rumynin, D.: Localization of modules for a semisimple Lie algebra in prime characteristic. Ann. Math. 167, 945–991 (2008)

    Article  MathSciNet  Google Scholar 

  4. Caldero, P.: Isomorphisms of finite invariants for enveloping algebras, semisimple case. Adv. Math. 134(2), 294–307 (1998)

    Article  MathSciNet  Google Scholar 

  5. Grothendieck, A., Raynaud, M.: Revêtements Étales et Groupe Fondamental, Lecture Notes in Mathematics, vol. 224 (1971)

  6. Kac, V., Radul, A.: Poisson Structures for Restricted Lie Algebras, The Gelfand Mathematical Seminars (1996–1999)

  7. Mirkovic, I., Rumynin, D.: Centers of reduced enveloping algebras. Math. Z. 231(1), 123–132 (1999)

    Article  MathSciNet  Google Scholar 

  8. Montgomery, S.: Fixed Rings of Finite Automorphism Groups of Associative Rings, Lecture Notes in Math (1980)

  9. Tange, R.: The Zassenhaus variety of a reductive Lie algebra in positive characteristic. Adv. Math. 224(1), 340–354 (2010)

    Article  MathSciNet  Google Scholar 

  10. Tikaradze, A.: On Automorphisms of Enveloping Algebras, IMRN (2019)

  11. Tikaradze, A.: The Weyl algebra as the fixed ring. Adv. Math. 345, 756–766 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

I am grateful to R.Tange for several helpful comments. I would also like to thank the anonymous referee for many helpful suggestions that led to improvement of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Toledo, Toledo, OH, 43606, USA

    Akaki Tikaradze

Authors
  1. Akaki Tikaradze
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Akaki Tikaradze.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tikaradze, A. Derived invariants of the fixed ring of enveloping algebras of semisimple Lie algebras. Math. Z. 297, 475–481 (2021). https://doi.org/10.1007/s00209-020-02521-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-020-02521-9

Search

Navigation