Skip to main content
SpringerLink
Log in

Commuting variety of Witt algebra

  • Research Article
  • Published:
Frontiers of Mathematics in China Aims and scope Submit manuscript

Abstract

Let \(\mathfrak{g} = W_1 \) be the Witt algebra over an algebraically closed field k of characteristic p > 3; and let be the commuting variety of g. In contrast with the case of classical Lie algebras of P. Levy [J. Algebra, 2002, 250: 473–484], we show that the variety is reducible, and not equidimensional. Irreducible components of and their dimensions are precisely given. As a consequence, the variety is not normal.

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. Chang H J. Uber Wittsche Lie-Ringe. Abh Math Semin Univ Hambg, 1941, 14: 151–184

    Article  MATH  Google Scholar 

  2. Demushkin S P. Cartan subalgebras of simple Lie p-algebras W n and S n. Sibirsk Mat Zh, 1970, 11(2): 310–325

    Article  MathSciNet  Google Scholar 

  3. Humphreys J. Linear Algebraic Groups. Grad Texts in Math, Vol 21. New York: Springer-Verlag, 1975

    Book  MATH  Google Scholar 

  4. Levy P. Commuting varieties of Lie algebras over fields of prime characteristic. J Algebra, 2002, 250: 473–484

    Article  MathSciNet  MATH  Google Scholar 

  5. Mygind M. Orbit closure in the Witt algebra and its dual space. J Algebra Appl, 2014, 13(5): 1350146

    Article  MathSciNet  MATH  Google Scholar 

  6. Ree R. On generalized Witt algebras. Trans Amer Math Soc, 1956, 83: 510–546

    Article  MathSciNet  MATH  Google Scholar 

  7. Richardson R W. Commuting varieties of semisimple Lie algebras and algebraic groups. Compos Math, 1979, 38: 311–327

    MathSciNet  MATH  Google Scholar 

  8. Strade H, Farnsteiner R. Modular Lie Algebras and Their Representations. Monographs and Textbooks in Pure and Applied Mathematics, Vol 116. New York: Marcel Dekker Inc, 1988

    MATH  Google Scholar 

  9. Tauvel P, Yu R. Lie Algebras and Algebraic Groups. Springer Monogr Math. Berlin: Springer, 2005

    MATH  Google Scholar 

  10. Wilson R. Automorphisms of graded Lie algebras of Cartan type. Comm Algebra, 1975, 3(7): 591–613

    Article  MathSciNet  MATH  Google Scholar 

  11. Yao Y-F, Chang H. The nilpotent commuting variety of the Witt algebra. J Pure Appl Algebra, 2014, 218(10): 1783–1791

    Article  MathSciNet  MATH  Google Scholar 

  12. Yao Y-F, Shu B. Nilpotent orbits in the Witt algebra W 1. Comm Algebra, 2011, 39(9): 3232–3241

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to express their thanks to the referees for many useful suggestions and comments on the manuscript. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11771279, 11801204) and the Natural Science Foundation of Shanghai (Grant No. 16ZR1415000).

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai Maritime University, Shanghai, 201306, China

    Yu-Feng Yao

  2. School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

    Hao Chang

Authors
  1. Yu-Feng Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hao Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hao Chang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yao, YF., Chang, H. Commuting variety of Witt algebra. Front. Math. China 13, 1179–1187 (2018). https://doi.org/10.1007/s11464-018-0725-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11464-018-0725-9

Keywords

MSC

Search

Navigation