Skip to main content
SpringerLink
Log in

Representations of Lie algebras of vector fields on affine varieties

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

For an irreducible affine variety X over an algebraically closed field of characteristic zero we define two new classes of modules over the Lie algebra of vector fields on X—gauge modules and Rudakov modules, which admit a compatible action of the algebra of functions. Gauge modules are generalizations of modules of tensor densities whose construction was inspired by non-abelian gauge theory. Rudakov modules are generalizations of a family of induced modules over the Lie algebra of derivations of a polynomial ring studied by Rudakov [23]. We prove general simplicity theorems for these two types of modules and establish a pairing between them.

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. A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear Physics. B 241 (1984), 333–380.

    Article  MathSciNet  Google Scholar 

  2. Y. Billig, A category of modules for the full toroidal Lie algebra, International Mathematics Research Notices (2006), Article no. 68395.

    Google Scholar 

  3. Y. Billig, Jet modules, Canadian Journal of Mathematics 59 (2007), 712–729.

    Article  MathSciNet  Google Scholar 

  4. Y. Billig and V. Futorny, Representations of Lie algebra of vector fields on a torus and chiral de Rham complex, Transactions of the American Mathematical Society 366 (2014), 4697–4731.

    Article  MathSciNet  Google Scholar 

  5. Y. Billig and V. Futorny, Classification of irreducible representations of Lie algebra of vector fields on a torus, Journal für die Reine und Angewandte Mathematk 2016 (2016), 199–216.

    MathSciNet  MATH  Google Scholar 

  6. Y. Billig and V. Futorny, Lie algebras of vector fields on smooth affine varieties, Communications in Algebra 46 (2018), 3413–3429.

    Article  MathSciNet  Google Scholar 

  7. Y. Billig and J. Nilsson, Representations of the Lie algebra of vector fields on a sphere, Journal of Pure and Applied Algebra 223 (2019), 3581–3593.

    Article  MathSciNet  Google Scholar 

  8. E. Cartan, Sur la structure des groups des groups de transformations finis et continus, Thesis, Université de Paris, 1894.

    MATH  Google Scholar 

  9. E. Cartan, Les groups de transformations continus, infinis, simples, Annales Scientifiques de l’École Normale Supérieure 26 (1909), 93–161.

    Article  Google Scholar 

  10. A. Cavaness and D. Grantcharov, Bounded weight modules of the Lie algebra of vector fields on ℂ2, Journal of Algebra and its Applications 16 (2017), Article no. 1750236.

    Article  MathSciNet  Google Scholar 

  11. S. Eswara Rao, Irreducible representations of the Lie-algebra of the diffeomorphisms of a d-dimensional torus, Journal of Algebra 182 (1996), 401–421.

    Article  MathSciNet  Google Scholar 

  12. S. Eswara Rao, Partial classification of modules for Lie algebra of diffeomorphisms of d-dimensional torus, Journal of Mathematical Physics 45 (2004), 3322–3333.

    Article  MathSciNet  Google Scholar 

  13. D. Jordan, On the ideals of a Lie algebra of derivations, Journal of the London Mathematical Society 33 (1986), 33–39.

    Article  MathSciNet  Google Scholar 

  14. D. Jordan, On the simplicity of Lie algebras of derivations of commutative algebras, Journal of Algebra, 228 (2000), 580–585.

    Article  MathSciNet  Google Scholar 

  15. W. Killing, Die Zusammensetzung der stetigen endlichen Transformationsgruppen, Mathematische Annalen 33 (1889), 1–48.

    Article  MathSciNet  Google Scholar 

  16. I. M. Krichever and S. P. Novikov, Algebras of Virasoro type, Riemann surfaces and structures of the theory of solitons, Functional Analysis and its Applications 21 (1987), 126–142.

    Article  Google Scholar 

  17. T. A. Larsson, Conformal fields: A class of representations of Vect(N), International Journal of Modern Physics. A 7 (1992), 6493–6508.

    Article  MathSciNet  Google Scholar 

  18. G. Liu, R. Lu and K. Zhao, Irreducible Witt modules from Weyl modules and gln-modules, Journal of Algebra 511 (2018), 164–181.

    Article  MathSciNet  Google Scholar 

  19. O. Mathieu, Classification of Harish-Chandra modules over the Virasoro algebra, Inventiones Mathematicae 107 (1992), 225–234

    Article  MathSciNet  Google Scholar 

  20. V. Mazorchuk and K. Zhao, Supports of weight modules over Witt algebras, Proceedings of the Royal Society of Edinburgh 141 (2011), 155–170.

    Article  MathSciNet  Google Scholar 

  21. S. Montgomery, Hopf Algebras and their Actions on Rings, CBMS Regional Conference Series in Mathematics, Vol. 82, American Mathematical Society, Providence, RI, 1993.

  22. I. Penkov and V. Serganova, Weight representations of the polynomial Cartan type Lie algebras W n and S n, Mathematical Research Letters 6 (1999), 397–416.

    Article  MathSciNet  Google Scholar 

  23. A. N. Rudakov, Irreducible representations of infinite-dimensional Lie algebras of Cartan type, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 8 (1974), 836–866.

    MathSciNet  MATH  Google Scholar 

  24. M. Schlichenmaier, From the Virasoro algebra to Krichever-Novikov type algebras and beyond, in Harmonic and Complex Analysis and its Applications, Trends in Mathematics, Birkhäuser/Springer, Cham, 2014, pp. 325–358.

    Chapter  Google Scholar 

  25. I. R. Shafarevich, Basic Algebraic Geometry. Vol. 1, Springer, Heidelberg, 2013.

  26. G. Shen, Graded modules of graded Lie algebras of Cartan type. I. Mixed products of modules, Scientia Sinica. Series A 29 (1986), 570–581.

    MathSciNet  MATH  Google Scholar 

  27. T. Siebert, Lie algebras of derivations and affine algebraic geometry over fields of characteristic 0, Mathematische Annalen 305 (1996), 271–286.

    Article  MathSciNet  Google Scholar 

  28. A. Tsuchiya, K. Ueno and Y. Yamada, Conformal field theory on universal family of stable curves with gauge symmetries, in Integrable Systems in Quantum Field Theory and Statistical Mechanics, Advanced Studies in Pure Mathematics, Vol. 19, Academic Press, Boston, MA, 1989, pp. 459–566.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, ON, K1S 5B6, Canada

    Yuly Billig

  2. Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, CEP 05508-090, São Paulo, SP, Brasil

    Vyacheslav Futorny

  3. Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, SE-412 96, Gothenburg, Sweden

    Jonathan Nilsson

Authors
  1. Yuly Billig
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vyacheslav Futorny
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jonathan Nilsson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vyacheslav Futorny.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Billig, Y., Futorny, V. & Nilsson, J. Representations of Lie algebras of vector fields on affine varieties. Isr. J. Math. 233, 379–399 (2019). https://doi.org/10.1007/s11856-019-1909-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-019-1909-z

Search

Navigation