Skip to main content

Group Gradings on Lie Algebras, with Applications to Geometry. I

  • Chapter
  • First Online:
Developments and Retrospectives in Lie Theory

Part of the book series: Developments in Mathematics ((DEVM,volume 38))

Abstract

In this article, which is the first part of a sequence of two, we discuss modern approaches to the classification of group gradings on simple and nilpotent Lie algebras. In the second article we discuss applications and related topics in differential geometry.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Bahturin, Yuri; Identical Relations in Lie Algebras. VNU Science Press, Utrecht, 1987, x+309pp.

    Google Scholar 

  2. Bahturin, Yuri; Brešar, Matej, Lie gradings on associative algebras, J. Algebra, 321 (2009), 264–283.

    Article  MathSciNet  MATH  Google Scholar 

  3. Bahturin, Yuri; Brešar, Matej; Kochetov, Mikhail, Group gradings on finitary simple Lie algebras, Int.J. Algebra Comp, 22 (2012), 125–146.

    Google Scholar 

  4. Bahturin, Y.; Kochetov, M. Classification of group gradings on simple Lie algebras of types A, B, C and D. J. Algebra 324 (2010), 2971–2989.

    Article  MathSciNet  MATH  Google Scholar 

  5. Bahturin, Yuri; Kochetov, Mikhail. Group gradings on the Lie algebra \(\mathfrak{p}\mathfrak{s}\mathfrak{l}_{n}\) in positive characteristic. J. Pure Appl. Algebra 213 (2009), no. 9, 1739–1749.

    Google Scholar 

  6. Bahturin, Yuri; Kochetov, Mikhail. Classification of group gradings on simple Lie algebras of types A,B,C and D. J. Algebra 324 (2010), no. 11, 2971–2989.

    Google Scholar 

  7. Bahturin, Yuri; Kochetov, Mikhail, Group gradings on restricted Cartan-type Lie algebras, Pacific J. Math. 253 (2011), no. 2, 289–319.

    Google Scholar 

  8. Bahturin, Y.; Kochetov, M.; Montgomery, S. Group gradings on simple Lie algebras in positive characteristic. Proc. Amer. Math. Soc., 137 (2009), no. 4, 1245–1254.

    Article  MathSciNet  MATH  Google Scholar 

  9. Bahturin, Y.; Shestakov, I.; Zaicev, M. Gradings on simple Jordan and Lie algebras. J. Algebra, 283 (2005), no. 2, 849–868.

    Article  MathSciNet  MATH  Google Scholar 

  10. Bahturin, Y. A.; Sehgal, S. K.; Zaicev, M. V. Group gradings on associative algebras, J. Algebra 241 (2001), no. 2, 677–698.

    Article  MathSciNet  MATH  Google Scholar 

  11. Bahturin, Yuri; Zaicev, Mikhail, Group gradings on simple Lie algebras of type “A”, J. Lie Theory 16(2006), 719–742.

    MathSciNet  MATH  Google Scholar 

  12. Bahturin, Y.; Zaicev, M. Gradings on simple algebras of finitary matrices, J. Algebra 324 (2010), no. 6, 1279–1289.

    Article  MathSciNet  MATH  Google Scholar 

  13. Baranov, A. A. Finitary simple Lie algebras. J. Algebra, 219 (1999), no. 1, 299–329.

    Article  MathSciNet  MATH  Google Scholar 

  14. Baranov, A. A.; Strade, H. Finitary Lie algebras. J. Algebra, 254 (2002), no. 1, 173–211.

    Article  MathSciNet  MATH  Google Scholar 

  15. Beidar, K. I.; Brešar, M.; Chebotar, M. A.; Martindale, 3rd,W. S. On Herstein’s Lie map conjectures. I, Trans. Amer. Math. Soc. 353 (2001), no. 10, 4235–4260 (electronic).

    Google Scholar 

  16. to3em___ , On Herstein’s Lie map conjectures. II, J. Algebra 238 (2001), no. 1, 239–264.

    Google Scholar 

  17. to3em_ , On Herstein’s Lie map conjectures. III, J. Algebra 249 (2002), no. 1, 59–94.

    Google Scholar 

  18. Brešar, M.; Chebotar, M. A.; Martindale, 3rd, W. S. Functional identities, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007.

    Google Scholar 

  19. Elduque, Alberto; Kochetov, Mikhail. Gradings on Simple Lie Algebras AMS Mathematical Surveys and Monographs, 189 (2013), 336 pp.

    Google Scholar 

  20. Goze, Michel; Khakimdjanov, Yusupdjan. Some nilpotent Lie algebras and its applications. Algebra and operator theory (Tashkent, 1997), 4964, Kluwer Acad. Publ., Dordrecht, 1998.

    Google Scholar 

  21. Goze, Michel; Hakimjanov, Yusupdjan. Sur les algèbres de Lie nilpotentes admettant un tore de dérivations. Manuscripta Math. 84 (1994), no. 2, 115–124.

    Google Scholar 

  22. Havliček, Miloslav; Patera, Jiři; Pelantova, Edita, On Lie gradings. II, Linear Algebra Appl. 277 (1998), no. 1–3, 97–125.

    Article  MathSciNet  MATH  Google Scholar 

  23. Jacobson, N. Structure of Rings. Colloquium Publications, 37, American Math. Society, Providence, RI, 1964.

    Google Scholar 

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

    Google Scholar 

  25. Năstăsescu, C.; Van Oystaeyen, F.; Methods of graded rings, Lecture Notes in Mathematics, Vol. 1836, Springer-Verlag, Berlin, 2004.

    Google Scholar 

  26. Patera, Jiri; Zassenhaus, Hans, On Lie gradings. I, Linear Algebra. Appl., 112 (1989), 87–159

    Google Scholar 

  27. Platonov, Vladimir P. Subgroups of algebraic groups over a local or global field containing a maximal torus. C. R. Acad. Sci. Paris Sr. I Math. 318 (1994), no. 10, 899–903.

    Google Scholar 

  28. Springer, T.A. and Steinberg, R. Conjugacy classes. Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69), Lecture Notes in Mathematics, 131, Springer, Berlin, 1970, pp. 167–266.

    Google Scholar 

  29. Strade, H.; Simple Lie algebras over fields of positive characteristic. I, de Gruyter Expositions in Mathematics, Vol. 38, Walter de Gruyter & Co., Berlin, 2004, Structure theory.

    Google Scholar 

  30. to3em__________ , Simple Lie algebras over fields of positive characteristic. II, de Gruyter Expositions in Mathematics, Vol. 42, Walter de Gruyter & Co., Berlin, 2009, Classifying the absolute toral rank two case.

    Google Scholar 

  31. Vergne, Michèle. Cohomologie des algèbres de Lie nilpotentes. Application l’étude de la variété des algèbres de Lie nilpotentes. C. R. Acad. Sci. Paris, Sr. A-B 267 1968, A867A870.

    Google Scholar 

  32. Waterhouse, W.C., Introduction to affine group schemes, GTM, Vol 66, Springer Verlag, New York, 1979.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yuri Bahturin .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Bahturin, Y., Goze, M., Remm, E. (2014). Group Gradings on Lie Algebras, with Applications to Geometry. I. In: Mason, G., Penkov, I., Wolf, J. (eds) Developments and Retrospectives in Lie Theory. Developments in Mathematics, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-319-09804-3_1

Download citation

Publish with us

Policies and ethics