Skip to main content
SpringerLink
Log in

Group-Graded Algebras, Extensions of Infinitesimal Groups, and Applications

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

Using results on algebras that are graded by p-groups, we study representations of infinitesimal groups \( \mathcal{G} \) that possess a normal subgroup \( \mathcal{N}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \triangleleft } \mathcal{G} \) with a diagonalizable factor group \( \mathcal{G} \mathord{\left/ {\vphantom {\mathcal{G} \mathcal{N}}} \right. \kern-\nulldelimiterspace} \mathcal{N} \). When combined with rank varieties, Auslander-Reiten theory and Premet's work on SL(2)1-modules, these techniques lead to the determination of the indecomposable modules of the infinitesimal groups of domestic representation type.

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. J. Alperin, Local Representation Theory, Cambridge Studies in Advanced Mathematics, Vol. 11, Cambridge University Press, Cambridge, 1986.

    Google Scholar 

  2. I. Assem, D. Simson, A. Skowronski, Elements of the Representation Theory of Associative Algebras, London Mathematical Society Student Texts, Vol. 65, Cambridge University Press, Cambridge, 2006.

    Google Scholar 

  3. M. Auslander, I. Reiten, S. Smalø, Galois actions on rings and finite Galois coverings, Math. Scand. 65 (1989), 5–32.

    MATH  MathSciNet  Google Scholar 

  4. M. Auslander, I. Reiten, S. Smalø, Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics, Vol. 36, Cambridge University Press, Cambridge, 1995.

    Google Scholar 

  5. D. Benson, Representation and Cohomology, I, Cambridge Studies in Advanced Mathematics, Vol. 30, Cambridge University Press, Cambridge, 1991.

    Google Scholar 

  6. M. Butler, C. Ringel, Auslander-Reiten sequences with few middle terms and applications to string algebras, Comm. Algebra 15 (1987), 145–179.

    Article  MATH  MathSciNet  Google Scholar 

  7. J. Carlson, E. Friedlander, J. Pevtsova, Modules of constant Jordan type, J. Reine Angew. Math. 614 (2008), 191–234.

    MATH  MathSciNet  Google Scholar 

  8. J. Carlson, D. Nakano, K. Peters, On the vanishing of extensions of modules over reduced enveloping algebras, Math. Ann. 302 (1995), 541–560.

    Article  MATH  MathSciNet  Google Scholar 

  9. C. Curtis, I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Wiley Interscience, New York, 1962. Russian transl.: Ч. Кэртис, И. Райнер, Теория представлений конечных групп и ассоциативных алгебр, Наука, М., 1969.

  10. E. Dade, Group-graded rings and modules, Math. Z. 174 (1980), 241–262.

    Article  MATH  MathSciNet  Google Scholar 

  11. J. de la Peña, R. Martinez-Villa, Automorphisms of representation-finite algebras, Invent. Math. 72 (1983), 359–362.

    Article  MATH  MathSciNet  Google Scholar 

  12. M. Demazure, P. Gabriel, Groupes Algébriques I, Masson/North-Holland, Paris, 1970.

  13. Yu. Drozd, On the representations of the Lie algebra sl 2, Vīsn. Kiïv. Unīv., Mat. i Mekh. 25 (1983), 70–77 (Ukrainian).

  14. K. Erdmann, Blocks of Tame Representation Type and Related Algebras, Lecture Notes in Mathematics, Vol. 1428, Springer-Verlag, New York, 1990.

  15. K. Erdmann, A. Skowroński, On Auslander-Reiten components of blocks and self-injective biserial algebras, Trans. Amer. Math. Soc. 330 (1992), 165–189.

    Article  MATH  MathSciNet  Google Scholar 

  16. R. Farnsteiner, Periodicity and representation type of modular Lie algebras, J. Reine Angew. Math. 464 (1995), 47–65.

    MATH  MathSciNet  Google Scholar 

  17. R. Fansteiner, On the Auslander-Reiten quiver of an infinitesimal group, Nagoya Math. J. 160 (2000), 103–121.

    MathSciNet  Google Scholar 

  18. R. Farnsteiner, Auslander-Reiten components for G 1 T-modules, J. Algebra Appl. 4 (2005), 739–759.

    Article  MATH  MathSciNet  Google Scholar 

  19. R. Farnsteiner, Polyhedral groups, McKay quivers, and the finite algebraic groups with tame principal blocks, Invent. Math. 166 (2006), 27–94.

    Article  MATH  MathSciNet  Google Scholar 

  20. R. Farnsteiner, D. Voigt, On infinitesimal groups of tame representation type, Math. Z. 244 (2003), 479–513.

    MATH  MathSciNet  Google Scholar 

  21. R. Farnsteiner, A. Skowroński, Classification of restricted Lie algebras with tame principal block, J. Reine Angew. Math. 546 (2002), 1–45.

    MATH  MathSciNet  Google Scholar 

  22. R. Farnsteiner, A. Skowroński, The tame infinitesimal groups of odd characteristic, Adv. in Math. 205 (2006), 229–274.

    Article  MATH  Google Scholar 

  23. R. Farnsteiner, A. Skowroński, Galois actions and blocks of tame infinitesimal group schemes, Trans. Amer. Math. Soc. 359 (2007), 5867–5898.

    Article  MATH  MathSciNet  Google Scholar 

  24. G. Fischer, Die Köcher der Frobeniuskerne der SL2, Dissertation, Universität Bielefeld, 1983.

  25. D. Fischmann, S. Montgomery, H. Schneider, Frobenius extensions of subalgebras of Hopf algebras, Trans. Amer. Math. Soc. 349 (1996), 4857–4895.

    Article  Google Scholar 

  26. E. Friedlander, B. Parshall, Support varieties for restricted Lie algebras, Invent. Math. 86 (1986), 553–562.

    Article  MATH  MathSciNet  Google Scholar 

  27. E. Friedlander, J. Pevtsova, A. Suslin, Generic and maximal Jordan types, Invent. Math. 168 (2007), 485–522.

    Article  MATH  MathSciNet  Google Scholar 

  28. R. Gordon, E. Green, Graded Artin algebras, J. Algebra 76 (1982), 111–137.

    Article  MATH  MathSciNet  Google Scholar 

  29. R. Gordon, E. Green, Representation theory of graded Artin algebras, J. Algebra 76 (1982), 138–152.

    Article  MATH  MathSciNet  Google Scholar 

  30. A. Heller, Indecomposable representations and the loop space operation, Proc. Amer. Math. Soc. 12 (1961), 640–643.

    Article  MATH  MathSciNet  Google Scholar 

  31. J. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, Vol. 21, Springer-Verlag, New York, 1981.

  32. N. Jacobson, Lie Algebras, Dover, New York 1979.

    Google Scholar 

  33. J. Jantzen, Über Darstellungen höherer Frobenius-Kerne halbeinfacher algebraischer Gruppen, Math. Z. 164 (1979), 271–292.

    Article  MATH  MathSciNet  Google Scholar 

  34. J. Jantzen, Kohomologie von p-Lie-Algebren und nilpotente Elemente, Abh. Math. Sem. Univ. Hamburg 56 (1986), 191–219.

    Article  MATH  MathSciNet  Google Scholar 

  35. J. Jantzen, Modular representations of reductive Lie algebras, J. Pure Appl. Algebra 152 (2000), 133–185.

    Article  MATH  MathSciNet  Google Scholar 

  36. J. Jantzen, Representations of Algebraic Groups, Mathematical Surveys and Monographs, Vol. 107, Amer. Math. Soc., Providence, RI, 2003.

  37. G. Karpilovsky, The Algebraic Structure of Crossed Products, Mathematics Studies, Vol. 142, North-Holland, Amsterdam, 1987.

  38. S. Montgomery, Hopf Algebras and their Actions on Rings, Regional Conference Series in Mathematics, Vol. 82, Amer. Math. Soc., Providence, RI, 1993.

  39. R. Pollack, Restricted Lie algebras of bounded type, Bull. Amer. Math. Soc. 74 (1968), 326–331.

    Article  MATH  MathSciNet  Google Scholar 

  40. А. Премет, Кольцо Грина простой трёхмерной р-алгебры Ли, Изв. высш. уч. завед., мат. 10 (1991), 56–67. English transl.: A. Premet, Green ring for simple three-dimensional Lie-p-algebra, Soviet Math. (Izv. VUZ) 35 (1991), no. 10, 51–60.

  41. C. Riedtmann, Algebren, Darstellungsköcher, Überlagerungen und zurück, Comment. Math. Helv. 55 (1980), 199–224.

    Article  MATH  MathSciNet  Google Scholar 

  42. C. Ringel, Finite-dimensional hereditary algebras of wild representation type, Math. Z. 161 (1978), 235–255.

    Article  MATH  MathSciNet  Google Scholar 

  43. C. Ringel, Tame algebras (On algorithms for solving vector space problems II), in: Representation Theory I, Proc. 2nd Internat. Conf., Ottawa 1979, Lecture Notes in Mathematics, Vol. 831, Springer-Verlag, New York, 1980, pp. 137–287.

  44. А. Рудаков, Приводимые р-представления простой трёхмерной р-алгебры Ли, Вестн. Моск. ун-та, сер. I, (1982), no. 8, 45–49. English transl.: A. Rudakov, Reducible p-representations of a simple three-dimensional Lie-p-algebra, Moscow Univ. Math. Bull. 37 (1982), no. 6, 51–56.

  45. H. Strade, R. Farnsteiner, Modular Lie Algebras and their Applications, Pure and Applied Mathematics, Vol. 116, Marcel Dekker, New York, 1988.

  46. H. Tachikawa, Representations of trivial extensions of hereditary algebras, in: Representation Theory II, Proc. 2nd Internat. Conf., Ottawa 1979, Lecture Notes in Mathematics, Vol. 832, Springer-Verlag, New York, 1980, pp. 579–599.

  47. D. Voigt, Induzierte Darstellungen in der Theorie der endlichen, algebraischen Gruppen, Lecture Notes in Mathematics, Vol. 592, Springer-Verlag, New York, 1977.

  48. W. Waterhouse, Introduction to Affine Group Schemes, Graduate Texts in Mathematics, Vol. 66, Springer-Verlag, New York, 1979.

Download references

Author information

Author notes

  1. R. Farnsteiner

    Present address: Department of Mathematics, University of Kiel, Ludewig-Meyn-Str. 4, 24098, Kiel, Germany

Authors and Affiliations

  1. Department of Mathematics, University of Bielefeld, P.O. Box 10 01 31, 33501, Bielefeld, Germany

    R. Farnsteiner

Authors
  1. R. Farnsteiner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Farnsteiner.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Farnsteiner, R. Group-Graded Algebras, Extensions of Infinitesimal Groups, and Applications. Transformation Groups 14, 127–162 (2009). https://doi.org/10.1007/s00031-009-9049-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-009-9049-x

Keywords

Search

Navigation