Mathematische Zeitschrift

, Volume 206, Issue 1, pp 153–168 | Cite as

Shapiro's lemma and its consequences in the cohomology theory of modular Lie algebras

  • Rolf Farnsteiner
  • Helmut Strade


Cohomology Group Verma Module Cohomology Theory Contact Algebra Frobenius Extension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Berkson, A.J.: Theu-algebra of a restricted Lie algebra is Frobenius. Proc. Am. Math. Soc.15, 14–15 (1964)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cartan, H., Eilenberg, S.: Homological algebra. Princeton: Princeton University Press 1956MATHGoogle Scholar
  3. 3.
    Chang, H.J.: Über Wittsche Lie Ringe. Abh. Math. Semin. Univ. Hamb.14, 151–184 (1941)Google Scholar
  4. 4.
    Chevalley, C., Eilenberg, S.: Cohomology theory of Lie groups and Lie algebras. Trans. Am. Math. Soc.63, 85–124 (1948)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chwe, B.S.: On the commutativity of restricted Lie algebras. Proc. Am. Math. Soc.16, 547 (1965)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Dzhumadil'daev, A.S.: Cohomology of modular Lie algebras. Math. USSR, Sb.47, 127–143 (1984)MATHCrossRefGoogle Scholar
  7. 7.
    Faith, C., Walker, E.A.: Direct sum decompositions of injective modules. J. Algebra5, 203–221 (1967)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Farnsteiner, R.: On ad-semisimple Lie algebras. J. Algebra83, 510–519 (1983)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Farnsteiner, R.: Conditions for the commutativity of restricted Lie algebras. Commun. Algebra13, 1475–1489 (1985)MATHMathSciNetGoogle Scholar
  10. 10.
    Farnsteiner, R.: Central extensions and invariant forms of graded Lie algebras. Algebras Groups Geom.3, 431–455 (1986)MATHMathSciNetGoogle Scholar
  11. 11.
    Farnsteiner, R.: Dual space derivations andH 2(L, F) of modular Lie algebras. Can. J. Math.39, 1078–1106 (1987)MATHMathSciNetGoogle Scholar
  12. 12.
    Farnsteiner, R.: On the cohomology of associative algebras and Lie algebras. Proc. Am. Math. Soc.99, 415–420 (1987)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Farnsteiner, R.: On the vanishing of homology and cohomology groups of associative algebras. Trans. Am. Math. Soc.306, 651–665 (1988)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Farnsteiner, R.: Cohomology groups of infinite dimensional algebras. Math. Z.199, 407–423 (1988)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Friedlander, E.M., Parshall, B.J.: Geometry ofp-unipotent Lie algebras. J. Algebra109, 25–45 (1987)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Friedlander, E.M., Parshall, B.J.: Support varieties for restricted Lie algebras. Invent. Math.86, 553–562 (1986)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Hilton, P.J., Stammbach, U.: A course in homological algebra. (Graduate Texts, vol. 4) Berlin Heidelberg New York: Springer 1970Google Scholar
  18. 18.
    Hochschild, G.P., Serre, J.P.: Cohomology of Lie algebras. Ann. Math.57, 591–603 (1953)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Hochschild, G.P.: On the cohomology groups of an associative algebra. Ann. Math.46, 58–67 (1945)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Hochschild, G.P.: Cohomology of restricted Lie algebras. Am. J. Math.76, 555–580 (1954)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Humphreys, J.E.: Restricted Lie algebras (and beyond). Contemp. Math.13, 91–98 (1982)MATHMathSciNetGoogle Scholar
  22. 22.
    Pareigis, B.: Einige Bemerkungen über Frobenius-Erweiterungen. Math. Ann.153, 1–13 (1964)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Seligman, G.B.: Modular Lie algebras. (Ergeb. Math. Grenzgeb. vol. 40), Berlin Heidelberg New York: Springer 1967MATHGoogle Scholar
  24. 24.
    Sen, C., Shen, G.: Cohomology of graded Lie algebras of Cartan type of characteristic p. Abh. Math. Semin. Univ. Hamb.57, 139–156 (1987)MATHCrossRefGoogle Scholar
  25. 25.
    Strade, H.: Representations of the Witt algebra. J. Algebra49, 595–605 (1977)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Strade, H.: Darstellungen Auflösbarer Lie Algebren. Math. Ann.232, 15–32 (1978)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Strade, H., Farnsteiner, R.: Modular Lie algebras and their representations. (Textbooks and Monographs, vol. 116), New York: Dekker 1988MATHGoogle Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Rolf Farnsteiner
    • 1
  • Helmut Strade
    • 2
  1. 1.Department of MathematicsUniversity of WisconsinMilwaukeeUSA
  2. 2.Mathematisches Seminar der Universität HamburgHamburg 13Federal Republic of Germany

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