Abstract
We determine the structure of restricted Lie algebras with bounded cohomology over arbitrary fields of prime characteristic. As a byproduct a classification of the serial restricted Lie algebras and the restricted Lie algebras of finite representation type is obtained. In addition, we derive complete information on the finite dimensional indecomposable restricted modules of these algebras over algebraically closed fields.
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References
A.J. Berkson: The u-algebra of a restricted Lie algebra is Frobenius. Proc. Amer. Math. Soc.15, 14–15 (1964)
D. Eisenbud, Ph.A. Griffith: Serial rings. J. Algebra17, 389–400(1971)
J. Feldvoss: Homologische Aspekte der Darstellungstheorie modularer Lie-Algebren. Dissertation, Universität Hamburg, 1989
J. Feldvoss: On the cohomology of restricted Lie algebras. Comm. Algebra19 (10), 2865–2906 (1991)
E.M. Friedlander, B.J. Parshall: Geometry of p-unipotent Lie algebras. J. Algebra109, 25–45 (1987)
A. Heller: Indecomposable representations and the loop-space operation. Proc. Amer. Math. Soc.12, 640–643(1961)
G.P. Hochschild: Cohomology of restricted Lie algebras. Amer. J. Math.76, 555–580(1954)
L.E.P. Hupert: Homological characteristics of pro-uniserial rings. J. Algebra69, 43–66 (1981)
T. Nakayama: On Frobeniusean algebras II. Ann. of Math.42, 1–21 (1941)
B. Pareigis: Kohomologie von p-Lie-Algebren. Math. Z.104, 281–336 (1968)
W. Pfautsch, D. Voigt: The representation-finite algebraic groups of dimension zero. C. R. Acad. Sci. Paris306, 685–689(1988)
R.D. Pollack: Restricted Lie algebras of bounded type. Bull. Amer. Math. Soc.74, 326–331 (1968)
A.V. Roiter: Unboundedness of the dimensions of the indecomposable representations of an algebra which has infinitely many indecomposable representations. Izv. Akad. SSSR Math.32, 1275–1282(1968)
J.J. Rotman: An Introduction to Homological Algebra. London: Academic Press, Inc., 1979
R. Schulz: Boundedness and periodicity of modules over QF-rings. J. Algebra101, 450–469 (1986)
H. Strade: Darstellungen auflösbarer Lie-p-Algebren. Math. Ann.232, 15–32(1978)
H. Strade, R. Farnsteiner: Modular Lie Algebras and Their Representations. New York-Basel: Marcel Dekker, Inc., 1988
H. Zassenhaus: Representation theory of Lie algebras of characteristic p. Bull. Amer. Math. Soc.60, 463–470 (1954)
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Feldvoss, J., Strade, H. Restricted Lie algebras with bounded cohomology and related classes of algebras. Manuscripta Math 74, 47–67 (1992). https://doi.org/10.1007/BF02567657
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DOI: https://doi.org/10.1007/BF02567657