Abstract
The cohomology of restricted Lie algebras was first defined by Hochschild in 1954, cf.[11]. It was however only recently that one could get more precise information about these cohomology groups in non-trivial cases. The most fascinating result is still the theorem (proved by Friedlander and Parshall) that for large p the cohomology ring of the Lie algebra of a reductive algebraic group is the ring of regular functions on the nilpotent cone in this Lie algebra. It is the purpose of this article to give a survey of recent developments in this theory.
Throughout this paper let k be an algebraically closed field with char(k)=p≠0.
Keywords
- Hopf Algebra
- Algebraic Group
- Spectral Sequence
- Projective Resolution
- Forgetful Functor
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.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
H.H. Andersen,J.C. Jantzen: Cohomology of induced representations of algebraic groups, Math.Ann.269(1984), 487–525
D. Benson: Modular Representation Theory: New Trends and Methods, Lecture Notes in Mathematics 1081, Berlin/Heidelberg/New York/Tokyo 1984(Springer)
N. Bourbaki: Groupes et algèbres de Lie, chap.4,5 et 6, Paris 1968 (Hermann)
M. Demazure,P. Gabriel: Groupes Algébriques I, Paris/Amsterdam 1970 (Masson/North-Holland)
E. Friedlander,B. Parshall: Cohomology of algebraic and related finite groups, Invent.math.74(1983), 85–117
E. Friedlander,B. Parshall: Cohomology of Lie algebras and algebraic groups, Amer.J.Math.108(1986), 235–253
E. Friedlander,B. Parshall: Cohomology of infinitesimal and discrete groups, Math.Ann.273(1986), 353–374
E.Friedlander,B.Parshall: Geometry of p-unipotent Lie algebras, J.Algebra (to appear)
E.Friedlander,B.Parshall: Support varieties for restricted Lie algebras, to appear
W. Hesselink: Cohomology and the resolution of the nilpotent variety, Math.Ann.223(1976), 249–252
G. Hochschild: Cohomology of restricted Lie algebras, Amer.J.Math. 76(1954), 555–580
N. Jacobson: Lie Algebras, New York/London/Sydney 1962 (Intersciene/Wiley)
J.C. Jantzen: Kohomologie von p-Lie-Algebren und nilpotente Elemente, Abh.Math.Sem.Univ.Hamburg 76(1986) (demnächst)
J.C.Jantzen: Representations of algebraic groups (to appear)
A. Kerber: Representations of Permutation Groups, Lecture Notes in Mathematics 240, Berlin/Heidelberg/New York 1971(Springer)
B. Kostant: Lie group representations on polynomial rings, Amer.J. Math.85(1963), 327–404
T.A. Springer: The unipotent variety of a semi-simple algebraic group, pp.373–391 in: Algebraic Geometry (Proc. Bombay 1968), London 1969 (Oxford Univ.Press)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 Springer-Verlag
About this paper
Cite this paper
Jantzen, J.C. (1987). Restricted lie algebra cohomology. In: Cohen, A.M., Hesselink, W.H., van der Kallen, W.L.J., Strooker, J.R. (eds) Algebraic Groups Utrecht 1986. Lecture Notes in Mathematics, vol 1271. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0079234
Download citation
DOI: https://doi.org/10.1007/BFb0079234
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-18234-4
Online ISBN: 978-3-540-47834-8
eBook Packages: Springer Book Archive
