Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The nucleus for restricted Lie algebras


Authors: David J. Benson and Daniel K. Nakano
Journal: Proc. Amer. Math. Soc. 131 (2003), 3395-3405
MSC (2000): Primary 20G10, 20G05
DOI: https://doi.org/10.1090/S0002-9939-03-06939-9
Published electronically: March 25, 2003
MathSciNet review: 1990628
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The nucleus was a concept first developed in the cohomology theory for finite groups. In this paper the authors investigate the nucleus for restricted Lie algebras. The nucleus is explicitly described for several important classes of Lie algebras.


References [Enhancements On Off] (What's this?)

  • [BN] C. P. Bendel and D. K. Nakano, Complexes and vanishing of cohomology for group schemes, J. Algebra 214 (1999), 668-713. MR 2001a:16015
  • [Ben1] D. J. Benson, Representations and cohomology, II, Cambridge Univ. Press, 1991. MR 93g:20099
  • [Ben2] D. J. Benson, Cohomology of modules in the principal block of a finite group, New York J. Math. 1 (1995), 196-205. MR 96h:20095
  • [Ben3] D. J. Benson, The nucleus, and extensions between modules for a finite group, Representations of Algebras, Proceedings of the Ninth International Conference (Beijing 2000), Beijing Normal University Press, 2002.
  • [BCRi1] D. J. Benson, J. F. Carlson and J. Rickard, Complexity and varieties for infinitely generated modules, II, Math. Proc. Camb. Phil. Soc. 120 (1996), 597-615. MR 97f:20008
  • [BCRi2] D. J. Benson, J. F. Carlson and J. Rickard, Thick subcategories of the stable module category, Fundamenta Mathematicae 153 (1997), 59-80. MR 98g:20021
  • [BCR] D. J. Benson, J. F. Carlson and G. R. Robinson, On the vanishing of group cohomology, J. Algebra 131 (1990), 40-73. MR 91c:20073
  • [C1] J. F. Carlson, Varieties for cohomology with twisted coefficients, Acta Math. Sin. (Engl. Ser.) 15 (1999), 81-92. MR 2000f:20087
  • [C2] J. F. Carlson, The thick subcategory generated by the trivial module, Infinite Length Modules, ed. Henning Krause and Claus Michael Ringel, Trends in Mathematics, Birkhäuser Verlag (2000), 285-296. MR 2001k:20005
  • [CR] J. F. Carlson and G. R. Robinson, Varieties and modules with vanishing cohomology, Math. Proc. Camb. Phil. Soc. 116 (1994), 245-251. MR 95c:20073
  • [CM] D. H. Collingwood and W. M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold, 1993. MR 94j:17001
  • [CNP] J. F. Carlson, D. K. Nakano and K. M. Peters, On the vanishing of extensions of modules over reduced enveloping algebras, Math. Annalen 302 (1995), 541-560. MR 96f:17029
  • [FP1] E. M. Friedlander and B. J. Parshall, Support varieties for restricted Lie algebras, Invent. Math. 86 (1986), 553-562. MR 88f:17018
  • [FP2] E. M. Friedlander and B. J. Parshall, Geometry of $p$-unipotent Lie algebras, J. Algebra 109 (1987), 25-45. MR 89a:17017
  • [FS] E. M. Friedlander and A. A. Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), no. 2, 209-270. MR 98h:14055a
  • [Hum] J. E. Humphreys, Conjugacy Classes in Semisimple Algebraic Groups, American Mathematical Society (Mathematical Surveys and Monographs), 1995. MR 97i:20057
  • [Jan] J. C. Jantzen, Representations of algebraic groups, Academic Press, 1987. MR 89c:20001
  • [NPV] D. K. Nakano, B. J. Parshall and D. C. Vella, Support varieties for algebraic groups, J. Reine Angew. Math. 547 (2002), 15-49. MR 2003b:20063
  • [Qu] D. G. Quillen, The spectrum of an equivariant cohomology ring I, II, Ann. of Math. 94 (1971), 549-572, 573-602. MR 45:7743
  • [SFB1] A. A. Suslin, E. M. Friedlander and C. P. Bendel, Infinitesimal one-parameter subgroups and cohomology, Jour. AMS. 10 (1997), 693-728. MR 98h:14055b
  • [SFB2] A. A. Suslin, E. M. Friedlander and C. P. Bendel, Support varieties for infinitesimal group schemes, Jour. AMS. 10 (1997), 729-759. MR 98h:14055c

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 20G10, 20G05

Retrieve articles in all journals with MSC (2000): 20G10, 20G05


Additional Information

David J. Benson
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: djb@byrd.math.uga.edu

Daniel K. Nakano
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: nakano@math.uga.edu

DOI: https://doi.org/10.1090/S0002-9939-03-06939-9
Received by editor(s): February 20, 2002
Received by editor(s) in revised form: June 20, 2002
Published electronically: March 25, 2003
Additional Notes: The research of the first author was partially supported by NSF grant DMS-9988110
The research of the second author was partially supported by NSF grant DMS-0102225
Communicated by: Stephen D. Smith
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society