Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

On unipotent algebraic $ G$-groups and $ 1$-cohomology


Author: David I. Stewart
Journal: Trans. Amer. Math. Soc. 365 (2013), 6343-6365
MSC (2010): Primary 20G07, 20G10; Secondary 18G50
DOI: https://doi.org/10.1090/S0002-9947-2013-05853-9
Published electronically: July 10, 2013
MathSciNet review: 3105754
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider non-abelian $ 1$-cohomology for groups with coefficients in other groups. We prove versions of the `five lemma' arising from this situation. We go on to show that a connected unipotent algebraic group $ Q$ acted on morphically by a connected algebraic group $ G$ admits a filtration with successive quotients having the structure of $ G$-modules. From these results we deduce extensions to results due to Cline, Parshall, Scott and van der Kallen. First, if $ G$ is a connected, reductive algebraic group with Borel subgroup $ B$ and $ Q$ a unipotent algebraic $ G$-group, we show the restriction map $ H^1(G,Q)\to H^1(B,Q)$ is an isomorphism. We also show that this situation admits a notion of rational stability and generic cohomology. We use these results to obtain corollaries about complete reducibility and subgroup structure.


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

  • [BMRT09] M. Bate, B. Martin, G. Roehrle, and R. Tange, Closed orbits and uniform s-instability in geometric invariant theory.
  • [CPSvdK77] E. Cline, B. Parshall, L. Scott, and W. van der Kallen, Rational and generic cohomology, Invent. Math. 39 (1977), no. 2, 143-163. MR 0439856 (55:12737)
  • [DM91] François Digne and Jean Michel, Representations of finite groups of Lie type, London Mathematical Society Student Texts, vol. 21, Cambridge University Press, Cambridge, 1991. MR 1118841 (92g:20063)
  • [Hum75] James E. Humphreys, Linear algebraic groups, Springer-Verlag, New York, 1975, Graduate Texts in Mathematics, No. 21. MR 0396773 (53:633)
  • [Hum95] -, Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs, vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1343976 (97i:20057)
  • [Jan03] Jens Carsten Jantzen, Representations of algebraic groups, second ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057 (2004h:20061)
  • [LS96] Martin W. Liebeck and Gary M. Seitz, Reductive subgroups of exceptional algebraic groups, Mem. Amer. Math. Soc. 121 (1996), no. 580, vi+111. MR 1329942 (96i:20059)
  • [McN10] George J. McNinch, Levi decompositions of a linear algebraic group, Transform. Groups 15 (2010), no. 4, 937-964. MR 2753264 (2012b:20110)
  • [McN13] George J. McNinch, Linearity for vector groups, 2013 (preprint).
  • [Ric82] R. W. Richardson, On orbits of algebraic groups and Lie groups, Bull. Austral. Math. Soc. 25 (1982), no. 1, 1-28. MR 651417 (83i:14041)
  • [Rot09] Joseph J. Rotman, An introduction to homological algebra, second ed., Universitext, Springer, New York, 2009. MR 2455920 (2009i:18011)
  • [Ser88] Jean-Pierre Serre, Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, Springer-Verlag, New York, 1988, Translated from the French. MR 918564 (88i:14041)
  • [Ser94] -, Cohomologie galoisienne, fifth ed., Lecture Notes in Mathematics, vol. 5, Springer-Verlag, Berlin, 1994. MR 1324577 (96b:12010)
  • [Ser98] -, 1998 Moursund lectures at the University of Oregon.
  • [Spr98] T. A. Springer, Linear algebraic groups, second ed., Progress in Mathematics, vol. 9, Birkhäuser Boston Inc., Boston, MA, 1998. MR 1642713 (99h:20075)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 20G07, 20G10, 18G50

Retrieve articles in all journals with MSC (2010): 20G07, 20G10, 18G50


Additional Information

David I. Stewart
Affiliation: Department of Mathematics and Statistics, New College, Oxford, OX1 3BN, United Kingdom

DOI: https://doi.org/10.1090/S0002-9947-2013-05853-9
Received by editor(s): September 26, 2011
Received by editor(s) in revised form: March 17, 2012, April 2, 2012, and April 9, 2012
Published electronically: July 10, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society