Available in electronic format
Available in print format		 Transacrions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Nonlinearizable actions of dihedral groups on affine space

Author(s): Kayo Masuda
Journal: Trans. Amer. Math. Soc. 356 (2004), 3545-3556.
MSC (2000): Primary 14R20; Secondary 14L30, 14D20
Posted: December 15, 2003
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: Let $G$ be a reductive, non-abelian, algebraic group defined over $\mathbb{C} $. We investigate algebraic $G$-actions on the total spaces of non-trivial algebraic $G$-vector bundles over $G$-modules with great interest in the case that $G$ is a dihedral group. We construct a map classifying such actions of a dihedral group in some cases and describe the spaces of those non-linearizable actions in some examples.

References:

1.
H. Bass, W. Haboush, Some equivariant K-theory of affine algebraic group actions, Comm. Algebra 15 (1987), 181-217. MR 88g:14013

2.
A. Bialynicki-Birula, Remarks on the action of an algebraic torus on $k^n$, I and II, Bull. Acad. Pol. Sci. 14 (1966), 177-181, and 15 (1967), 123-125.

3.
G. Freudenburg, L. Moser-Jauslin, A nonlinearizable action of $S_3$ on $\mathbb{C} ^4$, Ann. Inst. Fourier, Grenoble 52 (2002), 133-143. MR 2002j:14068

4.
S. Kaliman, M. Koras, L. Makar-Limanov, P. Russell, $\mathbb{C} ^*$-actions on $\mathbb{C} ^3$ are linearizable, Electronic Research Announcements of the AMS 3 (1997), 63-71. MR 98i:14046

5.
F. Knop, Nichtlinearisierbare Operationen halbeinfacher Gruppen auf affinen Räumen, Invent. Math. 105 (1991), 217-220. MR 92c:14046

6.
H. Kraft, $G$-vector bundles and the Linearization Problem, in Group Actions and Invariant Theory, CMS Conference Proceedings, vol. 10, Amer. Math. Soc., Providence, RI, 1989, pp. 111-123. MR 90j:14062

7.
H. Kraft, G.W. Schwarz, Reductive group actions with one-dimensional quotient, Publ. Math. IHES 76 (1992), 1-97. MR 94e:14065

8.
K. Masuda, Moduli of equivariant algebraic vector bundles over affine cones with one dimensional quotient, Osaka J. Math. 32 (1995), 1065-1085. MR 97k:14049

9.
K. Masuda, Moduli of equivariant algebraic vector bundles over a product of affine varieties, Duke Math. J. 88 (1997), 181-199. MR 99b:14011

10.
K. Masuda, Moduli of algebraic $SL_3$-vector bundles over adjoint representaion, Osaka J. Math. 38 (2001), 501-506. MR 2002g:14069

11.
M. Masuda, L. Moser-Jauslin, T. Petrie, Equivariant algebraic vector bundles over representations of reductive groups: Applications, Proc. Natl. Acad. Sci. 88 (1991), 9065-9066. MR 92j:14059b

12.
M. Masuda, L. Moser-Jauslin, T. Petrie, The equivariant Serre problem for abelian groups, Topology 35 (1996), 329-334. MR 97a:20074

13.
M. Masuda, L. Moser-Jauslin, T. Petrie, Equivariant algebraic vector bundles over cones with smooth one dimensional quotient, J. Math. Soc. Japan 50 (1998), 379-414. MR 99k:14079

14.
M. Masuda, T. Nagase, Equivariant algebraic vector bundles over adjoint representations, Osaka J. Math. 32 (1995), 701-708. MR 96k:14014

15.
M. Masuda, T. Petrie, Equivariant algebraic vector bundles over representations of reductive groups: Theory, Proc. Nat. Acad. Sci. 88 (1991), 9061-9064. MR 92j:14059a

16.
M. Masuda, T. Petrie, Stably trivial equivariant algebraic vector bundles, J. Amer. Math. Soc. 8 (1995), 687-714. MR 95j:14067

17.
M. Masuda, T. Petrie, Algebraic families of $O(2)$-actions on affine space $\mathbb{C} ^4$, Proc. of Symp. in Pure Math., vol. 56, Amer. Math. Soc., Providence, RI, 1994, pp. 347-354. MR 95f:14092

18.
K. Mederer, Moduli of $G$-equivariant vector bundles, Ph.D. thesis, Brandeis University (1995).

19.
D. Quillen, Projective modules over polynomial rings, Invent. Math., 36 (1976), 167-171. MR 55:337

20.
G.W. Schwarz, Exotic algebraic group actions, C. R. Acad. Sci. Paris 309 (1989), 89-94. MR 91b:14066

21.
A. Suslin, Projective modules over a polynomial ring, Soviet Math. Doklady 17 (1976), 1160-1164.

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14R20, 14L30, 14D20

Retrieve articles in all Journals with MSC (2000): 14R20, 14L30, 14D20

Additional Information:

Kayo Masuda
Affiliation: Mathematical Science II, Himeji Institute of Technology, 2167 Shosha, Himeji 671-2201, Japan
Email: kayo@sci.himeji-tech.ac.jp

DOI: 10.1090/S0002-9947-03-03405-6
PII: S 0002-9947(03)03405-6
Keywords: Algebraic group action, linearization problem
Received by editor(s): April 3, 2003
Posted: December 15, 2003
Additional Notes: Supported by Grant-in-Aid for Young Scientists, The Ministry of Education, Culture, Sports, Science and Technology, Japan
Copyright of article: Copyright 2003, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google