Nonabelian cohomology with coefficients in Lie groups
Authors:
Jinpeng An and Zhengdong Wang
Journal:
Trans. Amer. Math. Soc. 360 (2008), 3019-3040
MSC (2000):
Primary 20J06, 22E15, 57S15, 57S20
DOI:
https://doi.org/10.1090/S0002-9947-08-04278-5
Published electronically:
January 25, 2008
MathSciNet review:
2379785
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: In this paper we prove some properties of the nonabelian cohomology of a group
with coefficients in a connected Lie group
. When
is finite, we show that for every
-submodule
of
which is a maximal compact subgroup of
, the canonical map
is bijective. In this case we also show that
is always finite. When
and
is compact, we show that for every maximal torus
of the identity component
of the group of invariants
,
is surjective if and only if the
-action on
is
-semisimple, which is also equivalent to the fact that all fibers of
are finite. When
, we show that
is always surjective, where
is a maximal compact torus of the identity component
of
. When
is cyclic, we also interpret some properties of
in terms of twisted conjugate actions of
.
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Additional Information
Jinpeng An
Affiliation:
School of Mathematical Science, Peking University, Beijing, 100871, People’s Republic of China
Address at time of publication:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email:
anjinpeng@gmail.com
Zhengdong Wang
Affiliation:
School of Mathematical Science, Peking University, Beijing, 100871, People’s Republic of China
Email:
zdwang@pku.edu.cn
DOI:
https://doi.org/10.1090/S0002-9947-08-04278-5
Keywords:
Nonabelian cohomology,
Lie group,
twisted conjugate action.
Received by editor(s):
September 17, 2005
Received by editor(s) in revised form:
March 14, 2006
Published electronically:
January 25, 2008
Additional Notes:
This work was supported by the 973 Project Foundation of China (#TG1999075102).
Article copyright:
© Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.