Low Order Cohomology and Applications
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877 Joachim Erven Bernd-Jurqen Falkowski
Low Order Cohomology and Applications
Springer-Verlag Berlin Heidelberg New York 1981
Authors
Joachim Erven Siemens AG/Forschungslaboratorien Otto-Hahn-Ring 6, 8000 MUnchen 83 Federal Republic of Germany Bernd-JUrgen Falkowski Hochschule der Bundeswehr MUnchen, FB Informatik Werner-Heisenberg-Weg 39, 8014 Neubiberg Federal Republic of Germany
AMS Subject Classifications (1980): 22-XX
ISBN 3-540-10864-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-10864-5 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1981 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
CONTENTS
v
Introduction
I.
Continuous Cohomology of Lie Groups and Lie Algebras 2.
Basic Definitions Some Applications of
3.
An Application of
1.
II.
IV.
1
5
2
9
Continuous Tensor Products, Infinitely Divisible and Factorizable Representations
11
1. 2.
Continuous Tensor Products (CTPs) Infinitely Divisible Projective Representations and First Order Cocycles Necessary and Sufficient Conditions for the Existence of a CTP of Projective Representations
11
4. 5.
CTPs of Representations of C:OR,G} Factorizable Projective Representations of Current Groups and Fock Space
18
6. 7.
Coboundaries and their Associated Representations Factorizable Representations and CTPs
26 30
First Order Cohomology Groups for Certain Semi-Direct Products
32
1.
The General Theory
32
2.
The Cohomology of the Euclidean Motion Groups
38
3.
The Cohomology of the First Compact Lie-Groups
40
4.
The First
3.
III.
H
1 H
Leibniz-Extension of
Leibniz-Extension of
First Order Cohomology for
SL (2;
JR)
SL(2,R)
and
Preliminaries
2.
The Construction of the Principal Series for
16
19
42
SL (2; C )
1.
14
SU(1,1)
48 48
51
IV
V.
3.
Necessary Conditions for the Unitarity of Induced Representations of SU(1,1)
4.
The Complementary and the Discrete Series of
5.
The First Order Cocycles of
SU(1,1)
66
6.
The First Order Cocycles of
SL(2,C)
74
Further Results on Semi-Simple Lie Groups
78
1.
Kazdan's Result
78
2.
Spherical Functions
84
3.
The Connection between the Cohomology of the Lie Algebra and Lie Group with Applications to SU(ni1) and SO(ni1)
VI.
"Genuine" Infinitely Divisible Representations General Definitions
2.
SO (n) @ lR
Divisible Positive Functions for , n 3
54
SU(1,1) 61
100 104 104 105
Infinitely Divisible Positive Functions on the First Leibniz-Extensions of Certain Compact Lie Groups
108
4.
Infinitely Divisible Positive Functions on the First Le
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