The Geometry of Jordan and Lie Structures
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Wolfgang Bertram
The Geometry of Jordan and Lie Structures
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Author Wolfgang Bertram Institut Elie Cartan Université Henri Poincaré (Nancy 1) Faculté des Sciences B.P. 239 54506 Vandoeuvre-lès-Nancy, France e-mail: [email protected]
Cover illustration from R. Courant, H. Robbins: Was ist Mathematik? 3. Aufl., Springer, Berlin Heidelberg New York (1973)
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Mathematics Subject Classification (2000): 17C36, 17C30, 17C37, 53B35, 53C15, 53C35, 53C55, 22E15, 53A40, 53B05 ISSN 0075-8434 ISBN 3-540-41426-6 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH © Springer-Verlag Berlin Heidelberg 2000 Printed in Germany Typesetting: Camera-ready TEX output by the author SPIN: 10759910 41/3142-543210 - Printed on acid-free paper
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t h e o r y o n a n s y s t e m a t i c s t u d y t h e r m o n i c a n a l y s i s . I n f a c t , t h e a p p l i c a t i o n s o f e t h e h a r m o n i c a n a l y s i s o n s y m m e t r i c c o n e s ( c f . e m o n i g i n o f t h e a u t h o r ' s w o r k i n t h i s a r e a . T h e n r d a n u d y o f m a n y c a u s a l s y m m e t r i c s p a c e s ( s e e c t i o n X I . 3 ) , a n d s o o n i t b e c a m e c l e a r t h a t " g e n e r i c a l l y " a l l s y m m e t r i c s p a c e s v e a s i g n i f i c a n t r e l a t i o n t o J o r d a n t h e o r y . S i n c e J o r d a n t h e o r y d o e s n o t ( y e t ) l o n g t o t h e s t a n d a r d t o o l s i n h a r m o n i c a n a l y s i s , t h e p r e s e n t t e x t is d e s i g n e d p r o v i d e a s e l f - c o n t a i n e d i n t r o d u c t i o n t o J o r d a n t h e o r y f o r r e a d e r s h a v i n g b a s i c k n o w l e d g e o n L i e g r o u p s a n d s y m m e t r i c s p a c e s . O u r p o i n t o f v i e w m e g e o m e t r i c : t h r o u g h o u t w e i n t r o d u c e first t h e r e l e
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