ASSOCIATED LIE ALGEBRAS AND UNIT GROUPS OF GROUP ALGEBRAS
MEENA SAHAI
A thesis submitted to the
Indian Institute of Technology, Delhi for the award of the degree of
DOCTOR OF PHILOSOPHY
ouTIIN
DEPARTMENT OF MATHEMATICS
INDIAN INSTITUTE OF TECHNOLOGY
NEW DELHI-110016 APRIL 1993
CERTIFICATE
This is to certify that the thesis entitled, "ASSOCIATED LIE ALGEBRAS AND UNIT GROUPS OF GROUP ALGEBRAS", which is being submitted by Ms Meena Sahai for the award of the degree, DOCTOR OF PHILOSOPHY, to the Indian Institute of Technology, Delhi, is a bonafide record of research work done under my guidance and supervision.
The thesis has reached the stage of fulfilment of the requirements and regulations related to the degree. The results obtained in this thesis have not been submitted to any other Institute or University for the award of any degree or diploma.
(J.B.Srivastava) Professor,
Department of Mathematics,
Indian Institute of Technology, New Delhi - 110 016.
TO MY PARENTS
ACKNOWLEDGEMENTS
I wish to express my profound gratitude to my teacher, Professor J.B. Srivastava, for introducing me to this subject and for sharing some of his knowledge with me. I have benefited from his influence; mathematically and personally.
I gratefully acknowledge the support I received from the Department of Mathematics, Indian Institute of Technology, Delhi.
I thank all my friends and colleagues for being very nice to me throughout my stay. Particularly, I am indebted to Mr. Kalpdrum Passi and Ms Meera Krishna for the help that they extended to me.
This thesis is dedicated to my parents for the constant support and encouragement that I received from them.
Finally, I would like to mention, my husband, Dr. Vivek Sahai without whose tremendous cooperation and inspiration it would not have been possible for me to complete this work.
Meerta. Sakai
CONTENTS
NOTATIONS 1
:HAPTER I
INTRODUCTION 4
1. Associated Lie Ring and
Unit Group of an Associative Ring 5
2. Group Algebras 12
3. Main Problems 20
AFTER II
ASSOCIATED LIE ALGEBRAS OF GROUP ALGEBRAS 24 1. Lie Centrally Metabelian Group Algebras 26 2. Lie Solvable Group Algebras of Derived Length Three 40 3. Group Algebras KG with T3(81(L(KG))) = (0) 55
\PTER III
LIE PROPERTIES AND UNIT GROUPS 74 1. Group Algebras with Centrally Metabelian Unit Groups 75
2. Unit Group of a Ring 96
3. Subgroups Associated with the Derived Chain 100 PTER IV
JACOBSON RADICAL AND UNIT GROUPS 104 1. Unit Group of Group Algebras 105
2. p-Solvable Groups 113
3. Some Examples 116
CHAPTER V
REPRESENTATION THEORY 119
1. Representations of Bounded Degree 120
2. Some Finite Groups 122
3. Construction of Units 126
BIBLIOGRAPHY 128
