Orthogonal Decompositions and Integral Lattices
Alexei Kostrikin, Pham Huu TiepThe present book is the result of investigations carried out by algebraists at Moscow
University over the last fifteen years. It is written for mathematicians interested in
Lie algebras and groups, finite groups, Euclidean integral lattices, combinatorics
and finite geometries. The authors have used material available to all, and have
attempted to widen as far as possible the range of familiar ideas, thus making the
object of Euclidean lattices in complex simple Lie algebras even more attractive.
It is worth mentioning that orthogonal decompositions of Lie algebras have not
been investigated before now, for purely accidental reasons. However, automorphism groups of the integral lattices associated with them could not be investigated
properly until finite group theory had reached an appropriate stage of development.
No special theoretical preparation is required for reading and understanding
the first two chapters of the book, though the material of these chapters enables
the reader to form rather a clear notion of the subject-matter. The subsequent
chapters are intended for the reader who is familiar with the basics of the theories
of Lie algebras, Lie groups and finite groups, and is more-or-less acquainted with
integral Euclidean lattices. As a rule, undergraduates receive such information
from special courses delivered at the Faculty of Mathematics and Mechanics of
Moscow University. In any case, it is essential to have in mind a small collection
of classic books on the above-mentioned themes: [SSL], [SAG], [Gor 1], [Ser 1]
and [CoS 7],
Our teaching experience shows that the material of Part I and some chapters
from Part II can be used as a basis for special courses on Lie algebras and finite
groups. The material on integral lattices enrich the lecture course to a considerable
extent. The interest of the audience and the readers grows, due to the great number
of concrete unsolved problems on orthogonal decompositions and lattice geometry.
In connection with integral lattice theory, we mention here a comprehensive book
[CoS 7] and an interesting survey [Pie 3], where one can find much information
contiguous with our book.
It is a pleasure to acknowledge the contributions of the many people from whose
insights, assistance and encouragement we have profited greatly. First of all we
wish to express our thanks to Igor Kostrikin and Victor Ufnarovskii, who were
among the first to investigate orthogonal decompositions, and whose enthusiasm
has promoted the popularisation of this new research area. Their impetus was kept up by the concerted efforts of K. S. Abdukhalikov, A. I. Bondal, V. P. Burichenko
and D. N. Ivanov, to whom the authors are sincerely grateful. We are particularly
indebted to D. N. Ivanov and K. S. Abdukhalikov: Chapter 7 is based on the results
of D. N. Ivanov's C. Sc. Thesis, and the first five paragraphs of Chapter 10 are
taken from K. S. Abdukhalikov's C. Sc. Thesis. Some brilliant ideas came from
A. I. Bondal and V. P. Burichenko. We would like to thank Α. V. Alekseevskii,
Α. V. Borovik, S. V. Shpektorov, K. Tchakerian, A. D. Tchanyshev and Β. B.
Venkov, who have made contributions to progress in this area of mathematics. A
significant part of the book has drawn upon the Doctor of Sciences Thesis of the
second author. We are indebted to our colleagues W. Hesselink, P. E. Smith and
J. G. Thompson for a number of valuable ideas mentioned in the book.
Our sincere thanks go to Walter de Gruyter & Co, and especially to Prof.
Otto H. Kegel, for the opportunity of publishing our book. We would also like
to express our gratitude to Prof. James Wiegold for his efforts in improving the
English. We are grateful to Professor W. M. Kantor for many valuable comments.
The authors wish to state that the writing of the book and its publication were
greatly promoted by the creative atmosphere in the Faculty of Mathematics and
Mechanics of Moscow University.
The present work is partially supported by the Russian Federation Science Committee's Foundation Grant # 2.11.1.2 and the Russian Foundation of Fundamental
Investigations Grant # 93-011-1543. The final preparation of this book was completed when the second author stayed in Germany as an Alexander von Humboldt
Fellow. He wishes to express his sincere gratitude to the Alexander von Humboldt
Foundation and to Prof. Dr. G. O. Michler for their generous hospitality and
support.
A. I. Kostrikin
Pham Huu Tiep