Table Of ContentThe Sophus Lie Memorial Conference
OSLO 1992 •PROCEEDINGS
Edited by
OLAV ARNFINN LAUDAL and BJ0RN JAHREN
Scandinavian University Press, Oslo
© Universitetsforlaget AS 1994
ISBN 82-00-21646-2
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Contents
INTRODUCTION VII
MARIUS SOPHUS LIE XI
OPENING ADDRESSES
Sigurdur Helgason: Sophus Lie, the mathematician 3
Thomas Hawkins: The Birth of Lie's Theory of Groups 23
LECTURE SERIES
N.H. Ibragimov: Lie group analysis of differential equations 53
R.S. Palais: Symmetry, criticality, and condition C 77
B. Totaro: A sketch of Lie's work on differential invariant theory 99
RESEARCH PAPERS
A. Bialynicki-Birula and]. Swiecicka: A combinatorial approach
to geometric invarl.ant theory 115
S. Chem: Sophus Lie and differential geometry 129
Wu-Yi Hsiang and E. Straume: A brief introduction to the
topology of compact Lie transformation groups 139
A.A. Kirillov: Combinatorics of coadjoint orbits 153
F. Kirwan: Geometric invariant theory and the Atiyah-Jones conjecture 161
B. Komrakov: Primitive actions and the Sophus Lie problem 187
V. Lychagin: Differential Geometry via Differential Equations 2 71
A.L. Onishchik: Flag supermanifolds, their automorphisms and deformations 289 -
V. P. Palamodov: Lie algebras and deformation theory 303
V. Popov: Sections in invariant theory 315
T.A. Springer: A combinatorial result on K-orbits on a flag manifold 363
PROGRAM 373
LIST OF PARTICIPANTS 377
BIBLIOGRAPHY 37 9
v
Introduction
The Sophus Lie Memorial Conference, in commemoration of the150th anniversary of
the birth of Sophus Lie, was organized by The University of Oslo and the Norwegian
Academy of Science and Letters in Oslo, from 17th to 21st August 1992, as part of the
arrangements during The Sophus Lie Memorial Week.
The objective of the Memorial Week was twofold; to contribute to a better public
understanding, particularly among Norwegians, of the life and work of Sophus Lie, and
to offer to the Scandinavian mathematical community an overview of the present research
activity in Lie theory, particularly that which is not directly related to the standard
mathematical university curriculums.
The first objective was, we believe, partly attained through an exhibition illustrating the
life and work of Sophus Lie, commissioned by the University of Oslo, and through the
focusing of the national media on the legacy of Sophus Lie that resulted from the exhibition.
The purpose of the Sophus Lie Memorial Conference was to meet the second objective.
To that end, the conference was divided into two sections: one summer school section,
with 4 lecture series, covering: invariant theory, the Erlangen Program revisited, Symmetry
groups and Lie group analysis of differential equations, and a research section, where some
of the invited participants presented their own work.
These Proceedings contain the reworked texts of three of the lecture series, and all but
four of the research papers presented in the research section. In addition to the two
opening addresses, of the &hibition by Professor Sigurdur Helgason and of the Conference
by Professor Thomas Hawkins, we have included a text, written by Elin Strnm and
illustrated by original photos, where Sophus Lie speaks to us through his own letters.
The Conference took place in the conference halls of the Academy of Science and Letters,
and was attended by approximately one hundred pure and applied mathematicians and
physicists, from 15 nations. Through the lectures and the sometimes vivid discussions
following the presentations of the many interesting research papers, the organizers feel
VII
that part of the objective we had in mind was attained. The excellent social climate, at
the conference, as well as during the excursion to Nordfjordeid, the birthplace of Sophus
Lie, strengthened this feeling.
The Organizing Committee would like to thank the Norwegian Academy of Science and
Letters for hosting the Sophus Lie Memorial Conference, the Norwegian Research
Council for Sciences and the Humanities for the initial grant, and the Institute of
Mathematics at the University of Oslo for financial and organizational support.
Finally, we extend our gratitude to the participants for their contribution to the conference,
to these Proceedings, and to the sesquicentennial commemoration of Sophus Lie.
On behalf of the Organizing Committee,
Olav Arnfinn Laudal BjorriJahren
VIII
Marius Sophus Lie
ELIN STR0M
The author wishes to thank professor Eldar Straume, Troms0, and Bernd Fritzsche, Ph.D., Leipzig.
Translated from Norwegian by Mary Bjrerum. As for sources, see Bibliography p. 379. Photographs: Page XI: Oslo
City Museum (detail); page XVI: University of Oslo Library; page XX!: kindly lent us by Dr. B. Fritzsche. The
remaining photos in this biographical essay are taken from Sophus Lie's private album.
M A R I u s s 0 p H u s L I E
/<f4,i- /<f~e9
The early days
Marius Sophus Lie was born on 17 December 1.842 in the vicarage at Eid, in Nordfjord,
where his father was rector of the parish. In 1851 the family moved from Eid to Moss, a
small town on the Oslofjord .. He completed his secondary education as a student at Nissens
Latin-og Realskole in Kristiania (Oslo), in 1859. Between 1859 and 1865,
he studied science at Det kg!. Frederiks Universitet in Kristiania. The teachers
included O.I. Broch, Vilhelm Bjerknes, and Ludvig Sylow, who was one
of the first university professors to lecture on the theory ofs ubstitutions, the
group theory created by Abel and Galois.
In 1869 Sophus Lie printed, at his own and his friend Ernst Motzfeldt's THE FIRST PHOTO OF
SOPHUS LIE, TAKEN IN
expense, his first scientific paper: Reprdsentation der Imaginiiren der Plangeainetrie.
1864. HE IS 21
YEARS OLD AND HAS JUST
He submitted the paper to the Society of Science and Letters in Kristiania,
BECOME A SECOND
requesting that it be printed in the society's series of reports. Though it LIEUTENANT
seems that no one in Kristiania understood the real mathematical content
of the paper, a sufficient number understood that Sophus Lie was a great talent. The
paper was therefore accepted, and more importantly, the Academic Collegium of the
University gave Sophus Lie a travel grant for studies in Germany and France.
XI
c
/cf e9 - /cf 7 /
The mathematician
At the tiine, Berlin was the centre of the mathematical world, dominated by Weierstrass,
Kummer and Kronecker. Sophus Lie travelled to Berlin in the autumn of 1869, giving
his first impressions of the city in a letter to his friend Ernst Motzfeldt: "I have started to
attend lectures. I have no difficulties at all with the langu.age. On the other hand, I am no master
ofc onversation. Strangely enough, several oft he things in which I am particularly interested are
not included in the lectures. The advantage oft his is that I have a lot oft ime for my own work.
However, my stay here is ofg reat benefit to me because oft he conversations with my colleagu.es.
In Germany the professors in Berlin consitute a scientific aristocracy that believes itself to be
possessed ofa ll wisdom. So far I have only talked about mathematics on one single occasion with
each ofB erlin's three leadingfigu.res. One feels as ifo ne is standing before a King in the kingdom
ofs cience. No matter what one thinks oft he Prussians, they can think - there is no doubt about
that."
In Berlin, Sophus Lie met the mathematician Felix Klein, who was seven years younger
than he. Klein had already, at the age of 20, published Plucker's geometrical works. To
Ernst Motzfeldt Sophus Lie wrote: "Klein and I met each other a week ago and have already
become good friends. We were acquainted with each other's works, and there are several points
ofc ontact between them. I have also made a few other good scientific acquaintances. "
• KUMMER'S SEMINAR
Sophus Lie and Felix Klein attended Kummer;s geometry seminar. To Motzfeldt,
December 2, 1869: "Today I had a triumph which I am sure you will be interested to hear
about. Professor Kummer suggested that we test our powers on a discussion ofa ll line congruences
oft he 3rd degree. Fortunately, a couple ofm onths ago, I had already solved a problem which was
in a way a special example oft he above, but was nevertheless much more general.
Kummer, who knew that I had been working on line congruences, asked me to give a lecture. So
yesterday, my friend Klein presented my work in the seminar. After Kummer, in a very flattering
manner, had expressed his thanks for the lecture, he suggested that I should undertake a more
thorough study ofa specific congruence oft he 3rd degree which he himself had been working on
recently. He asked whether an approach based on the same or similar principles as those I had
described would be easy. His method, on the other hand, had presented major problems which he
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