Deformation Theory of Algebras and Structures and Applications

1023 Pages·1988·31.83 MB·English

by  Michiel Hazewinkel (auth.)

Deformation Theory of Algebras and Structures
and Applications
NATO ASI Series
Advanced Science Institutes Series
A Series presentmg the results of activities sponsored by the NA TO Science Committee,
which aims at the dissemination of advanced scientific and technological knowledge,
with a view to strengthening links between scientific communities.
The Series IS published by an international board of publishers In conjunction with
the NATO Scientific Affairs Division
A Life Sciences Plenum Publishing Corporation
B Physics London and New York
C Mathematical Kluwer Academic Publishers
and Physical Sciences Dordrecht, Boston and London
D Behavioural and Social Sciences
E Applied Sciences
F Computer and Systems Sciences Springer-Verlag
G Ecological Sciences Berlin, Heidelberg, New York, London,
H Cell Biology Paris and Tokyo
Series C: Mathematical and Physical Sciences -Vol. 247
Deformation Theory of Algebras
and Structures and Applications
edited by
Michiel Hazewinkel
CWI, Amsterdam and
University of Utrecht, The Netherlands
and
Murray Gerstenhaber
University of Pennsylvania,
Philadelphia, PA, U.S.A.
Kluwer Academic Publishers
Dordrecht / Boston / London
Published in cooperation with NATO Scientific Affairs Division
Based on the NATO Advanced Study Institute on
Deformation Theory of Algebras and Structures and Applications
'/I Ciocco', Castelvecchio-Pascoli, Tuscany, Italy
June 1-14. 1986
Library of Congress Cataloging in Publication Data
NATO Advanced Study Institute (1986 : Tuscany, Italy)
Defonnation theory of algebf'as and structures and
applications.
(NATO ASI series. Series C, Mathematical and physical
sciences ; no. 247)
"Based on the NATO Advanced Study Institute held at
11 Ciocco, Castelrecchio-Pascoli, Tuscany, Italy."
Includes index.
1. Homotopy theory--Congresses. 2. Perturbation
(Mathemat ics) --Congresses. 3. Algebra. I. Hazewinkel,
Michiel. II. Gerstenhaber, Murray, 1927-
III. Title. IV. Series.
QA612.7.N38 1986 514' .24 88-27389
ISBN-I 3: 978-94-010-7875-7 e-ISBN-I 3: 978-94-009-3057-5
DOl: 10. I 007/978-94-009-3057-5
Published by Kluwer Academic Publishers,
PO Box 17, 3300 AA Dordrecht, The Netherlands.
Kluwer Academic Publishers incorporates the publishing programmes of
D. Reidel, Martinus Nijhoff, Dr W Junk, and MTP Press.
Sold and distributed in the U.S.A and Canada
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In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
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All Rights Reserved
© 1988 by Kluwer Academic Publishers.
Softcover reprint of the hardcover 1s t edition 1988
No part of the material protected by this copyright notice may be reproduced or utilized
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or by any information storage and retrieval system, without written permission from the
copyright owner.
TABLE OF CONTENTS
Preface vu
M. Hazewinkel, The philosophy of deformations: introductory remarks and a
guide to this volume
Part A. Defonnations of algebras
M. Gerstenhaber & S.D. Schack, Algebraic cohomology and deformation theory II
M. Goze, Perturbations of Lie algebra structures 265
C. Roger, Cohomology of current Lie algebras 357
A. Fialowski, An example of formal deformations of Lie algebras 375
J.M. Ancochea Bermudez, On the rigidity of solvable Lie algebras 403
M. Gerstenhaber & S.D. Schack, Triangular algebras 447
Part B. Perturbations of algebras in functional analysis and operator theory
R Rochberg, Deformation theory for algebras of analytic functions 501
E. Christensen, Close operator algebras 537
K. Jarosz, Perturbations of function algebras 557
B.E. Johnson, Perturbations of multiplication and homomorphisms 565
Part C. Defonnations and moduli in geometry and differential equations, and algebras
D.G. Babbitt & v.s. Varadarajan, Local isoformal deformation theory for meromorphic 583
differential equations near an irregular singularity
R Hermann, Geometric and Lie-theoretic principles in pure and applied deformation theory 701
J. Gasqui & H. Goldschmidt, Complexes of differential operators and symmetric spaces 797
J.F. Pommaret, Deformation theory of geometric and algebraic structures 829
J. Gasqui & H. Goldschmidt, Some rigidity results in the deformation theory of symmetric spaces 839
Part D. Deformations of algebras and mathematical and quantum physics
A. Lichnerowicz, Applications of the deformations of the algebraic structures to geometry 855
and mathematical physics
M. de Wilde & P. Lecomte, Formal deformations of the Poisson Lie algebra of a symplectic 897
manifold and star-products. Existence, equivalence, derivations.
D. Melotte, Invariant deformations of the Poisson Lie algebra of a symplectic 961
manifold and star-products
vi
Part E. Deformations elsewhere
F. Calogero, A remarkable matrix 975
R. Vilela Mendes, Deformation stability of periodic and quasi periodic motion in 981
dissipative systems
List of participants 1015
Index 1017
PREFACE
This volume is a result of a meeting which took place in June 1986 at 'll Ciocco" in Italy entitled
'Deformation theory of algebras and structures and applications'. It appears somewhat later than is
perhaps desirable for a volume resulting from a summer school. In return it contains a good many
results which were not yet available at the time of the meeting. In particular it is now abundantly
clear that the Deformation theory of algebras is indeed central to the whole philosophy of
deformations/perturbations/stability. This is one of the main results of the 254 page paper below
(practically a book in itself) by Gerstenhaber and Shack entitled "Algebraic cohomology and defor
mation theory".
Two of the main philosphical-methodological pillars on which deformation theory rests are the fol
lowing
• (Pure) To study a highly complicated object, it is fruitful to study the ways in which it can arise as
a limit of a family of simpler objects: "the unraveling of complicated structures" .
• (Applied) If a mathematical model is to be applied to the real world there will usually be such
things as coefficients which are imperfectly known. Thus it is important to know how the behaviour of
a model changes as it is perturbed (deformed).
Both these themes also lead to the question of what quantities remain invariant or change continu
ously under deformations. This leads to immediately related matters such as invariants and canonical
forms. For more detailed remarks on all this cf. the introductory article below: "The philosophy of
deformations: introductory remarks and a guide to this volume".
As a result of these and related considerations there have appeared a number of subdisciplines in
various quite separate areas of mathematics (and its applications), which are all concerned with ques
tions of deformations, perturbations, (structural) stability, bifurcations, sensitivity, and which thus all
are instances of the even more general idea that it is wise to study also families of objects instead of
just single objects. An idea, by the way, which has also resulted in major advances at the technical
level in mathematics. Thus there are
Quantum mechanics and relativety theory (classical and semi-classical limits, quantum groups,
quantization)
Algebraic geometry and several complex variables theory (= analytic geometry)
(deformations of singularities, moduli)
Differential geometry (deformations of complex and other structures, moduli, applications to
(overdetermined) (partial) differential equations).
Functional analysis (deformations of operator and C' -algebras, families of operators)
Special functions (q-special functions and q-orthogonal polynomals and their links with
quantum groups)
Algebraic and differential theory (homotopies, isotopies, ... )
Numerical mathematics (the homotopy or continuation method for solving equations and
differential equations)
Global analysis and particle mechanics (completely integrable systems, bifurcations, (structural)
stability)
and quite a few more.
All are related and it is now (some time after the meeting) totally clear that the deformation theory
of algebras and the corresponding cohomology tools are indeed central to the whole topic.
vii
viii
To do full justice to the whole theme of deformation theory would probably require some 10
volumes or so (even without detailed proofs). So some choices had to be made. The main victims are
the large and well-developed theories of deformations of singularities and of moduli in algebraic
geometry (and analytic geometry C several complex variables). However, at the meeting itself, there
were two beautiful sets of lectures on these themes by R.O. Buchweitz and K. Behnke. The somewhat
separate (as far as I can see) theory of deformation of solids, crystals, and mechanical structures
received no attention at all.
Organizing a meeting such as the one which caused this volume takes time, energy, and funds. It is
a pleasure to thank the NATO office for scientific affairs for almost all of the funds, to thank Dr. C.
Sinclair (of that office), the late Dr. M. di Lullo, and Dr. and Mrs. Tilo Kester of various bits of
advice and help, to thank the NSF for providing transport funds for two young promising American
mathematicians, to thank the staff of 'll Ciocco' for providing a marvelous ambience, to thank my
secretary, Nada Mitrovi~ for organizational and typing help, to thank my institute, the CWI in
Amsterdam, for support in various ways, and finally and above all, to thank the participants and lec
turers at this summer institute for all the efforts, energy and enthousiasm which went into it.
In each of the first four parts of the book below the more expository lintroductory papers tend to
occur first.
Amsterdam, May 1988 Michiel Hazewinkel
The philosophy of deformations: introductory remarks
and a guide to this volume
Michiel Hazewinkel
Centre for Mathematics and Computer Science
p.o. Box 4079, 1009 AB Amsterdam, The Netherlands
1. INTRODUCTION. THE IDEA OF DEFORMATIONS
One of the more prominent, specifically modern, and pervasive trends in mathematics has to do with
perturbations and deformations. Instead of studying one particular model, e.g. one differential equa
tion, or one particular algebra of operators, one is as least as interested in families of these things,
and the question of how various properties change as the object under consideration is varied. One
reason of this is no doubt the modem emphasis on the tenuous relation (logically speaking) between a
mathematical model and the phenomena it is designed to deal with. Thus, to paraphrase Arnol'd,
when dealing with models intended to apply to the real world, the .question soon arises of choosing
those properties of the model which are not very sensitive to small changes in the model and which
thus have a chance of representing some properties of the real process.
Intuitively a deformation of a mathematical object is a family of the same kind of objects depend
ing on some parameter(s). Thus for example one could have a family of differential equations
x
= f(x,t,X)
depending on a real parameter A. or for example a family of real three dimensional algebras defined
by
Of course the parameter on which the family under consideration depends need not be one dimen
sional but can be a vector and it need not vary over a vectorspace but can also vary over various sub
sets of a vectorspace or over more general objects such as algebraic schemes. By viewing the object
associated to the parameter(s) value X as 'lying over X' one obtains a 'fibre object' picture
x
~
B
where the fibre over X E B is the object from the family labelled by X. For example in the case of the
family of diff'erential equations the fibre over X E B is the n-dimensional vector space Rn with a flow
M. Hazewinkel and M. Gerstenhaber (eds.),
Deformation Theory ofA lgebras and Siructures and Applications, 1· 7.
© 1988 by Kluwer Academic Publishers.
2
x
given by == f(x,I,'A) and for the 3-dimensional algebra example the fibre over 'A E B is the commu
tative algebra
R[XI/ (X3 -AX).
Thus for the geometric type categories such as topological spaces, schemes, manifolds with singular
ities, ... , a deformation is simply a (surjective) morphism with, however, special stress on aspects and
questions which involve how the fibres, i.e. the inverse images of points b in B, vary with b.
There is no clear cut dividing line between perturbations and deformations, the subject of this book.
Perhaps the difference can be indicated roughly by saying that perturbations consider unstructured
neighborhoods of a given object: all nearby objects are considered on an equal footing. while defor
mation theory is concerned with the (detailed) structure of the set of (isomorphism) classes of objects
of the kind under consideration and how they fit into (smooth) families. Also deformation theory and
its applications are not necessarily concerned only with small neighborhoods. For example one of
Poincare's favourite techniques (the continuation method) consisted of imbedding the problem in a
one-parameter family of problems depending on an auxiliary parameter s and to consider the solubil
ity of the problem as s varies. 1bis also makes it clear that deformation theoretic ideas have very old
roots. Indeed, the idea of "moduli", originally the number of parameters on which a given kind of
structure depends, goes back: to Riemann, as do some other deformation theoretic ideas.
2. DEFOJlMATION THEORETIC QUESTIONS
Let ns consider some typical deformation theoretic questions.
A first one, no doubt, is rigidity. Intuitively, an object Xo is rigid if for every deformation ~ into
which it fits, it is true that XI is isomorphic to Xo. Depending on context this intuitive idea must be
made precise in various ways. For instance in the theory of deformations of algebras one important
way in which to make precise the idea of a deformation of an associative algebra A over a field k is as
follows. A deformation of A is an associative algebra AI over the power series ring k[[t]] such that
Ao == AI®k[I])k is isomorphic to A. Two deformations AI and A'I are equivalent if AI and A'l are
isomorphic as kill)] algebras; and the trivial deformation is A ®kk[[t]]. An algebra is rigid
if every deformation is equivalent to the trivial one.
There are both local and global aspects to rigidity though the local ones have received far more
attention. Thus in a geometric setting of, say, a deformation 'IT: X -+ B of manifolds with a
diffeomorphism on them, parameterized by a topological space B, it may very well be the case that for
every b E B the fibres 'IT-1(b) are isomorphic geometric objects, i.e. isomorphic discrete dynamical
systems in this case, without it being the case that 'IT: X -+ B is isomorphic to the trivial deformation
X 0 X B -+ B. 1bis depends on whether the isomorphisms <Pb: Xb ~ X 0 can be chosen in such a way
that they depend continuously on b E B. The example of (locally trivial) vectorbundles shows that
this need not be the case. The matter is related to the distinction between coarse and fine moduli
spaces in algebraic geometry.
For discrete dynamical systems and differential equations on a manifold M (local) rigidity is some
thing like structural stability, the property which says that nearby systems have "the same" phase por
trait. (Where, of course, there are several meanings which can be given to the phrase "the same".)
Typically, deformations come in various guises, ranging from infinitesimal ones (-= possible defor
mation directions), to formal ones, to true families. For algebras e.g. an infinitesimal deformation A.
is an algebra over k[£] /~; a (one dimensional) formal deformation would be an algebra over the
power series ring k[[t)] and a true family could be an algebra over the polynomials kIt] or, an inter
mediate case which makes sense e.g. when k == C or R, an algebra defined over the rings of conver
gent power series R{ {I}} or C{ {t}}.
Typically infinitesimal deformations are classified by a suitable 2-nd cohomology group like