Transformation Groups and Lie Algebras

Description

This book is based on the extensive experience of teaching for mathematics, physics and engineering students in Russia, USA, South Africa and Sweden. The author provides students and teachers with an easy to follow textbook spanning a variety of topics. The methods of local Lie groups discussed in the book provide universal and effective method for solving nonlinear differential equations analytically. Introduction to approximate transformation groups also contained in the book helps to develop skills in constructing approximate solutions for differential equations with a small parameter.

replacement of his Introduction to Modern Group Analysis , Tau, Ufa, 2000, ref. #15 of his Practical Course , of which Transformation Groups and Lie Algebras is the sequel (focusing primarily on Lie symmetries)

Includes solutions to problems, too!

22-01 (11R32 17B66 22E05 22E60 34A26 34C14 58J70)

Inspired by Abel's and Galois' celebrated application of group theory to the solvability of algebraic equations, Sophus Lie formulated the theory of Lie transformation groups in order to solve differential equations. While not the proper generalization of Galois theory—this is the Picard-Vessiot theory, a.k.a. differential Galois theory [I. Kaplansky, An introduction to differential algebra, second edition, Hermann, Paris, 1976; MR0460303; J. F. Ritt, Differential algebra, Dover, New York, 1966; MR0201431]—Lie symmetry analysis of differential equations has proved to be an incredibly fertile field for present day research. The modern era began with Garrett Birkhoff's book on hydrodynamics [Hydrodynamics: a study in logic, fact and similitude, revised edition, Princeton Univ. Press, Princeton, NJ, 1960; MR0122193], which was the first to emphasize the applicability of Lie's methods outside of quantum mechanics, and then these methods were forged into a powerful tool of modern applied mathematics by L. V. Ovsyannikov and his Soviet school [L. V. Ovsyannikov, Group analysis of differential equations, English translation, Academic Press, New York, 1982; MR0668703]. Nail Ibragimov has assumed the role of Ovsyannikov's principal disciple, carrying on the tradition in this active area of contemporary research.
The text under review is the latest in a series of books, at various levels, that Ibragimov has written on the subject, and is designed to be a basic introduction to the area for the beginning student. The book is divided into two parts, with each chapter therein including a number of exercises without solutions. Furthermore, each part concludes with a multi-question "assignment'' that is followed by complete solutions.
Part I provides an elementary introduction to some of the basic aspects of symmetry analysis, directed towards ordinary differential equations. Following a short motivational chapter, Chapter 2 describes the basics of local transformation groups on Euclidean space, their infinitesimal generators, and Lie's method for calculating their invariants through solving the linear partial differential equations governing their infinitesimal invariance. (Other practical approaches to the calculation of invariants, e.g. moving frames [M. E. Fels and P. J. Olver, Acta Appl. Math. 55 (1999), no. 2, 127–208; MR1681815; E. L. Mansfield, A practical guide to the invariant calculus, Cambridge Monogr. Appl. Comput. Math., 26, Cambridge Univ. Press, Cambridge, 2010; MR2656212], are not mentioned.) The chapter ends with a proof of the standard infinitesimal criterion for invariance of a system of equations.
Chapter 3 introduces Lie's infinitesimal calculus for determining the symmetry group of a differential equation. The idea is to view the differential equation as a submanifold of the space coordinatized by the derivative coordinates appearing in the equation; in modern terminology (not used here), these are known as jet spaces or bundles [P. J. Olver, Applications of Lie groups to differential equations, second edition, Grad. Texts in Math., 107, Springer, New York, 1993; MR1240056; Equivalence, invariants, and symmetry, Cambridge Univ. Press, Cambridge, 1995; MR1337276]. One prolongs the infinitesimal generators to jet space using Lie's prolongation formula, and applies the infinitesimal invariance criterion established in Chapter 2. The fact that one can systematically calculate the infinitesimal symmetry group of (most) differential equations is the reason Lie's symmetry analysis has become so useful in a wide range of applications. Indeed, there are a number of software packages available, in Mathematica, Maple, Reduce, etc., that will automatically perform the required computations, although Ibragimov does not reference such widely available tools, possibly leaving the reader with the mistaken impression it has to be "done by hand''. More precisely, the infinitesimal calculation determines the connected component of the identity of the full symmetry group; additional discrete symmetries, also of interest, can often be found by applying methods developed by P. E. Hydon [Symmetry methods for differential equations, Cambridge Texts Appl. Math., Cambridge Univ. Press, Cambridge, 2000; MR1741548], again not mentioned in the book.
Chapter 4 is concerned with the basic theory of Lie algebras of vector fields, viewed as infinitesimal generators of multi-parameter transformation groups. (Despite the title of the chapter, "Lie algebras of operators'', there is no discussion of Lie algebras of more general differential operators, of importance in quantum mechanics [A. González-López, N. Kamran and P. J. Olver, J. Phys. A 24 (1991), no. 17, 3995–4008; MR1126643].) Some structure theory is presented, leading to the definition of a solvable Lie algebra, although there is no mention of semi-simple Lie algebras and hence no development of or references to the general structure theory. Classifications of low-dimensional Lie algebras in R2 and R3 are presented. However, Lie's complete classification of all possible (nonsingular) finite-dimensional (and infinite-dimensional) Lie algebras of vector fields in one and two dimensions, both real and complex [S. Lie, in Gesammelte Abhandlungen. 5. Bd.: Abhandlungen über die Theorie der Transformationsgruppen, 767–772, Teubner, Leipzig, 1924; see JFM 50.0002.01; P. J. Olver, op. cit.; MR1337276], does not appear. Furthermore, Ibragimov's tables of "non-similar'' two- and three-dimensional Lie algebras contain cases that are, in fact, locally isomorphic to the same entry in Lie's complete classification table.
The final chapter of the first part attempts to use Lie's theory of transformation groups as the basis for Galois theory. Oddly, because its primary focus is on differential equations, this chapter only treats the Galois theory of polynomial equations and does not get into differential Galois theory. Here things go badly wrong. Ibragimov proposes a misleading and, in general, incorrect definition of the Galois group of a polynomial equation as the subgroup of the symmetric group induced by the action of its Lie symmetry group, consisting of those projective (linear fractional) transformations that preserve the submanifold defined by the equation, on the roots of the equation. In simple examples such as those presented in the text, this works. However, in general there is very little relation between the Galois group and the group of projective symmetries of the roots. A similar situation arises in differential Galois theory: while they both can be viewed as groups of symmetries taking solutions to solutions, there is, in general, no evident relation between the Lie symmetry group of a differential equation and its differential Galois group; see [W. R. Oudshoorn and M. van der Put, Math. Comp. 71 (2002), no. 237, 349–361; MR1863006] for further discussion of this point.
Strangely, the first part of the text includes no applications. For example, there is no discussion of how one uses symmetries to integrate ordinary differential equations; no analysis of the Noether correspondence between symmetries and conservation laws; and no development of the methods of calculating group-invariant solutions of partial differential equations. Lie's motivation for the definition of a solvable Lie algebra was that it can be used to solve ordinary differential equations, but this key application is not discussed. So, while the calculation of symmetry groups is, perhaps, of some intrinsic interest, the reader may well be left wondering what the point is, and what are they good for.
Part II of the book is devoted to the author's theory of approximate transformation groups and approximate symmetries. (Competing approaches and further developments are not mentioned.) Here one includes a small parameter ε, regarded as a perturbation, and imposes the symmetry conditions, exponentiation of infinitesimal generators, commutator relations, etc., typically, to first order in ε. This part does contain some genuine applications to partial differential equations, including integration of equations with a small parameter, determination of approximately invariant solutions, and construction of approximate conservation laws, making the omission of similar applications from the first part, which could be used to motivate their approximate counterparts, all the more strange.
The book tends to adhere to nineteenth century notation, terminology, and standards of rigor, at times omitting hypotheses required for the rigorous validity of results. A case in point is Theorem 3.2 on infinitesimal invariance of differential equations, which does not explicitly include the conditions of maximal rank and local solvability, which are required in order that the group preserve the solution space [P. J. Olver, op. cit.; MR1240056]. Although the former is implied by the earlier treatment of infinitesimal invariance of algebraic equations, the latter, arising in the study of overdetermined systems of partial differential equations or Lewy-type counterexamples, is not.
I was struck by the complete absence of the term "Lie group'' in the text and index, which only considers them in the context of their (local) action on some open subset of Euclidean space. This is very much in the original spirit of Lie, for whom the abstract concept of a Lie group did not exist in the modern sense of being a smooth manifold equipped with a smooth group multiplication and inversion, which, I believe, first appears in Élie Cartan's 1930 text [La théorie des groupes finis et continus et l'analysis situs, Mem. Sci. Math., 42, Gauthier-Villars, Paris, 1930; JFM 56.0370.08]. Lie thus was able to deal with both finite-dimensional Lie groups and infinite-dimensional Lie pseudo-groups on an equal footing. (Despite arising as symmetry groups of many partial differential equations of importance in a broad range of applications, the latter do not appear in this text.) Surprisingly, to this day there is no well-accepted abstract concept of an infinite-dimensional Lie pseudo-group, which only arises in the context of its action on a space [P. J. Olver, in XVIII International Fall Workshop on Geometry and Physics, 35–63, AIP Conf. Proc., 1260, Amer. Inst. Phys., Melville, NY, 2010; MR3024544; J.-F. Pommaret, Systems of partial differential equations and Lie pseudogroups, Mathematics and its Applications, 14, Gordon & Breach, New York, 1978; MR0517402].
Finally, the bibliography is completely inadequate, citing only nineteenth and early twentieth century texts by Lie, Bianchi, and Campbell, plus Ovsyannikov's 1962 Russian text (which was later translated by Bluman but never published in the West), and, with but two exceptions, papers written by the author. A novice reader will be ill prepared to make contact with and benefit from the extensive modern literature and achievements—Russian, Western, and beyond—that have proliferated over the last 60 years.
In summary, Ibragimov's book is an elementary, but ultimately inadequate introduction to what is a genuinely interesting, highly active, and widely applicable subject. While, for many of the basic applications, the author's approach suffices, it paints a very restrictive picture of the true extent of modern Lie theory, even in the context of differential equations. The beginning student would do much better to consult other basic introductory texts, e.g. [B. J. Cantwell, Introduction to symmetry analysis, Cambridge Texts Appl. Math., Cambridge Univ. Press, Cambridge, 2002; MR1929892; P. E. Hydon, op. cit.; MR1741548]. Readers wishing for more in-depth treatments, including overviews of many of the fascinating modern developments, can also profitably consult [G. W. Bluman, A. F. Cheviakov and S. C. Anco, Applications of symmetry methods to partial differential equations, Appl. Math. Sci., 168, Springer, New York, 2010; MR2561770; A. V. Bocharov et al., Symmetries and conservation laws for differential equations of mathematical physics, English translation, Transl. Math. Monogr., 182, Amer. Math. Soc., Providence, RI, 1999; MR1670044; P. J. Olver, op. cit.; MR1240056; op. cit.; MR1337276; L. V. Ovsyannikov, op. cit.; MR0668703].

Reviewed by Peter J. Olver