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Camille Laurent-Gengoux, Anne Pichereau, and Pol Vanhaecke
Publisher: Springer (2012)
Details: 461 pages, Hardcover
Series: Grundlehren der mathematischen Wissenschaften 347
Topics: Ring Theory, Lie Algebras, Classical Mechanics, Algebra, Algebraic Geometry
MAA Review[Reviewed by Michael Berg, on 11/27/2012]
The first chapter of the book under review leads off with the sentence, “In this chapter, we give the basic definitions of a Poisson algebra, of a Poisson variety, of a Poisson manifold and of a Poisson morphism.” This is obviously the right way to start, given that the authors are intending to present something perhaps a bit more ineffable, Poisson structures, to a broad audience. They note (cf. the Preface) that
So, what’s the skinny on these things? Well, back to the first chapter:
And then there’s the other side of this coin:
The stage is set, therefore, and we already recognize a number of familiar themes in the shadows. Indeed, in the book’s Introduction, the authors cite Hamiltonian mechanics and integrable systems and deformation theory and quantization by way of a (quite persuasive) motivation: none of this is a surprise in view of the preceding excerpts.
Thus, the book under review deals with very exciting (and current) material presented from a fascinating vantage point and should be welcomed by any scholar whose work touches upon the matters cited above, or other related themes; certainly the presence of, e.g., symplectic geometry in the game points to any number of situations where Poisson structures should easily find a role to play.
As an entry in the venerable Springer Grundlehren series, this book is not meant to be for rookies. Nonetheless, it is still offered as a textbook properly so-called: its thirteen chapters are peppered with sets of exercises and each chapter comes equipped with supplemental notes that go a bit beyond the text, introduce some historical material, and point to other relevant sources. The book also employs the sound pedagogical device of presenting its material in three parts: “Theoretical Background,” “Examples,” and “Applications.” Given the authors’ goal of spreading the word on all things Poisson, this orchestration is obviously quite sound: it should aid substantially in taking mature workers in an appropriate field (e.g. symplectic geometry) expeditiously to a marvelous new view of not only their bailiwicks but of much else besides.
Poisson Structures should succeed very easily in its goals and make a positive impact.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.