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Poisson Structures

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MAA Review

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

Poisson structures naturally appear in very different forms and contexts [with s]ymplectic manifolds, Lie algebras, singularity theory, r-matrices … all lead[ing] to a certain type of Poisson structure, sharing several features … despite the distances between the mathematics they originate from.

So, what’s the skinny on these things? Well, back to the first chapter:

Geometrically speaking, a Poisson structure on a smooth manifold … associates [a vector field] to every smooth function on [that manifold] … [which, i]n the context of a mechanical system … yields the equations of motion, when [the smooth function] is taken as the Hamiltonian. The Poisson bracket is … a Lie bracket, which amounts to demanding that that Poisson’s theorem is valid … that the Poisson bracket of two constants of motion is itself a constant of motion.

And then there’s the other side of this coin:

Algebraically speaking, one considers on a (typically infinite dimensional) vector space … two different algebra structures: (1) a commutative, associative multiplication, (2) a Lie bracket. It results in the following definition: Poisson algebra := Comm[utative] accoc[iative] algebra + Lie algebra + Compatibility…

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.

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