Topics in noncommutative algebra. The theorem of Campbell, Baker, Hausdorff and Dynkin.

*(English)*Zbl 1231.17001
Lecture Notes in Mathematics 2034. Berlin: Springer (ISBN 978-3-642-22596-3/pbk; 978-3-642-22597-0/ebook). xxii, 439 p. (2011).

The central topic of this book, the mathematical result named after the mathematicians Henry Frederick Baker, John Edward Campbell, Eugene Borisovich Dynkin and Felix Hausdorff (named henceforth CBHD), shares, together with the Fundamental Theorem of Algebra, the remarkable property to cross, by its nature, the realms of many mathematical disciplines: Algebra, Analysis and Geometry. The interest in this result straddling the decades, the very nature of it ranging over those disciplines, its fields of application stretching across so many branches of Mathematics and Physics, the proofs so variegated and rich in ideas and the engrossing history of the early contributions are the facts that have seemed to the authors to be a sufficient incentive and stimulus in devoting this book to such a fascinating Theorem.

The main results are the following. After a historical preamble given in Chapter 1, the book is divided into two parts. Part I includes Chapters 2–6. Chapter 2 is entirely devoted to recalling algebraic prerequisites and notations to help non-specialist readers. Chapter 3 illustrates a proof of the CBHD, obtained from general results of algebra. In Chapter 4, several shorter, but specialized, proofs of this theorem are presented. Throughout Chapter 5, the authors exploit identities implicitly contained in the theorem and Chapter 6 classifies the deep intertwinement occurring between the CBHD and Poincaré-Birkhoff-Witt Theorem.

Part II includes Chapters 7–10. Chapter 7 consists of a collection of the missing proofs from Chapter 2 and Chapter 8 is used to complete some results dealing with the existence of the free Lie algebra related to a certain set. An algebraic approach to formal power series can be found in Chapter 9 and, finally, Chapter 10 contains all the machinery about symmetric algebras which is needed in Chapter 6.

The main results are the following. After a historical preamble given in Chapter 1, the book is divided into two parts. Part I includes Chapters 2–6. Chapter 2 is entirely devoted to recalling algebraic prerequisites and notations to help non-specialist readers. Chapter 3 illustrates a proof of the CBHD, obtained from general results of algebra. In Chapter 4, several shorter, but specialized, proofs of this theorem are presented. Throughout Chapter 5, the authors exploit identities implicitly contained in the theorem and Chapter 6 classifies the deep intertwinement occurring between the CBHD and Poincaré-Birkhoff-Witt Theorem.

Part II includes Chapters 7–10. Chapter 7 consists of a collection of the missing proofs from Chapter 2 and Chapter 8 is used to complete some results dealing with the existence of the free Lie algebra related to a certain set. An algebraic approach to formal power series can be found in Chapter 9 and, finally, Chapter 10 contains all the machinery about symmetric algebras which is needed in Chapter 6.

Reviewer: Juan Núñez Valdés (Sevilla)