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## Über dieses Buch

Motivated by the importance of the Campbell, Baker, Hausdorff, Dynkin Theorem in many different branches of Mathematics and Physics (Lie group-Lie algebra theory, linear PDEs, Quantum and Statistical Mechanics, Numerical Analysis, Theoretical Physics, Control Theory, sub-Riemannian Geometry), this monograph is intended to: fully enable readers (graduates or specialists, mathematicians, physicists or applied scientists, acquainted with Algebra or not) to understand and apply the statements and numerous corollaries of the main result, provide a wide spectrum of proofs from the modern literature, comparing different techniques and furnishing a unifying point of view and notation, provide a thorough historical background of the results, together with unknown facts about the effective early contributions by Schur, Poincaré, Pascal, Campbell, Baker, Hausdorff and Dynkin, give an outlook on the applications, especially in Differential Geometry (Lie group theory) and Analysis (PDEs of subelliptic type) and quickly enable the reader, through a description of the state-of-art and open problems, to understand the modern literature concerning a theorem which, though having its roots in the beginning of the 20th century, has not ceased to provide new problems and applications.

The book assumes some undergraduate-level knowledge of algebra and analysis, but apart from that is self-contained. Part II of the monograph is devoted to the proofs of the algebraic background. The monograph may therefore provide a tool for beginners in Algebra.

## Inhaltsverzeichnis

### Chapter 1. Historical Overview

The period since the CBHD Theorem first came to life, about 120 years ago, 3 can be divided into two distinct phases. The range 1890–1950, beginning 4 with Schur’s paper [154] and ending with Dynkin’s [56], and the remaining 5 60-year range, up to the present day. The first range comprises, besides 6 the contributions by the authors whose names are recalled in our acronym 7 CBHD, other significant papers (often left unmentioned) by Ernesto Pascal, 8 Jules Henri Poincaré, Friedrich Heinrich Schur.

Andrea Bonfiglioli, Roberta Fulci

### Chapter 2. Background Algebra

The aim of this chapter is to recall the main algebraic prerequisites and all the notation and definitions used throughout the Book. All main proofs are deferred to Chap. 7. This chapter (and its counterpart Chap. 7) is intended for a Reader having only a basic undergraduate knowledge in Algebra; a Reader acquainted with a more advanced knowledge of Algebra may pass directly to Chap. 3.

Andrea Bonfiglioli, Roberta Fulci

### Chapter 3. The Main Proof of the CBHD Theorem

THE aim of this chapter is to present the main proof of the Campbell- Baker-Hausdorff-Dynkin Theorem (CBHD for short), the topic of this Book. The proof is split into two very separate parts.

Andrea Bonfiglioli, Roberta Fulci

### Chapter 4. Some “Short” Proofs of the CBHD Theorem

THE aim of this chapter is to give all the details of five other proofs (besides the one given in Chap. 3) of the Campbell, Baker, Hausdorff Theorem, stating that ; x♦y := Log(Exp(x) ·Exp(y)) is a series of Lie polynomials in x, y. As we showed in Chap. 3, this is the “qualitative” part of the CBHD Theorem, and the actual formula expressing xthat x♦y as an explicit series (that is, Dynkin’s Formula) can be quite easily derived from this qualitative counterpart as exhibited in Sect. 3.3.

Andrea Bonfiglioli, Roberta Fulci

### Chapter 5. Convergence of the CBHD Series and Associativity of the CBHD Operation

THE aim of this chapter is twofold. On the one hand, we aim to study the 4 convergence of the Dynkin series $$\begin{array}{*{20}c} {uv: = } & {\sum\limits_{j = 1}^\infty {\left( {\sum\limits_{n = 1}^j {\frac{{\left( { - 1} \right)^{n + 1} }}{n}\,\sum\limits_{\begin{array}{*{20}c} {(h_1,k_1 ), \cdots (h_n,k_n ) \ne (0,0)} \\ {h_1 + k_1 + \cdots + h_n + k_n = j} \\ \end{array}} \times \frac{{(ad\,u)^{h_1 } (ad\,\upsilon )^{k_1 } \cdots (ad\,u)^{h_n } (ad\,\upsilon )^{k_n - 1} (\upsilon )}}{{h_1 ! \cdots h_n !k_1 ! \cdots k_n !(\sum\nolimits_{i = 1}^n {(h_i + k_i )} )}}} } \right),} } \\ \end{array}$$ in various contexts. For instance, this series can be investigated in any nilpotent Lie algebra (over a field of characteristic zero) where it is actually a finite sum, or in any finite dimensional real or complex Lie algebra and, more generally, its convergence can be studied in any normed Banach-Lie algebra (over R or C). For example, the case of the normed Banach algebras (becoming normed Banach-Lie algebras if equipped with the associated commutator) will be extensively considered here.

Andrea Bonfiglioli, Roberta Fulci

### Chapter 6. Relationship Between the CBHD Theorem, the PBW Theorem and the Free Lie Algebras

THE aim of this chapter is to unravel the close relationship existing = between the Theorems of CBHD and of Poincaré-Birkhoff-Witt (“PBW” for short) and to show how the existence of free Lie algebras intervenes.We have analyzed, in Chap. 3, how the PBWTheoremintervenes in the classical approach to the proof of CBHD, in that PBWcan be used to prove in a simple way Friedrichs’s characterization of L(V ) (see Theorem 3.13 on page 133). Also, as for the proofs of CBHD in Chap. 4, the rÔle of the free Lie algebras was broadly manifest.

Andrea Bonfiglioli, Roberta Fulci

### Chapter 7. Proofs of the Algebraic Prerequisites

THE aim of this chapter is to collect all the missing proofs of the results in 3 Chap. 2. The chapter is divided into several sections, corresponding to 4 those of Chap. 2. Finally, Sect. 7.8 collects some proofs from Chaps. 4 and 6 5 too, considered as less crucial in favor of economy of presentation in the 6 chapters they originally belonged to.

Andrea Bonfiglioli, Roberta Fulci

### Chapter 8. Construction of Free Lie Algebras

THE aim of this chapter is twofold. On the one hand (Sect. 8.1), we complete the missing proof from Chap. 2 concerning the existence of a free Lie algebra Lie(X) related to a set X. This proof relies on the direct construction of Lie(X) as a quotient of the free non-associative algebra Lib(X). Furthermore, we prove that Lie(X) is isomorphic to L(K_X_), and the latter provides a free Lie algebra over X.

Andrea Bonfiglioli, Roberta Fulci

### Chapter 9. Formal Power Series in One Indeterminate

THE aim of this chapter is to collect some prerequisites on formal power series in one indeterminate, needed in this Book. One of the main aims is to furnish a purely algebraic proof of the fact that, by substituting into each other – in any order – the two series $$\sum^\infty_{n=1} \frac{x^n}{n!} \quad {\rm and} \quad \sum^\infty_{n=1} \frac{(-1)^{(n+1)}x^n}{n}$$ one obtains the result x.

Andrea Bonfiglioli, Roberta Fulci

### Chapter 10. Symmetric Algebra

IN this chapter, we recall the basic facts we needed about the so-called symmetric algebra (of a vector space), which we used in Chap. 6 in exhibiting the relationship between the CBHD Theorem and the PBW Theorem.

Andrea Bonfiglioli, Roberta Fulci

### Backmatter

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