Topics in Noncommutative Algebra: The Theorem of Campbell, Baker, Hausdorff and Dynkin (Lecture Notes in Mathematics Book 2034)

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. 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: 1) 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; 2) provide a wide spectrum of proofs from the modern literature, comparing different techniques and furnishing a unifying point of view and notation; 3) 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; 4) give an outlook on the applications, especially in Differential Geometry (Lie group theory) and Analysis (PDEs of subelliptic type); 5) 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 Front Matter....Pages i-xxii Historical Overview....Pages 1-45 Front Matter....Pages 47-47 Background Algebra....Pages 49-114 The Main Proof of the CBHD Theorem....Pages 115-172 Some “Short” Proofs of the CBHD Theorem....Pages 173-264 Convergence of the CBHD Series and Associativity of the CBHD Operation....Pages 265-369 Relationship Between the CBHD Theorem, the PBW Theorem and the Free Lie Algebras....Pages 371-389 Front Matter....Pages 391-391 Proofs of the Algebraic Prerequisites....Pages 393-457 Construction of Free Lie Algebras....Pages 459-477 Formal Power Series in One Indeterminate....Pages 479-499 Symmetric Algebra....Pages 501-521 Back Matter....Pages 523-539 Suitable for graduates or specialists, mathematicians, physicists or applied scientists, acquainted with Algebra or not, this title provides a wide spectrum of proofs from the modern literature, comparing different techniques and furnishing a unifying point of view and notation. Andrea Bonfiglioli, Roberta Fulci. Includes Bibliographical References (p. 529-536) And Index.