This book collects a series of contributions addressing the various contexts in which the theory of Lie groups is applied. A preliminary chapter serves the reader both as a basic reference source and as an ongoing thread that runs through the subsequent chapters. From representation theory and Gerstenhaber algebras to control theory, from differential equations to Finsler geometry and Lepage manifolds, the book introduces young researchers in Mathematics to a wealth of different topics, encouraging a multidisciplinary approach to research. As such, it is suitable for students in doctoral courses, and will also benefit researchers who want to expand their field of interest.
About the Author
Giovanni Falcone received his Ph.D. in Erlangen, Germany in 1998, and has been an Assistant Professor in Palermo, Italy since 1999. His work mainly focuses on Lie algebras, Lie groups and algebraic groups, and (in combinatorics) on block designs. He has been invited to give lectures at the University of Technology in Budapest (2017), at Nankai University in Tianjin (2016), and at the Nesin Mathematics Village in Izmir (2015). At the University of Palermo, he coordinated the project “Lie groups, Differential equations, and Geometry,” supported by the Marie Curie Action nr. 317721. This book is one of the outcomes of the project.
Table of Contents
Preface. - Introduction.- 1 A short survey on Lie theory and Finsler Geometry.- 2 Remarks on infinite-dimensional representations of the Heisenberg algebra.- 3 Character, Multiplicity and Decomposition Problems in the Representation Theory of complex Lie Algebras.- 4 The BCH-Formula and Order Conditions for Splitting Methods Winfried Auzinger, Wolfgang Herfort, Othmar Koch, and Mechthild Thalhammer.- 5 Cohomology Operations Defining Cohomology Algebra of the Loop Space.- 6 Half-Automorphisms of Cayley-Dickson Loops.- 7 Invariant control systems on Lie groups.- 8 An Optimal Control Problem for an Nonlocal Problem on the Plane.- 9 On the geometry of the domain of the solution of nonlinear Cauchy.- 10 Reduction of some semi-discrete schemes for an evolutionary equation to two-layer schemes and estimates for the approximate solution error.- 11 Hilbert’s Fourth Problem and Projectively Flat Finsler Metrics.- 12 Holonomy theory of Finsler manifolds.- 13 Lepage Manifolds.