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Multi-body Kinematics and Dynamics with Lie Groups explores the use of Lie groups in the kinematics and dynamics of rigid body systems. The first chapter reveals the formal properties of Lie groups on the examples of rotation and Euclidean displacement groups. Chapters 2 and 3 show the specific algebraic properties of the displacement group, explaining why dual numbers play a role in kinematics (in the so-called screw theory). Chapters 4 to 7 make use of those mathematical tools to expound the kinematics of rigid body systems and in particular the kinematics of open and closed kinematical chains. A complete classification of their singularities demonstrates the efficiency of the method. Dynamics of multibody systems leads to very big computations. Chapter 8 shows how Lie groups make it possible to put them in the most compact possible form, useful for the design of software, and expands the example of tree-structured systems. This book is accessible to all interested readers as no previous knowledge of the general theory is required.

Group theory --- Lie groups.

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The geometry and analysis that is discussed in this book extends to classical results for general discrete or Lie groups, and the methods used are analytical but have little to do with what is described these days as real analysis. Most of the results described in this book have a dual formulation; they have a 'discrete version' related to a finitely generated discrete group, and a continuous version related to a Lie group. The authors chose to centre this book around Lie groups but could quite easily have pushed it in several other directions as it interacts with opetators, and probability theory, as well as with group theory. This book will serve as an excellent basis for graduate courses in Lie groups, Markov chains or potential theory.

Lie groups. --- Geometry, Differential.

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Lie algebras --- Lie groups --- Lie algebras. --- Lie groups.

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Lie groups --- Lie, Groupes de

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Let mathbb{B} be a Lie group admitting a left-invariant negatively curved Kählerian structure. Consider a strongly continuous action alpha of mathbb{B} on a Fréchet algebra mathcal{A}. Denote by mathcal{A}^infty the associated Fréchet algebra of smooth vectors for this action. In the Abelian case mathbb{B}=mathbb{R}^{2n} and alpha isometric, Marc Rieffel proved that Weyl's operator symbol composition formula (the so called Moyal product) yields a deformation through Fréchet algebra structures {star_{heta}^alpha}_{hetainmathbb{R}} on mathcal{A}^infty. When mathcal{A} is a C^*-algebra, every deformed Fréchet algebra (mathcal{A}^infty,star^alpha_heta) admits a compatible pre-C^*-structure, hence yielding a deformation theory at the level of C^*-algebras too. In this memoir, the authors prove both analogous statements for general negatively curved Kählerian groups. The construction relies on the one hand on combining a non-Abelian version of oscillatory integral on tempered Lie groups with geom,etrical objects coming from invariant WKB-quantization of solvable symplectic symmetric spaces, and, on the second hand, in establishing a non-Abelian version of the Calderón-Vaillancourt Theorem. In particular, the authors give an oscillating kernel formula for WKB-star products on symplectic symmetric spaces that fiber over an exponential Lie group.

Lie groups. --- Kählerian structures.