New Developments In Lie Theory And Geometry
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New Developments In Lie Theory And Geometry
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Author : Carolyn Gordon
language : en
Publisher: American Mathematical Soc.
Release Date : 2009
New Developments In Lie Theory And Geometry written by Carolyn Gordon and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with Mathematics categories.
This volume is an outgrowth of the Sixth Workshop on Lie Theory and Geometry, held in the province of Cordoba, Argentina in November 2007. The representation theory and structure theory of Lie groups play a pervasive role throughout mathematics and physics. Lie groups are tightly intertwined with geometry and each stimulates developments in the other. The aim of this volume is to bring to a larger audience the mutually beneficial interaction between Lie theorists and geometers that animated the workshop. Two prominent themes of the representation theoretic articles are Gelfand pairs and the representation theory of real reductive Lie groups. Among the more geometric articles are an exposition of major recent developments on noncompact homogeneous Einstein manifolds and aspects of inverse spectral geometry presented in settings accessible to readers new to the area.
Lie Groups And Algebras With Applications To Physics Geometry And Mechanics
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Author : D.H. Sattinger
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-11-11
Lie Groups And Algebras With Applications To Physics Geometry And Mechanics written by D.H. Sattinger and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-11 with Mathematics categories.
This book is intended as an introductory text on the subject of Lie groups and algebras and their role in various fields of mathematics and physics. It is written by and for researchers who are primarily analysts or physicists, not algebraists or geometers. Not that we have eschewed the algebraic and geo metric developments. But we wanted to present them in a concrete way and to show how the subject interacted with physics, geometry, and mechanics. These interactions are, of course, manifold; we have discussed many of them here-in particular, Riemannian geometry, elementary particle physics, sym metries of differential equations, completely integrable Hamiltonian systems, and spontaneous symmetry breaking. Much ofthe material we have treated is standard and widely available; but we have tried to steer a course between the descriptive approach such as found in Gilmore and Wybourne, and the abstract mathematical approach of Helgason or Jacobson. Gilmore and Wybourne address themselvesto the physics community whereas Helgason and Jacobson address themselves to the mathematical community. This book is an attempt to synthesize the two points of view and address both audiences simultaneously. We wanted to present the subject in a way which is at once intuitive, geometric, applications oriented, mathematically rigorous, and accessible to students and researchers without an extensive background in physics, algebra, or geometry.
Compact Lie Groups
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Author : Mark R. Sepanski
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-12-19
Compact Lie Groups written by Mark R. Sepanski and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-12-19 with Mathematics categories.
Blending algebra, analysis, and topology, the study of compact Lie groups is one of the most beautiful areas of mathematics and a key stepping stone to the theory of general Lie groups. Assuming no prior knowledge of Lie groups, this book covers the structure and representation theory of compact Lie groups. Coverage includes the construction of the Spin groups, Schur Orthogonality, the Peter-Weyl Theorem, the Plancherel Theorem, the Maximal Torus Theorem, the Commutator Theorem, the Weyl Integration and Character Formulas, the Highest Weight Classification, and the Borel-Weil Theorem. The book develops the necessary Lie algebra theory with a streamlined approach focusing on linear Lie groups.
Advances In Analysis And Geometry
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Author : Tao Qian
language : en
Publisher: Birkhäuser
Release Date : 2012-12-06
Advances In Analysis And Geometry written by Tao Qian and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
On the 16th of October 1843, Sir William R. Hamilton made the discovery of the quaternion algebra H = qo + qli + q2j + q3k whereby the product is determined by the defining relations ·2 ·2 1 Z =] = - , ij = -ji = k. In fact he was inspired by the beautiful geometric model of the complex numbers in which rotations are represented by simple multiplications z ----t az. His goal was to obtain an algebra structure for three dimensional visual space with in particular the possibility of representing all spatial rotations by algebra multiplications and since 1835 he started looking for generalized complex numbers (hypercomplex numbers) of the form a + bi + cj. It hence took him a long time to accept that a fourth dimension was necessary and that commutativity couldn't be kept and he wondered about a possible real life meaning of this fourth dimension which he identified with the scalar part qo as opposed to the vector part ql i + q2j + q3k which represents a point in space.
Emergence Of The Theory Of Lie Groups
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Author : Thomas Hawkins
language : en
Publisher: Springer Science & Business Media
Release Date : 2000-07-19
Emergence Of The Theory Of Lie Groups written by Thomas Hawkins and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000-07-19 with Mathematics categories.
The great Norwegian mathematician Sophus Lie developed the general theory of transformations in the 1870s, and the first part of the book properly focuses on his work. In the second part the central figure is Wilhelm Killing, who developed structure and classification of semisimple Lie algebras. The third part focuses on the developments of the representation of Lie algebras, in particular the work of Elie Cartan. The book concludes with the work of Hermann Weyl and his contemporaries on the structure and representation of Lie groups which serves to bring together much of the earlier work into a coherent theory while at the same time opening up significant avenues for further work.
Lie Groups Lie Algebras And Representations
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Author : Brian Hall
language : en
Publisher: Springer
Release Date : 2015-05-11
Lie Groups Lie Algebras And Representations written by Brian Hall and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-05-11 with Mathematics categories.
This textbook treats Lie groups, Lie algebras and their representations in an elementary but fully rigorous fashion requiring minimal prerequisites. In particular, the theory of matrix Lie groups and their Lie algebras is developed using only linear algebra, and more motivation and intuition for proofs is provided than in most classic texts on the subject. In addition to its accessible treatment of the basic theory of Lie groups and Lie algebras, the book is also noteworthy for including: a treatment of the Baker–Campbell–Hausdorff formula and its use in place of the Frobenius theorem to establish deeper results about the relationship between Lie groups and Lie algebras motivation for the machinery of roots, weights and the Weyl group via a concrete and detailed exposition of the representation theory of sl(3;C) an unconventional definition of semisimplicity that allows for a rapid development of the structure theory of semisimple Lie algebras a self-contained construction of the representations of compact groups, independent of Lie-algebraic arguments The second edition of Lie Groups, Lie Algebras, and Representations contains many substantial improvements and additions, among them: an entirely new part devoted to the structure and representation theory of compact Lie groups; a complete derivation of the main properties of root systems; the construction of finite-dimensional representations of semisimple Lie algebras has been elaborated; a treatment of universal enveloping algebras, including a proof of the Poincaré–Birkhoff–Witt theorem and the existence of Verma modules; complete proofs of the Weyl character formula, the Weyl dimension formula and the Kostant multiplicity formula. Review of the first edition: This is an excellent book. It deserves to, and undoubtedly will, become the standard text for early graduate courses in Lie group theory ... an important addition to the textbook literature ... it is highly recommended. — The Mathematical Gazette
Geometry And Representation Theory Of Real And P Adic Groups
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Author : Juan Tirao
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Geometry And Representation Theory Of Real And P Adic Groups written by Juan Tirao and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
The representation theory of Lie groups plays a central role in both clas sical and recent developments in many parts of mathematics and physics. In August, 1995, the Fifth Workshop on Representation Theory of Lie Groups and its Applications took place at the Universidad Nacional de Cordoba in Argentina. Organized by Joseph Wolf, Nolan Wallach, Roberto Miatello, Juan Tirao, and Jorge Vargas, the workshop offered expository courses on current research, and individual lectures on more specialized topics. The present vol ume reflects the dual character of the workshop. Many of the articles will be accessible to graduate students and others entering the field. Here is a rough outline of the mathematical content. (The editors beg the indulgence of the readers for any lapses in this preface in the high standards of historical and mathematical accuracy that were imposed on the authors of the articles. ) Connections between flag varieties and representation theory for real re ductive groups have been studied for almost fifty years, from the work of Gelfand and Naimark on principal series representations to that of Beilinson and Bernstein on localization. The article of Wolf provides a detailed introduc tion to the analytic side of these developments. He describes the construction of standard tempered representations in terms of square-integrable partially harmonic forms (on certain real group orbits on a flag variety), and outlines the ingredients in the Plancherel formula. Finally, he describes recent work on the complex geometry of real group orbits on partial flag varieties.
New Trends In Sub Riemannian Geometry
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Author : Fabrice Baudoin
language : en
Publisher: American Mathematical Society
Release Date : 2025-01-27
New Trends In Sub Riemannian Geometry written by Fabrice Baudoin and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-01-27 with Mathematics categories.
This volume contains the proceedings of the AMS-EMS-SMF Special Session on Sub-Riemannian Geometry and Interactions, held from July 18–20, 2022, at the Université de Grenoble-Alpes, Grenoble, France. Sub-Riemannian geometry is a generalization of Riemannian one, where a smooth metric is defined only on a preferred subset of tangent directions. Under the so-called Hörmander condition, all points are connected by finite-length curves, giving rise to a well-defined metric space. Sub-Riemannian geometry is nowadays a lively branch of mathematics, connected with probability, harmonic and complex analysis, subelliptic PDEs, geometric measure theory, optimal transport, calculus of variations, and potential analysis. The articles in this volume present some developments of a broad range of topics in sub-Riemannian geometry, including the theory of sub-elliptic operators, holonomy, spectral theory, and the geometry of the exponential map.
Lie Theory And Its Applications In Physics
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Author : Vladimir Dobrev
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-09
Lie Theory And Its Applications In Physics written by Vladimir Dobrev and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-09 with Mathematics categories.
Traditionally, Lie Theory is a tool to build mathematical models for physical systems. Recently, the trend is towards geometrisation of the mathematical description of physical systems and objects. A geometric approach to a system yields in general some notion of symmetry which is very helpful in understanding its structure. Geometrisation and symmetries are meant in their broadest sense, i.e., classical geometry, differential geometry, groups and quantum groups, infinite-dimensional (super-)algebras, and their representations. Furthermore, we include the necessary tools from functional analysis and number theory. This is a large interdisciplinary and interrelated field. Samples of these new trends are presented in this volume, based on contributions from the Workshop “Lie Theory and Its Applications in Physics” held near Varna, Bulgaria, in June 2011. This book is suitable for an extensive audience of mathematicians, mathematical physicists, theoretical physicists, and researchers in the field of Lie Theory.
Recent Advances In Representation Theory Quantum Groups Algebraic Geometry And Related Topics
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Author : Pramod M. Achar
language : en
Publisher: American Mathematical Society
Release Date : 2014-08-27
Recent Advances In Representation Theory Quantum Groups Algebraic Geometry And Related Topics written by Pramod M. Achar and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-08-27 with Mathematics categories.
This volume contains the proceedings of two AMS Special Sessions "Geometric and Algebraic Aspects of Representation Theory" and "Quantum Groups and Noncommutative Algebraic Geometry" held October 13–14, 2012, at Tulane University, New Orleans, Louisiana. Included in this volume are original research and some survey articles on various aspects of representations of algebras including Kac—Moody algebras, Lie superalgebras, quantum groups, toroidal algebras, Leibniz algebras and their connections with other areas of mathematics and mathematical physics.