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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.

Mathematics. --- Topological Groups, Lie Groups. --- Non-associative Rings and Algebras. --- Manifolds and Cell Complexes (incl. Diff.Topology). --- Algebra. --- Topological Groups. --- Cell aggregation --- Mathématiques --- Algèbre --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Calculus --- Nonassociative rings. --- Rings (Algebra). --- Topological groups. --- Lie groups. --- Manifolds (Mathematics). --- Complex manifolds. --- Aggregation, Cell --- Cell patterning --- Cell interaction --- Microbial aggregation --- Mathematical analysis --- Groups, Topological --- Continuous groups --- Representations of Lie groups. --- Representations of Lie algebras. --- Lie algebras. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Analytic spaces --- Manifolds (Mathematics) --- Geometry, Differential --- Topology --- Algebraic rings --- Ring theory --- Algebraic fields --- Rings (Algebra) --- Lie groups --- Representations of Lie algebras --- Representations of Lie groups --- Cell aggregation_xMathematics --- Topological Groups --- Complex manifolds --- Nonassociative rings --- Electronic books

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This book is about the computational aspects of invariant theory. Of central interest is the question how the invariant ring of a given group action can be calculated. Algorithms for this purpose form the main pillars around which the book is built. There are two introductory chapters, one on Gröbner basis methods and one on the basic concepts of invariant theory, which prepare the ground for the algorithms. Then algorithms for computing invariants of finite and reductive groups are discussed. Particular emphasis lies on interrelations between structural properties of invariant rings and computational methods. Finally, the book contains a chapter on applications of invariant theory, covering fields as disparate as graph theory, coding theory, dynamical systems, and computer vision. The book is intended for postgraduate students as well as researchers in geometry, computer algebra, and, of course, invariant theory. The text is enriched with numerous explicit examples which illustrate the theory and should be of more than passing interest. More than ten years after the first publication of the book, the second edition now provides a major update and covers many recent developments in the field. Among the roughly 100 added pages there are two appendices, authored by Vladimir Popov, and an addendum by Norbert A'Campo and Vladimir Popov. .

Algebra --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Mathematics. --- Topological groups. --- Algorithms. --- Algorism --- Groups, Topological --- Math --- Lie groups. --- Topological Groups, Lie Groups. --- Arithmetic --- Continuous groups --- Science --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Foundations --- Topological Groups.

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This book provides quick access to the theory of Lie groups and isometric actions on smooth manifolds, using a concise geometric approach. After a gentle introduction to the subject, some of its recent applications to active research areas are explored, keeping a constant connection with the basic material. The topics discussed include polar actions, singular Riemannian foliations, cohomogeneity one actions, and positively curved manifolds with many symmetries. This book stems from the experience gathered by the authors in several lectures along the years, and was designed to be as self-contained as possible. It is intended for advanced undergraduates, graduate students, and young researchers in geometry, and can be used for a one-semester course or independent study.

Mathematics. --- Differential Geometry. --- Topological Groups, Lie Groups. --- Algebraic Topology. --- Topological Groups. --- Global differential geometry. --- Algebraic topology. --- Mathématiques --- Géométrie différentielle globale --- Topologie algébrique --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Lie groups. --- Isometrics (Mathematics) --- Groups, Lie --- Topological groups. --- Differential geometry. --- Lie algebras --- Symmetric spaces --- Topological groups --- Transformations (Mathematics) --- Topology --- Groups, Topological --- Continuous groups --- Geometry, Differential --- Differential geometry

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This exposition provides the state-of-the art on the differential geometry of hypersurfaces in real, complex, and quaternionic space forms. Special emphasis is placed on isoparametric and Dupin hypersurfaces in real space forms as well as Hopf hypersurfaces in complex space forms. The book is accessible to a reader who has completed a one-year graduate course in differential geometry. The text, including open problems and an extensive list of references, is an excellent resource for researchers in this area. Geometry of Hypersurfaces begins with the basic theory of submanifolds in real space forms. Topics include shape operators, principal curvatures and foliations, tubes and parallel hypersurfaces, curvature spheres and focal submanifolds. The focus then turns to the theory of isoparametric hypersurfaces in spheres. Important examples and classification results are given, including the construction of isoparametric hypersurfaces based on representations of Clifford algebras. An in-depth treatment of Dupin hypersurfaces follows with results that are proved in the context of Lie sphere geometry as well as those that are obtained using standard methods of submanifold theory. Next comes a thorough treatment of the theory of real hypersurfaces in complex space forms. A central focus is a complete proof of the classification of Hopf hypersurfaces with constant principal curvatures due to Kimura and Berndt. The book concludes with the basic theory of real hypersurfaces in quaternionic space forms, including statements of the major classification results and directions for further research.

Geometry --- Mathematics --- Physical Sciences & Mathematics --- Hypersurfaces. --- Mathematics. --- Topological groups. --- Lie groups. --- Differential geometry. --- Hyperbolic geometry. --- Differential Geometry. --- Topological Groups, Lie Groups. --- Hyperbolic Geometry. --- Hyperspace --- Surfaces --- Global differential geometry. --- Topological Groups. --- Geometry, Differential --- Groups, Topological --- Continuous groups --- Hyperbolic geometry --- Lobachevski geometry --- Lobatschevski geometry --- Geometry, Non-Euclidean --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Differential geometry

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This is a textbook that derives the fundamental theories of physics from symmetry.   It starts by introducing, in a completely self-contained way, all mathematical tools needed to use symmetry ideas in physics. Thereafter, these tools are put into action and by using symmetry constraints, the fundamental equations of Quantum Mechanics, Quantum Field Theory, Electromagnetism, and Classical Mechanics are derived. As a result, the reader is able to understand the basic assumptions behind, and the connections between the modern theories of physics. The book concludes with first applications of the previously derived equations.

Physics. --- Mathematical Methods in Physics. --- Mathematical Physics. --- Particle and Nuclear Physics. --- Topological Groups, Lie Groups. --- Topological Groups. --- Mathematical physics. --- Physique --- Physique mathématique --- Physics - General --- Physics --- Physical Sciences & Mathematics --- Topological groups. --- Lie groups. --- Nuclear physics. --- Groups, Topological --- Continuous groups --- Physical mathematics --- Mathematics --- Symmetry (Physics) --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Atomic nuclei --- Atoms, Nuclei of --- Nucleus of the atom --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics