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D-modules continues to be an active area of stimulating research in such mathematical areas as algebra, analysis, differential equations, and representation theory. Key to D-modules, Perverse Sheaves, and Representation Theory is the authors' essential algebraic-analytic approach to the theory, which connects D-modules to representation theory and other areas of mathematics. Significant concepts and topics that have emerged over the last few decades are presented, including a treatment of the theory of holonomic D-modules, perverse sheaves, the all-important Riemann-Hilbert correspondence, Hodge modules, and the solution to the Kazhdan-Lusztig conjecture using D-module theory. To further aid the reader, and to make the work as self-contained as possible, appendices are provided as background for the theory of derived categories and algebraic varieties. The book is intended to serve graduate students in a classroom setting and as self-study for researchers in algebraic geometry, and representation theory.

Ordered algebraic structures --- Geometry --- algebra --- topologie (wiskunde) --- Topological groups. Lie groups --- Algebra --- landmeetkunde --- wiskunde --- Group theory --- Représentations d'algèbres de Lie --- Representations of Lie algebras --- Linear algebraic groups. --- D-modules. --- Representations of groups. --- Groupes algébriques linéaires --- D-modules, Théorie des --- Représentations de groupes --- Groupes algébriques linéaires --- D-modules, Théorie des --- Représentations de groupes

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The finite simple groups are the building blocks from which all the finite groups are made and as such they are objects of fundamental importance throughout mathematics. The classification of the finite simple groups was one of the great mathematical achievements of the twentieth century, yet these groups remain difficult to study which hinders applications of the classification. This textbook brings the finite simple groups to life by giving concrete constructions of most of them, sufficient to illuminate their structure and permit real calculations both in the groups themselves and in the underlying geometrical or algebraic structures. This is the first time that all the finite simple groups have been treated together in this way and the book points out their connections, for example between exceptional behaviour of generic groups and the existence of sporadic groups, and discusses a number of new approaches to some of the groups. Many exercises of varying difficulty are provided. The Finite Simple Groups is aimed at advanced undergraduate and graduate students in algebra as well as professional mathematicians and scientists who use groups and want to apply the knowledge which the classification has given us. The main prerequisite is an undergraduate course in group theory up to the level of Sylow’s theorems.

Mathematics. --- Group Theory and Generalizations. --- Topological Groups, Lie Groups. --- Group theory. --- Topological Groups. --- Mathématiques --- Groupes, Théorie des --- Finite simple groups --- Groupes simples finis --- Mathematics --- Algebra --- Physical Sciences & Mathematics --- Finite simple groups. --- Théorie des groupes --- Math --- Simple groups, Finite --- Algebra. --- Topological groups. --- Lie groups. --- Science --- Finite groups --- Linear algebraic groups --- Groups, Topological --- Continuous groups --- Groups, Theory of --- Substitutions (Mathematics) --- Mathematical analysis --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups

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The use of Clifford algebras in mathematical physics and engineering has grown rapidly in recent years. Whereas other developments have privileged a geometric approach, the author uses an algebraic approach which can be introduced as a tensor product of quaternion algebras and provides a unified calculus for much of physics. The book proposes a pedagogical introduction to this new calculus, based on quaternions, with applications mainly in special relativity, classical electromagnetism and general relativity. The volume is intended for students, researchers and instructors in physics, applied mathematics and engineering interested in this new quaternionic Clifford calculus.

Clifford algebras. --- Quaternions. --- Algebra, Universal --- Algebraic fields --- Curves --- Surfaces --- Numbers, Complex --- Vector analysis --- Geometric algebras --- Algebras, Linear --- Algebra. --- Group theory. --- Topological Groups. --- Mathematical physics. --- Classical and Quantum Gravitation, Relativity Theory. --- Associative Rings and Algebras. --- Group Theory and Generalizations. --- Topological Groups, Lie Groups. --- Mathematical Methods in Physics. --- Physical mathematics --- Physics --- Groups, Topological --- Continuous groups --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Mathematics --- Mathematical analysis --- Gravitation. --- Associative rings. --- Rings (Algebra). --- Topological groups. --- Lie groups. --- Physics. --- Field theory (Physics) --- Matter --- Antigravity --- Centrifugal force --- Relativity (Physics) --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Algebraic rings --- Ring theory --- Rings (Algebra) --- Properties

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Symmetry is a key ingredient in many mathematical, physical, and biological theories. Using representation theory and invariant theory to analyze the symmetries that arise from group actions, and with strong emphasis on the geometry and basic theory of Lie groups and Lie algebras, Symmetry, Representations, and Invariants is a significant reworking of an earlier highly-acclaimed work by the authors. The result is a comprehensive introduction to Lie theory, representation theory, invariant theory, and algebraic groups, in a new presentation that is more accessible to students and includes a broader range of applications. The philosophy of the earlier book is retained, i.e., presenting the principal theorems of representation theory for the classical matrix groups as motivation for the general theory of reductive groups. The wealth of examples and discussion prepares the reader for the complete arguments now given in the general case. Key Features of Symmetry, Representations, and Invariants: • Early chapters suitable for honors undergraduate or beginning graduate courses, requiring only linear algebra, basic abstract algebra, and advanced calculus • Applications to geometry (curvature tensors), topology (Jones polynomial via symmetry), and combinatorics (symmetric group and Young tableaux) • Self-contained chapters, appendices, comprehensive bibliography • More than 350 exercises (most with detailed hints for solutions) further explore main concepts • Serves as an excellent main text for a one-year course in Lie group theory • Benefits physicists as well as mathematicians as a reference work.

algebra --- topologie (wiskunde) --- Mathematical physics --- Topological groups. Lie groups --- Algebra --- wiskunde --- Group theory --- fysica --- Representations of groups --- Invariants --- Symmetry (Mathematics) --- Lie groups --- Représentations de groupes --- Analyse multidimensionnelle --- Symétrie (Mathématiques) --- Groupes de Lie --- Algèbre --- EPUB-LIV-FT LIVMATHE LIVSTATI SPRINGER-B --- Representations of groups. --- Invariants. --- Symmetry (Mathematics). --- Algebra. --- Lie groups. --- Group theory. --- Geometry. --- Mathematical physics. --- Topological Groups. --- Group Theory and Generalizations. --- Mathematical Methods in Physics. --- Topological Groups, Lie Groups. --- General Algebraic Systems. --- Groups, Topological --- Continuous groups --- Physical mathematics --- Physics --- Mathematics --- Euclid's Elements --- Mathematical analysis --- Groups, Theory of --- Substitutions (Mathematics) --- Physics. --- Topological groups. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics

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One of the mathematical challenges of modern physics lies in the development of new tools to efficiently describe different branches of physics within one mathematical framework. This text introduces precisely such a broad mathematical model, one that gives a clear geometric expression of the symmetry of physical laws and is entirely determined by that symmetry. The first three chapters discuss the occurrence of bounded symmetric domains (BSDs) or homogeneous balls and their algebraic structure in physics. It is shown that the set of all possible velocities is a BSD with respect to the projective group; the Lie algebra of this group, expressed as a triple product, defines relativistic dynamics. The particular BSD known as the spin factor is exhibited in two ways: first, as a triple representation of the Canonical Anticommutation Relations, and second, as a ball of symmetric velocities. The associated group is the conformal group, and the triple product on this domain gives a representation of the geometric product defined in Clifford algebras. It is explained why the state space of a two-state quantum mechanical system is the dual space of a spin factor. Ideas from Transmission Line Theory are used to derive the explicit form of the operator Mobius transformations. The book further provides a discussion of how to obtain a triple algebraic structure associated to an arbitrary BSD; the relation between the geometry of the domain and the algebraic structure is explored as well. The last chapter contains a classification of BSDs revealing the connection between the classical and the exceptional domains. With its unifying approach to mathematics and physics, this work will be useful for researchers and graduate students interested in the many physical applications of bounded symmetric domains. It will also benefit a wider audience of mathematicians, physicists, and graduate students working in relativity, geometry, and Lie theory.

Mathematics. --- Applications of Mathematics. --- Mathematical Methods in Physics. --- Geometry. --- Classical and Quantum Gravitation, Relativity Theory. --- Topological Groups, Lie Groups. --- Differential Geometry. --- Topological Groups. --- Global differential geometry. --- Mathematical physics. --- Mathématiques --- Géométrie --- Géométrie différentielle globale --- Physique mathématique --- Homogeneous spaces. --- Mathematical physics -- Mathematical models. --- Special relativity (Physics). --- Topological groups. --- Engineering & Applied Sciences --- Applied Mathematics --- Homogeneous spaces --- Mathematical physics --- Special relativity (Physics) --- Applied Physics --- Mathematical models --- Mathematical models. --- Ether drift --- Mass energy relations --- Relativity theory, Special --- Restricted theory of relativity --- Special theory of relativity --- Physical mathematics --- Physics --- Spaces, Homogeneous --- Mathematics --- Lie groups. --- Applied mathematics. --- Engineering mathematics. --- Differential geometry. --- Physics. --- Gravitation. --- Relativity (Physics) --- Lie groups --- Geometry, Differential --- Groups, Topological --- Continuous groups --- Euclid's Elements --- Math --- Science --- Differential geometry --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Field theory (Physics) --- Matter --- Antigravity --- Centrifugal force --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Engineering --- Engineering analysis --- Mathematical analysis --- Properties