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This book is an introduction to the theory of Lie groups and their representations at the advanced undergraduate or beginning graduate level. It covers the essentials of the subject starting from basic undergraduate mathematics. The correspondence between linear Lie groups and Lie algebras is developed in its local and global aspects. The classical groups are analyzed in detail, first with elementary matrix methods, then with the help of the structural tools typical of the theory of semisimple groups, such as Cartan subgroups, root, weights and reflections. The fundamental groups of the classical groups are worked out as an application of these methods. Manifolds are introduced when needed, in connection with homogeneous spaces, and the elements of differential and integral calculus on manifolds are presented, with special emphasis on integration on groups and homogeneous spaces. Representation theory starts from first principles, such as Schur's lemma and its consequences, and proceeds from there to the Peter-Weyl theorem, Weyl's character formula, and the Borel-Weil theorem, all in the context of linear groups.

512.81 --- 512.81 Lie groups --- Lie groups --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Lie groups. --- Groupes de Lie

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This book constitutes the proceedings of the 2000 Howard conference on "Infinite Dimensional Lie Groups in Geometry and Representation Theory". It presents some important recent developments in this area. It opens with a topological characterization of regular groups, treats among other topics the integrability problem of various infinite dimensional Lie algebras, presents substantial contributions to important subjects in modern geometry, and concludes with interesting applications to representation theory. The book should be a new source of inspiration for advanced graduate students and esta

Infinite dimensional Lie algebras --- Infinite groups --- Infinite-dimensional manifolds --- Lie groups

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Topological groups. Lie groups --- Ordered algebraic structures --- Lie groups. --- Compact groups. --- Root systems (Algebra) --- Lie, Groupes de --- Groupes compacts --- Systèmes de racines (algèbre) --- Compact groups --- Lie groups --- Systems of roots (Algebra) --- Lie algebras --- Groups, Lie --- Symmetric spaces --- Topological groups --- Groups, Compact --- Locally compact groups --- Lie, Groupes de. --- Groupes compacts.

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Symmetry analysis based on Lie group theory is the most important method for solving nonlinear problems aside from numerical computation. The method can be used to find the symmetries of almost any system of differential equations and the knowledge of these symmetries can be used to reduce the complexity of physical problems governed by the equations. This is a broad, self-contained, introduction to the basics of symmetry analysis for first and second year graduate students in science, engineering and applied mathematics. Mathematica-based software for finding the Lie point symmetries and Lie-Ba;cklund symmetries of differential equations is included on a CD along with more than forty sample notebooks illustrating applications ranging from simple, low order, ordinary differential equations to complex systems of partial differential equations. MathReader 4.0 is included to let the user read the sample notebooks and follow the procedure used to find symmetries.

Differential equations --- Symmetry (Physics) --- Invariance principles (Physics) --- Symmetry (Chemistry) --- Conservation laws (Physics) --- Physics --- 517.91 Differential equations --- Numerical solutions. --- Differential equations - Numerical solutions --- Lie groups. --- Symmetry (Physics).

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Kac-Moody Lie algebras 9 were introduced in the mid-1960s independently by V. Kac and R. Moody, generalizing the finite-dimensional semisimple Lie alge­ bras which we refer to as the finite case. The theory has undergone tremendous developments in various directions and connections with diverse areas abound, including mathematical physics, so much so that this theory has become a stan­ dard tool in mathematics. A detailed treatment of the Lie algebra aspect of the theory can be found in V. Kac's book [Kac-90l This self-contained work treats the algebro-geometric and the topological aspects of Kac-Moody theory from scratch. The emphasis is on the study of the Kac-Moody groups 9 and their flag varieties XY, including their detailed construction, and their applications to the representation theory of g. In the finite case, 9 is nothing but a semisimple Y simply-connected algebraic group and X is the flag variety 9 /Py for a parabolic subgroup p y C g.

Kac-Moody algebras. --- Representations of groups. --- Flag manifolds. --- Kac-Moody, Algèbres de --- Représentations de groupes --- Variétés de drapeaux (Géométrie) --- Representations of groups --- Flag manifolds --- Kac-Moody algebras --- Kac-Moody, Algèbres de --- Représentations de groupes --- Variétés de drapeaux (Géométrie) --- Algebraic topology. --- Topological groups. --- Lie groups. --- Algebra. --- Algebraic geometry. --- Group theory. --- Algebraic Topology. --- Topological Groups, Lie Groups. --- Algebraic Geometry. --- Group Theory and Generalizations. --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Algebraic geometry --- Geometry --- Mathematics --- Mathematical analysis --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Groups, Topological --- Continuous groups --- Topology