Developments and Trends in Infinite-Dimensional Lie Theory / edited by Karl-Hermann Neeb, Arturo Pianzola.

Springer eBooks

Preface -- Part A: Infinite-Dimensional Lie (Super-)Algebras -- Isotopy for Extended Affine Lie Algebras and Lie Tori -- Remarks on the Isotriviality of Multiloop Algebras -- Extended Affine Lie Algebras and Other Generalizations of Affine Lie Algebras – A Survey -- Tensor Representations of Classical Locally Finite Lie Algebras -- Lie Algebras, Vertex Algebras, and Automorphic Forms -- Kac–Moody Superalgebras and Integrability -- Part B: Geometry of Infinite-Dimensional Lie (Transformation) Groups -- Jordan Structures and Non-Associative Geometry -- Direct Limits of Infinite-Dimensional Lie Groups -- Lie Groups of Bundle Automorphisms and Their Extensions -- Gerbes and Lie Groups -- Part C: Representation Theory of Infinite-Dimensional Lie Groups Functional Analytic Background for a Theory of Infinite- Dimensional Reductive Lie Groups -- Heat Kernel Measures and Critical Limits -- Coadjoint Orbits and the Beginnings of a Geometric Representation Theory -- Infinite-Dimensional Multiplicity-Free Spaces I: Limits of Compact Commutative Spaces -- Index.

This collection of invited expository articles focuses on recent developments and trends in infinite-dimensional Lie theory, which has become one of the core areas of modern mathematics. The book is divided into three parts: infinite-dimensional Lie (super-)algebras, geometry of infinite-dimensional Lie (transformation) groups, and representation theory of infinite-dimensional Lie groups. Part (A) is mainly concerned with the structure and representation theory of infinite-dimensional Lie algebras and contains articles on the structure of direct-limit Lie algebras, extended affine Lie algebras and loop algebras, as well as representations of loop algebras and Kac–Moody superalgebras. The articles in Part (B) examine connections between infinite-dimensional Lie theory and geometry. The topics range from infinite-dimensional groups acting on fiber bundles, corresponding characteristic classes and gerbes, to Jordan-theoretic geometries and new results on direct-limit groups. The analytic representation theory of infinite-dimensional Lie groups is still very much underdeveloped. The articles in Part (C) develop new, promising methods based on heat kernels, multiplicity freeness, Banach–Lie–Poisson spaces, and infinite-dimensional generalizations of reductive Lie groups. Contributors: B. Allison, D. Belti??, W. Bertram, J. Faulkner, Ph. Gille, H. Glöckner, K.-H. Neeb, E. Neher, I. Penkov, A. Pianzola, D. Pickrell, T.S. Ratiu, N.R. Scheithauer, C. Schweigert, V. Serganova, K. Styrkas, K. Waldorf, and J.A. Wolf.

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