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This monograph gives an overview of various classes of infinite-dimensional Lie groups and their applications in Hamiltonian mechanics, fluid dynamics, integrable systems, gauge theory, and complex geometry. While infinite-dimensional groups often exhibit very peculiar features, this book describes unifying geometric ideas of the theory and gives numerous illustrations and examples, ranging from the classification of the Virasoro coadjoint orbits to knot theory, from optimal mass transport to moduli spaces of flat connections on surfaces.
The text includes many exercises and open questions, and it is accessible to both students and researchers in Lie theory, geometry, and Hamiltonian systems.
List of contents
Preface.- Introduction.- I Preliminaries.- II Infinite-dimensional Lie Groups: Their Geometry, Orbits and Dynamical Systems.- III Applications of Groups: Topological and Holomorphic Gauge Theories.- Appendices.- A1 Root Systems.- A2 Compact Lie Groups.- A3 Krichever-Novikov Algebras.- A4 Kähler Structures on the Virasoro and Loop Group Coadjoint Orbits.- A5 Metrics and Diameters of the Group of Hamiltonian Diffeomorphisms.- A6 Semi-Direct Extensions of the Diffeomorphism Group and Gas Dynamics.- A7 The Drinfeld-Sokolov Reduction.- A8 Surjectivity of the Exponential Map on Pseudo-Differential Symbols.- A9 Torus Actions on the Moduli Space of Flat Connections.- Bibliography.- Index
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From the reviews:
"The book under review is a welcome addition to the literature on infinite-dimensional Lie groups. ... the present monograph is to 'present the unifying ideas of the theory by concentrating on specific types and examples of infinite-dimensional Lie groups', in the authors' own words. The groups discussed here can be divided roughly into three classes ... . The main part of the book consists in the treatment of these groups, including their geometry, their coad-joint orbits, and their relationship to the Hamiltonian structures." (Daniel Beltita, Mathematical Reviews, Issue 2009 k)
"The book itself starts with (possibly infinite-dimensional) Lie groups and their algebras, defines the adjoint and co-adjoint representations, and then proceeds to central extensions ... . there are ample references to the enormous bibliography, which contains 393 listings, so the interested reader can easily delve further if he or she wishes. The book may be most useful as a way to get an overview of the subject ... or as a window through which to glimpse any one of the subjects ... ." (David G. Ebin, Bulletin of the American Mathematical Society, January, 2011)
