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El. knyga: Representations of Solvable Lie Groups: Basic Theory and Examples

(Saint Louis University, Missouri), (Université de Bourgogne, France)
  • Formatas: PDF+DRM
  • Serija: New Mathematical Monographs
  • Išleidimo metai: 16-Apr-2020
  • Leidėjas: Cambridge University Press
  • Kalba: eng
  • ISBN-13: 9781108651936
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  • Formatas: PDF+DRM
  • Serija: New Mathematical Monographs
  • Išleidimo metai: 16-Apr-2020
  • Leidėjas: Cambridge University Press
  • Kalba: eng
  • ISBN-13: 9781108651936
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"The theory of unitary group representations began with finite groups, and blossomed in the twentieth century both as a natural abstraction of classical harmonic analysis, and as a tool for understanding various physical phenomena. Combining basic theoryand new results, this monograph is a fresh and self-contained exposition of group representations and harmonic analysis on solvable Lie groups. Covering a range of topics from stratification methods for linear solvable actions in a finite-dimensional vector space, to complete proofs of essential elements of Mackey theory and a unified development of the main features of the orbit method for solvable Lie groups, the authors provide both well-known and new examples, with a focus on those relevant to contemporary applications. Clear explanations of the basic theory make this an invaluable reference guide for graduate students as well as researchers"--

Recenzijos

'There is the included background material on Lie theory, and there are also quite a lot of examples provided. Aspiring researchers in this area will likely find these features helpful, both aspiring and current researchers should also appreciate the wealth of material found here, as well as the extensive (five page, 92 entries) bibliography.' Mark Hunacek, MAA Reviews ' embeddings into matrix algebras and unitary representations are both possible and useful, and they are given a central role in this book.' M. Bona, Choice 'Throughout the book, the authors carefully explain the theory step by step and provide many concrete examples with computations which help the readers to understand. This book is a valuable exposition and an excellent research guide for the basic representation theory of solvable Lie groups for graduate students and researchers.' Junko Inoue, MathSciNet 'This monograph is a summary of a long career and experience of two great experts in their domain. Clear explanations of the basic theory make this an invaluable reference guide for graduate students as well as researchers. The concrete and example-based exposition is accessible to advanced graduate students and non-specialists.' Béchir Dali, zbMATH

Daugiau informacijos

A complete and self-contained account of the basic theory of unitary group representations for graduate students and researchers.
Preface ix
1 Basic theory of solvable Lie algebras and Lie groups
1(67)
1.1 Solvable Lie algebras
1(4)
1.2 Representations of a Lie algebra and weights
5(5)
1.3 The Lie theorem and its first consequences
10(9)
1.4 Adjoint and coadjoint representations
19(4)
1.5 Ado theorem for a solvable Lie algebra
23(5)
1.6 Lie groups
28(8)
1.7 Nilpotent and solvable Lie groups
36(16)
1.8 Exponential groups
52(4)
1.9 Finite-dimensional group representations
56(12)
2 Stratification of an orbit space
68(98)
2.1 Matrix normal form
69(15)
2.2 Layers for a representation
84(20)
2.3 Orbit structure for a completely solvable action
104(16)
2.4 Construction of rational supplementary elements
120(10)
2.5 Orbit structure for an action of exponential type
130(24)
2.6 Structure of the generic layer
154(12)
3 Unitary representations
166(103)
3.1 Unitary representations
166(24)
3.2 Decomposition of representations
190(28)
3.3 Induced representations
218(30)
3.4 Elements of Mackey theory
248(21)
4 Coadjoint orbits and polarizations
269(64)
4.1 Coadjoint orbits
269(9)
4.2 Polarizations
278(5)
4.3 Admissibility and the Pukanszky condition
283(6)
4.4 Isotropic subspaces associated to a flag
289(8)
4.5 Fine layering and Vergne polarizations
297(13)
4.6 Positivity and properties of polarizations
310(9)
4.7 Integral orbits
319(14)
5 Irreducible unitary representations
333(69)
5.1 Holomorphic induction
334(17)
5.2 Construction of irreducible representations
351(18)
5.3 Orbit method for solvable groups
369(33)
6 Plancherel formula and related topics
402(31)
6.1 Invariant measure on a coadjoint orbit
403(6)
6.2 Character formula
409(9)
6.3 Semicharacters and the Plancherel formula
418(15)
List of notations 433(7)
Bibliography 440(5)
Index 445
Didier Arnal is Emeritus Professor at the University of Burgundy and previously was Director of the Burgundy Mathematics Institute. He instituted and has worked over the past fifteen years on a cooperation project between France and Tunisia. He has authored papers on a diverse range of topics including deformation quantization, harmonic analysis, and algebraic structures. Bradley Currey III is a professor at Saint Louis University (SLU), Missouri. Formerly the Director of Graduate Studies in Mathematics at SLU, he has also served as a co-organizer in the Mathematics Research Communities program of the American Mathematical Society. Much of his recent research has explored the interplay of the theory of solvable Lie groups and applied harmonic analysis.