Quantization on Nilpotent Lie Groups

This book presents a consistent development of the Kohn-Nirenberg type global quantization theory in the setting of graded nilpotent Lie groups in terms of their representations. It contains a detailed exposition of related background topics on homogeneous Lie groups, nilpotent Lie groups, and the analysis of Rockland operators on graded Lie groups together with their associated Sobolev spaces. For the specific example of the Heisenberg group the theory is illustrated in detail. In addition, the book features a brief account of the corresponding quantization theory in the setting of compact Lie groups. The monograph is the winner of the 2014 Ferran Sunyer i Balaguer Prize.

Author Biography: Veronique Fischer is a Senior Lecturer in Analysis at the University of Bath. Michael Ruzhansky is a Professor of Pure Mathematics at Imperial College London. The research of this monograph was supported by the EPSRC Grant EP/K039407/1 when Veronique Fischer was employed at Imperial College London. It started when she was working at the University of Padua. The work was also supported by the Marie Curie FP7 project (PseudodiffOperatorS - 301599) and by the Leverhulme Trust (grant RPG-2014-02).

Promotional Information: This is an open access book, the electronic versions are freely accessible online.

Preface.- Introduction.- Notation and conventions.- 1 Preliminaries on Lie groups.- 2 Quantization on compact Lie groups.- 3 Homogeneous Lie groups.- 4 Rockland operators and Sobolev spaces.- 5 Quantization on graded Lie groups.- 6 Pseudo-differential operators on the Heisenberg group.- A Miscellaneous.- B Group C* and von Neumann algebras.- Schrodinger representations and Weyl quantization.- Explicit symbolic calculus on the Heisenberg group.- List of quantizations.- Bibliography.- Index.