Invariant differential operators. Volume 1: Noncompact semisimple Lie algebras and groups
type: Item
instance of: scholarly article
title: Invariant differential operators. Volume 1: Noncompact semisimple Lie algebras and groups
zbMATH Open document ID: 1348.22001
DOI: 10.1515/9783110427646
author: V. K. Dobrev
published in: De Gruyter Studies in Mathematical Physics
publication date: 2016-08-02T00:00:00Z
review text: The book under review presents a point of view on the theory of invariant differential operators that uses the perspective of Lie groups, Lie algebras and their representations. The present volume, which is the first one out of two, is divided into nine chapters. The first chapter is a fairly informal introduction which describes the general notion of symmetry, discusses the notion of invariant differential operators and clarifies the aims and the structure of the text. The second chapter is intended as a concise introduction to the general structure theory of Lie algebras. This chapter starts with basic definitions, covers classical results on nilpotent and solvable Lie algebras and then proceeds to the discussion of weights, roots, root systems and Weyl groups. All this leads naturally to the classification of semi-simple finite dimensional Lie algebras and the construction and realization of Kac-Moody Lie algebras, which are illustrated by many explicit examples, in particular, of all exceptional simple finite dimensional Lie algebras. The chapter finishes with a discussion of some basics on representations of Lie algebras, in particular on highest weight modules, Verma modules and characters. The third chapter is devoted to real semi-simple Lie algebras. It contains a classification of noncompact semi-simple Lie algebras, a discussion of parabolic subalgebras and description of real forms. This chapter contains many explicit examples where various classical real algebras are described in detail on a case-by-case basis. The fourth chapter of the book discusses invariant differential operators. It starts with generalities on Lie groups which include a description of classical groups, Cartan subgroups and Cartan and Iwasawa decomposition. This is followed by basic facts from representation theory, covering induced, elementary and unitary representations and the theory of singular vectors. The latter is illustrated by examples in all Dynkin types. Chapter~5 consider…
reviewed by: Volodymyr Mazorchuk
Mathematics Subject Classification ID: 22-01
Mathematics Subject Classification ID: 17-01
Mathematics Subject Classification ID: 22Exx
Mathematics Subject Classification ID: 17Bxx
zbMATH DE Number: 6608460
zbMATH Keywords: symmetry
zbMATH Keywords: differential operator
zbMATH Keywords: Lie algebra
zbMATH Keywords: group
zbMATH Keywords: Kac-Moody algebra
zbMATH Keywords: semi-simple Lie algebra
zbMATH Keywords: root system
zbMATH Keywords: real form
zbMATH Keywords: module
zbMATH Keywords: highest weight
zbMATH Keywords: Verma module
zbMATH Keywords: parabolic subalgebra
zbMATH Keywords: Lie group
zbMATH Keywords: representation
zbMATH Keywords: compact group
zbMATH Keywords: character
zbMATH Keywords: multiplet
zbMATH Keywords: singular vector
zbMATH Keywords: bilinear form
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1515/9783110427646
OpenAlex ID: W2521147961
description: scientific article; zbMATH DE number 6608460