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Lie groups. An approach through invariants and representations. (English) Zbl 1154.22001
Universitext. New York, NY: Springer (ISBN 0-387-26040-4/pbk). xxii, 596 p. (2007).
Lie theory, in our contemporary understanding, is one of the central areas in mathematics and mathematical physics. Initiated by Sophus Lie’s pioneering program of studying continuous transformation groups (Lie groups) in a systematic way, begun in the second half of the 19th century, Lie theory has rapidly grown into one of the most important, active, propelling and ubiquitous topics of modern mathematics in the 20th century. In fact, Lie groups, Lie algebras, linear algebraic groups, their representation theory, and their invariant theory have become part of the very foundations of modern mathematics, involving a virtually unique blend of algebraic, analytic, geometric, topological and combinatorial aspects and methods. Also, Lie theory has utmost significant applications in modern theoretical physics, especially in quantum theory and relativity theory.
Accordingly, there is a large number of textbooks and monographs on topics in Lie theory, which generally differ with regard to their viewpoint, purpose, coverage, methodology, level, and prerequisites.
The textbook under review, presented by one of the most renowned researchers and teachers in various fields of algebra and geometry, provides another introduction to the subject in a very original and uniquely multifarous way. Namely, in contrast to the majority of the majority existing introductory texts on Lie groups, the author has tried to present the various different aspects of Lie theory as a whole and in a unified manner, thereby combining the general theory with many concrete examples and applications. Simultaneously, the exposition is really meant to be introductory, with the prerequisites kept to a minimum and with a wealth of related background material provided throughout the text. Due to the author’s expertise, invariant theory and its relations to Lie theory has been chosen as the guiding pedagogical principle for this comprehensive account of the subject, which appears to be fairly unique in the relevant textbook literature.
After about 25 years of development from lecture notes and courses taught at various universities worldwide, the book under review represents the author’s approach to the subject in its complete and final form, thereby covering all the essential material in a single new primer of Lie theory.
The text consists of fifteen chapters, each of which is subdivided into several sections and subsections. The chapters are grouped in such a way that they represent the many different topics treated in the book, some of which can be taken as the subject of an entire, largely independent graduate course.
Chapter 1 gives a first introduction to group actions, orbit spaces, invariants, equivariant maps, and group representations, together with many classical examples. Chapter 2 treats, in this context, symmetric functions, resultants, discriminants, Bézoutians, Schur functions, and the Cauchy formulas as further classical background material. Chapter 3 touches upon another classical topic, namely the invariant theory of algebraic forms à la A. Capelli (1902). This material is presented in modern form and includes such basics like the polarization process, the Aronhold method, the Clebsch-Gordan formula, the Capelli identity, covariants, and the Gayley \(\Omega\)-process. In Chapter 4, this invariant-theoretic approach is taken as a pretext for introducing Lie groups and Lie algebras in a systematic way, together with their first fundamental structure theorems. Chapter 5 develops the necessary tensor calculus from scratch and uses this crucial framework to discuss the Clifford algebra, spin groups, some basic constructions in representation theory, the universal enveloping algebra, and free algebras in the sequel.
Chapter 6 provides a short introduction to semisimple algebras, together with an explanation of the related general methods of noncommutative algebra, including matrices over division rings, semisimple modules, Reynolds operators, the double centralizer theorem, Wedderburn’s theorem, primitive idempotents, and other related concepts.
Chapter 7 introduces algebraic groups, with a special emphasis on linearly reductive groups and Borel subgroups. This chapter requires some basic knowledge of the underlying elementary algebraic geometry, which the author freely refers to as for additional reading. The material presented here is to stress the parallel theory of reductive algebraic and compact (complex) Lie groups, on the one hand, and to prepare the ground for the general theory of group representations developed in the following chapters, on the other hand.
Chapter 8 begins the study of the representation theory of various groups with extra structure. This chapter essentially focuses on characters, matrix coefficients, the Peter-Weyl theorem, Weyl’s “unitary trick”, and representations of linear reductive groups. Hopf algebras, Hopf ideals, and the Tannaka-Krein duality are briefly introduced at the end of this chapter, mainly in order to describe the link between reductive groups and compact Lie groups more closely later on.
Chapter 9 returns to invariant theory and is titled “Tensor Symmetry”. Among the topics briefly touched upon here are the First Fundamental Theorem (FFT) of invariant theory for the linear group, Young symmetrizers and Young diagrams, representations of linear groups, polynomial functors and Schur functors, branching rules, and (semi-) standard diagrams.
Chapter 10 deals with the structure and classification of semisimple Lie algebras and their representations via root systems, decomposition methods, dual Hopf algebras, and Cartan-Weyl theory in general. This chapter also contains the corresponding theory of adjoint and simply connected algebraic groups and their compact (Lie) forms.
Chapter 11 offers a deeper study of the relationship between invariants and the representation theory of classical groups (à la H. Weyl), culminating in the corresponding First and Second Fundamental Theorems, spin representations, and Weyl’s character formula.
The last four chapters are meant to complement the theory developed so far. Chapter 12 explains some combinatorial aspects of representation theory via the theory of tableaux after Robinson-Schensted, Knuth, Schützenberger, and Littlewood-Richardson, whereas Chapter 13 briefly discusses standard monomials, Grassmannians, flag varieties, Schubert calculus, and further combinatorial tools in the study of invariants and representations of classical groups. Chapter 14 offers a glimpse into geometric invariant theory à la Hilbert-Mumford, and Chapter 15 returns to the classical invariant theory of binary forms in its modern setting (via Hilbert series) and in its computational aspects.
All in all, this unique textbook, combing invariant theory and Lie theory in a very natural way, refers to virtually every (classical and modern) aspect of this deep interrelation, though sometimes in a more concise and informal style. Such a broad panoramic view to the subject is certainly a novelty in the relevant textbook literature and cannot be found anywhere else. The author’s effort at providing such a rich source of insight must be seen as being highly welcome, rewarding, valuable and utmost masterly likewise.

MSC:
22-02 Research exposition (monographs, survey articles) pertaining to topological groups
22E10 General properties and structure of complex Lie groups
05E10 Combinatorial aspects of representation theory
05E15 Combinatorial aspects of groups and algebras (MSC2010)
14L24 Geometric invariant theory
14M15 Grassmannians, Schubert varieties, flag manifolds
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
20G05 Representation theory for linear algebraic groups
13A50 Actions of groups on commutative rings; invariant theory
MathOverflow Questions:
Representations of the Lorentz group
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