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Lie Algebras and Representation Theory Andreasˇcap

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Lie Algebras and Representation Theory Andreasˇcap Lie Algebras and Representation Theory Fall Term 2016/17 Andreas Capˇ Institut fur¨ Mathematik, Universitat¨ Wien, Nordbergstr. 15, 1090 Wien E-mail address: [email protected] Contents Preface v Chapter 1. Background 1 Group actions and group representations 1 Passing to the Lie algebra 5 A primer on the Lie group { Lie algebra correspondence 8 Chapter 2. General theory of Lie algebras 13 Basic classes of Lie algebras 13 Representations and the Killing Form 21 Some basic results on semisimple Lie algebras 29 Chapter 3. Structure theory of complex semisimple Lie algebras 35 Cartan subalgebras 35 The root system of a complex semisimple Lie algebra 40 The classification of root systems and complex simple Lie algebras 54 Chapter 4. Representation theory of complex semisimple Lie algebras 59 The theorem of the highest weight 59 Some multilinear algebra 63 Existence of irreducible representations 67 The universal enveloping algebra and Verma modules 72 Chapter 5. Tools for dealing with finite dimensional representations 79 Decomposing representations 79 Formulae for multiplicities, characters, and dimensions 83 Young symmetrizers and Weyl's construction 88 Bibliography 93 Index 95 iii Preface The aim of this course is to develop the basic general theory of Lie algebras to give a first insight into the basics of the structure theory and representation theory of semisimple Lie algebras. A problem one meets right in the beginning of such a course is to motivate the notion of a Lie algebra and to indicate the importance of representation theory. The simplest possible approach would be to require that students have the necessary background from differential geometry, present the correspondence between Lie groups and Lie algebras, and then move to the study of Lie algebras, which are easier to understand than the Lie groups themselves. This is unsatisfactory however, since in the further development the only necessary prerequisite is just a good knowledge of linear algebra, so requiring a background in differential geometry just for understanding the motivation seems rather strange. Therefore, I decided to start the course with an informal discussion of the back- ground. The starting point for this introduction is the concept of a group action, which is very intuitive when starting from the idea of a group of symmetries. Group represen- tations then show up naturally as actions by linear maps on vector spaces. In the case of a Lie group (with matrix groups being the main example) rather than a discrete group, one may linearize the concepts to obtain a Lie algebra and representations of this Lie algebra. The last part of the introduction is then a short discussion of the correspon- dence between Lie groups and Lie algebras, which shows that in spite of the considerable simplification achieved by passing to the Lie algebra, not too much information is lost. Most of the rest of the course is based on parts of the second chapter of my book \Parabolic geometries I: Background and General Theory" (a joint work with J. Slov´ak from Brno). Chapter 2 discusses the general theory of Lie algebras. We start by discussing nilpo- tent and solvable Lie algebras, and prove the fundamental theorems of Engel and Lie. Next we switch to the discussion of semisimple, simple and reductive Lie algebras. We discuss representations and the Killing form and prove Cartan's criteria for solvability and semisimplicity in terms of the Killing form. We give a proof of complete reducibil- ity of representations of semisimple Lie algebras which is independent of the structure theory of such algebras. This is used to prove that any semisimple Lie algebra is a direct sum of simple ideals. Finally, we describe a systematic way to produce examples of reductive and semisimple Lie algebras of matrices. Some background from linear algebra (in particular concerning Jordan decompositions) is reviewed in the text. Chapter 3 studies the structure theory of complex semisimple Lie algebras, which is also a fundamental ingredient for the study of representations of such algebras. Choosing a Cartan subalgebra, one obtains the root decomposition of the given Lie algebra into simultaneous eigenspaces under the adjoint action of the Cartan subalgebra. General results on Jordan decompositions show that the elements of the Cartan subalgebra are simultaneously diagonalizable in any finite dimensional representation, thus leading to the weight decomposition. The structure of the root decomposition can be analyzed v vi PREFACE using the representation theory of sl(2; C), which is easy to describe. In that way, one associates with any complex semisimple Lie algebra an abstract root system, which is simply a nice set of vectors in a finite dimensional inner product space. We conclude the chapter by briefly discussing the classification of irreducible root systems, and how this can be used to give a complete classification of complex simple Lie algebras. The basic theory of complex representations of complex semisimple Lie algebras is studied in chapter 4. With the background developed so far, we quickly arrive at a description of the possible weights of finite dimensional representations. Next, we study highest weight vectors and show that in a finite dimensional representation any such vector generates an irreducible subrepresentation. Using this, we arrive quickly at the result that a finite dimensional irreducible representation is determined up to isomorphism by its highest weight, which has to be dominant an algebraically integral. Next, we discuss two approaches to the proof of existence of finite dimensional irre- ducible representations with any dominant integral highest weight. The first approach is on a case{by{case basis, using fundamental representations and tensor products. We first discuss the necessary background from multilinear algebra, and then describe the fundamental representations (and some basic relations between them) for the classical simple Lie algebras. Secondly, we outline the general proof for existence of irreducible representations via Verma modules. The necessary background on universal enveloping algebras and induced modules is discussed. The final chapter offers a brief survey various tools that can be used to describe irreducible representations and to split general representations into irreducible pieces. The first part of the chapter deals with tools for general complex semisimple Lie al- gebras. We discuss the isotypical splitting, the Casimir element, and various formulae for multiplicities of weights and characters. As an important example, we discuss the decomposition of a tensor product of two irreducible representations. The second part is devoted to the relation between representations of gl(n; C) and representations of permutation groups. We discuss Young diagrams and Young symmetrizers, and Weyl's construction of irreducible representations of the classical simple Lie groups in terms of Schur functors. There are several good books on Lie algebras and representation theory available, which usually however are too detailed for serving as a basis for a relatively short course. Two particularly recommendable sources are the books \Lie groups beyond an introduction" by A.W. Knapp (which I will refer to as [Knapp]) and \Represen- tation Theory A First Course" by W. Fulton and J. Harris (which I will refer to as [Fulton-Harris]). Both these books do not only discuss Lie algebras but also Lie groups, and [Fulton-Harris] also discusses representations of finite groups. The two books also complement each other nicely from the approach taken by the authors: [Fulton-Harris] emphasizes examples and the concrete description of representations of the classical simple Lie algebras, [Knapp] contains a detailed account of the general theory and also discussed real Lie algebras and Lie groups. Two other recommendable texts which only discuss Lie algebras are the books \Introduction to Lie Algebras and Representation Theory" by J.E. Humphreys, and \Notes on Lie algebras" by H. Samel- son. A nice short text is the book \Lectures on Lie Groups and Lie Algebras" by R. Carter, G. Segal, and I. Mac Donald. Apart from a brief survey of the theory of complex semisimple Lie algebras, this also offers an introduction to Lie Groups with an emphasis on the compact case, and an introduction to complex algebraic groups. CHAPTER 1 Background Large parts of the theory of Lie algebras can be developed with very little back- ground. Indeed, mainly a good knowledge of linear algebra is needed. Apart from that, only a bit of Euclidean geometry shows up, which again can be traced back to linear algebra. On the other hand, Lie algebras usually not mentioned in introductory courses, so most of the students in this course probably have not heard the definition of a Lie algebra before. Moreover, this definition will probably sound rather strange to most beginners, since skew symmetry and the Jacobi identity are much less intuitive than commutativity and associativity. Hence I have decided not to start with the abstract definition of a Lie algebra and then develop the theory, but rather to indicate first where the concepts come from, and why it may be a good idea to study Lie algebras and their representations. In particular, I want to show how the idea of a group of symmetries leads (via actions and representations of groups) to Lie algebras and their representations. Moreover, I want to point out in this chapter some examples in which thinking in terms of representation theory is very helpful. Group actions and group representations 1.1. Symmetries and group actions. The idea of a symmetry is probably one of the basic concepts in mathematics. Usually this is understood as having a distinguished set of functions from a set X to itself which may be thought of as preserving some additional structure. The basic features are that the composition of two symmetries is again a symmetry and that any symmetry is a bijection, whose inverse is a symmetry, too.
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