In this chapter we put together an overview of the literature and some suggestions for further reading. The list is divided into three sections: textbooks (books suitable for readers just learning the theory), monographs (books that provide detailed coverage but which still can be classified as “core” theory of Lie groups and Lie algebras) and “Further reading”. Needless to say, this division is rather arbitrary and should not be taken too seriously.

Basic textbooks

There is a large number of textbooks on Lie groups and Lie algebras. Below we list some standard references which can be used either to complement the current book or to replace it.

Basic theory of Lie groups (subgroups, exponential map, etc.) can be found in any good book on differential geometry, such as Spivak or Warner. For more complete coverage, including discussion of representation theory, the classic references are Bröcker and tom Dieck or the book by Fulton and Harris. Other notable books in this category include Varadarajan, Onishchik and Vinberg. The latest (and highly recommended) additions to this list are Bump, Sepanski and Procesi. Each of these books has its own strengths and weaknesses; we suggest that the reader looks at them to choose the book which best matches his tastes.

In this chapter, we begin the study of semisimple Lie algebras and their representations. This is one of the highest achievements of the theory of Lie algebras, which has numerous applications (for example, to physics), not to mention that it is also one of the most beautiful areas of mathematics.

Throughout this chapter, g is a finite-dimensional semisimple Lie algebra (see Definition 5.35); unless specified otherwise, g is complex.

Properties of semisimple Lie algebras

Cartan's criterion of semimplicity, proved in Section 5.8, is not very convenient for practical computations. However, it is extremely useful for theoretical considerations.

Proposition 6.1. Let g be a real Lie algebra and gℂ – its complexification (see Definition 3.49). Then g is semisimple iff gℂ is semisimple.

Proof. Immediately follows from Cartan's criterion of semisimplicity.

Remark 6.2. This theorem fails if we replace the word “semisimple” by “simple”: there exist simple real Lie algebras g such that gℂ is a direct sum of two simple algebras.

Theorem 6.3. Let g be a semisimple Lie algebra, and I ⊂ g – an ideal. Then there is an ideal I' such that g = I ⊕ I'.

Proof. Let I⊥ be the orthogonal complement with respect to the Killing form. By Lemma 5.45, I⊥ is an ideal. Consider the intersection I ∩ I⊥. It is an ideal in g with zero Killing form (by Exercise 5.1). Thus, by Cartan criterion, it is solvable.

In this section, we give a sample syllabus of a one-semester graduate course on Lie groups and Lie algebras based on this book. This course is designed to fit the standard schedule of US universities: 14 week semester, with two lectures a week, each lecture 1 hour and 20 minutes long.

Lecture 1: Introduction. Definition of a Lie group; C1 implies analytic. Examples: ℝn, S1, SU(2). Theorem about closed subgroup (no proof). Connected component and universal cover.

Lecture 2: G/H. Action of G on manifolds; homogeneous spaces. Action on functions, vector fields, etc. Left, right, and adjoint action. Left, right, and bi-invariant vector fields (forms, etc).

Lecture 3: Classical groups: GL, SL, SU, SO, Sp – definition. Exponential and logarithmic maps for matrix groups. Proof that classical groups are smooth; calculation of the corresponding Lie algebra and dimension. Topological information (connectedness, π1). One-parameter subgroups in a Lie group: existence and uniqueness.

Lecture 4: Lie algebra of a Lie groups:

g = T1G = right-invariant vector fields = 1-parameter subgroups.

Exponential and logarithmic maps and their properties. Morphisms f:G1 → G2 are determined by f*:g1 → g2. Example: elements Jx, Jy, Jz ∈ so(3). Definition of commutator: exey = ex+y+½[x, y]+….

Lecture 5: Properties of the commutator. Relation with the group commutator; Ad and ad. Jacobi identity. Abstract Lie algebras and morphisms.

In any algebra textbook, the study of group theory is usually mostly concerned with the theory of finite, or at least finitely generated, groups. This is understandable: such groups are much easier to describe. However, most groups which appear as groups of symmetries of various geometric objects are not finite: for example, the group SO(3, ℝ) of all rotations of three-dimensional space is not finite and is not even finitely generated. Thus, much of material learned in basic algebra course does not apply here; for example, it is not clear whether, say, the set of all morphisms between such groups can be explicitly described.

The theory of Lie groups answers these questions by replacing the notion of a finitely generated group by that of a Lie group – a group which at the same time is a finite-dimensional manifold. It turns out that in many ways such groups can be described and studied as easily as finitely generated groups – or even easier. The key role is played by the notion of a Lie algebra, the tangent space to G at identity. It turns out that the group operation on G defines a certain bilinear skew-symmetric operation on g = T1G; axiomatizing the properties of this operation gives a definition of a Lie algebra.

Reminders from differential geometry

This book assumes that the reader is familiar with basic notions of differential geometry, as covered for example, in. For reader's convenience, in this section we briefly remind some definitions and fix notation.

Unless otherwise specified, all manifolds considered in this book will be C∞ real manifolds; the word “smooth” will mean C∞. All manifolds we will consider will have at most countably many connected components.

For a manifold M and a point m ∈ M, we denote by TmM the tangent space to M at point m, and by TM the tangent bundle to M. The space of vector fields on M (i.e., global sections of TM) is denoted by Vect(M). For a morphism f:X → Y and a point x ∈ X, we denote by f*:TxX → Tf(x)Y the corresponding map of tangent spaces.

Recall that a morphism f:X → Y is called an immersion if rank f* = dimX for every point x ∈ X; in this case, one can choose local coordinates in a neighborhood of x ∈ X and in a neighborhood of f(x) ∈ Y such that f is given by f (x1, … xn) = (x1, …, xn, 0, … 0).

In this section, we will discuss the representation theory of Lie groups and Lie algebras – as far as it can be discussed without using the structure theory of semisimple Lie algebras. Unless specified otherwise, all Lie groups, algebras, and representations are finite-dimensional, and all representations are complex. Lie groups and Lie algebras can be either real or complex; unless specified otherwise, all results are valid both for the real and complex case.

Basic definitions

Let us start by recalling the basic definitions.

Definition 4.1. A representation of a Lie group G is a vector space V together with a morphism ρ:G → GL(V).

A representation of a Lie algebra g is a vector space V together with a morphism ρ:g → gl(V).

A morphism between two representations V, W of the same group G is a linear map f:V → W which commutes with the action of G:f ρ(g) = ρ(g)f. In a similar way one defines a morphism of representations of a Lie algebra. The space of all G-morphisms (respectively, g-morphisms) between V and W will be denoted by HomG(V, W) (respectively, Homg(V, W)).

Remark 4.2. Morphisms between representations are also frequently called intertwining operators because they “intertwine” action of G in V and W.

The notion of a representation is completely parallel to the notion of module over an associative ring or algebra; the difference of terminology is due to historical reasons.

This book is an introduction to the theory of Lie groups and Lie algebras, with emphasis on the theory of semisimple Lie algebras. It can serve as a basis for a two-semester graduate course or – omitting some material – as a basis for a rather intensive one-semester course. The book includes a large number of exercises.

The material covered in the book ranges from basic definitions of Lie groups to the theory of root systems and highest weight representations of semisimple Lie algebras; however, to keep book size small, the structure theory of semisimple and compact Lie groups is not covered.

Exposition follows the style of famous Serre's textbook on Lie algebras: we tried to make the book more readable by stressing ideas of the proofs rather than technical details. In many cases, details of the proofs are given in exercises (always providing sufficient hints so that good students should have no difficulty completing the proof). In some cases, technical proofs are omitted altogether; for example, we do not give proofs of Engel's or Poincare–Birkhoff–Witt theorems, instead providing an outline of the proof. Of course, in such cases we give references to books containing full proofs.

It is assumed that the reader is familiar with basics of topology and differential geometry (manifolds, vector fields, differential forms, fundamental groups, covering spaces) and basic algebra (rings, modules).

In this chapter, we study representations of complex semisimple Lie algebras. Recall that by results of Section 6.3, every finite-dimensional representation is completely reducible and thus can be written in the form V = ⊕niVi, where Vi are irreducible representations and ni ∈ ℤ+ are the multiplicities. Thus, the study of representations reduces to classification of irreducible representations and finding a way to determine, for a given representation V, the multiplicities ni. Both of these questions have a complete answer, which will be given below.

Throughout this chapter, g is a complex finite-dimensional semisimple Lie algebra. We fix a choice of a Cartan subalgebra and thus the root decomposition g = h ⊕⊕R gα (see Section 6.6). We will freely use notation from Chapter 7; in particular, we denote by αi, i = 1 … r, simple roots, and by si ∈ W corresponding simple reflections. We will also choose a non-degenerate invariant symmetric bilinear form (,) on g.

All representations considered in this chapter are complex and unless specified otherwise, finite-dimensional.

Weight decomposition and characters

As in the study of representations of sl(2, ℂ) (see Section 4.8), the key to the study of representations of g is decomposing the representation into the eigenspaces for the Cartan subalgebra.

Definition 8.1. Let V be a representation of g. A vector u ∈ V is called a vector of weight λ ∈ h* if, for any h ∈ h, one has hu = 〈λ, h〉u.