Lectures on Lie Groups

The following sections are included:

• Preface
• Contents

The following sections are included:

• Basic Concepts and Definitions
• A Brief Overview
• Compact Groups, Haar Integral and the Averaging Method
• Frobenius–Schur Orthogonality and the Character Theory
• Classification of Irreducible Complex Representations of S³
• L₂(G) and Concluding Remarks

A Lie group G is, by definition, a differentiable group; it consists of a group structure and a manifold structure such that the multiplication map and the inversion map are differentiable. One might say that the vector space structure is a natural focal point of various branches of mathematics at its elementary level. The Lie group structure is another natural focal point at its higher ground. Intuitively speaking, the differentiability of the group structure should provide a way to “linearize” the group structure at the “infinitesimal level” and the “linear object” so obtained should be a useful invariant in analyzing the original Lie group structure. This decisive step was accomplished by S. Lie in the late nineteenth century. The linear object he obtained was originally called the “infinitesimal group” by himself and was later renamed to “Lie algebra” by H. Weyl…

Roughly speaking, the linear representation theory that we discussed in the first lecture is a kind of extrinsic linearization; and the basic Lie group theory that we discussed in the second lecture is a kind of intrinsic linearization. The former is based on the compactness of the group and the technique of averaging provided by the Haar integral; and the latter is based upon the differentiability of the group structure and technically relies on the existence-uniqueness theory of ordinary differential equations, in the form of Frobenius theorem on complete integrability of distributions. Therefore, in the case of compact connected Lie groups, one naturally expects that they can be combined to provide a rather satisfactory understanding of both the structural theory and the representation theory of this important family of groups. This shall be exactly the topic of our discussion for the next few lectures…

In this lecture, we shall continue the study of the orbital geometry of the adjoint transformation of G on both the manifold G and its Lie algebra 𝔊. Based upon the maximal tori theorem of É. Cartan and the Weyl reduction, i.e. G/Ad ≅ T/W and 𝔊/Ad ≅ 𝔥/W, it is rather natural to consider the Weyl transformation groups (W, T) and (W, 𝔥) as the “vital core” of the geometry of non-commutativity of G. On the one hand, they are farreaching simplifications of the adjoint actions of G on both G and 𝔊, and yet on the other hand, they retain the vital point of the orbit structures of the original adjoint transformations which is actually the geometric version of the totality of the non-commutativity of G. It is an added blessing that (W, 𝔥) are generated by reflections, namely, Coxeter groups. This naturally makes the basic geometry of Coxeter groups to become an important component of the structural theory of Lie groups.

A Lie algebra 𝔊 over ℝ is called a compact Lie algebra if it can be realized as the Lie algebra of a compact Lie group G. Let us analyze the algebraic implications of the above rather geometric definition in order to obtain algebraic characterization of compact Lie algebras.

Let G be a given compact connected Lie group and 𝔊 be its Lie algebra. Then, by Theorem 5.1, 𝔊 splits into the direct sum of its center and its simple ideals, namely

In this lecture, we shall study the basic structural theory of Lie algebras in general, namely, beyond the rather special case of compact Lie algebras. For example, the set of linear transformations of a vector space V over 𝕜 constitutes a Lie algebra with [A,B] = AB − BA, denoted by 𝔤𝔩(V) or 𝔤𝔩(n, 𝕜), while a sub-Lie algebra 𝔤 ⊂ 𝔤𝔩(V) will be regarded as a linear representation of 𝔤 on V. In the special, but of basic importance, case of k = ℂ, one has the well-known theorem of Jordan canonical form and Jordan decomposition for a single linear transformation A (cf. Section 1.1). It is natural to seek ways of generalizing such a wonderful result beyond a single linear transformation, thus asking what would be a proper setting for such a generalization? As one shall find out in the discussion of this lecture, representations of Lie algebras will provide an advantageous setting for such a task and it naturally leads to the basic structure theory of Lie algebras.

Historically, the classification theory of complex semi-simple Lie algebras is the monumental contribution of W. Killing, achieved at a very early stage of such studies. It has been traditionally the important core part of the theory of Lie groups and Lie algebras. In retrospect, the final result of his classification can be concisely restated as follows:

Theorem 1 (Killing). A complex semi-simple Lie-algebra g is always the complexification of a compact semi-simple Lie algebra, namely,

Historically, in the development of the theory of Lie algebras, the great work of W. Killing on the classification of complex semi-simple Lie algebras was soon followed up by É. Cartan, on the classification theory of real semi-simple Lie algebras. On the other hand, in the study of Riemannian geometry, spaces of constant sectional curvature are outstanding important examples which are characterized by the symmetry property that the local isometry groups, ISO(Mⁿ, pt), are of the maximal possibility of O(n) everywhere. In the case of simply connected ones, they are exactly those reflectionally symmetric spaces Eⁿ, Sⁿ, and Hⁿ. A far-reaching generalization will be those spaces which are centrally symmetric with respect to every point p ∈ Mⁿ, namely, ISO(Mⁿ, pt) ⊃ {±Id}, nowadays simply referred to as symmetric spaces. Such a seemingly rather weak symmetry property everywhere turns out to be already extremely restrictive and can be effectively studied via Lie group theory. In fact, such symmetric spaces can be classified and the classification theory of symmetric spaces is intimately correlated with that of real semi-simple Lie algebras. This is the great contribution of É. Cartan, one of the most wonderful performances of geometric and algebraic theories developing hand in hand, almost like a beautiful musical duet…