Introduction

Ab initio molecular dynamics has been called a “virtual matter laboratory” [510] and even a “virtual laboratory” [934]. This notion is justified in view of the obvious parallels to experiments, being performed typically in the real laboratory. In the virtual lab, ideally, a system is prepared in some initial state and then evolves according to the basic laws of (quantum-statistical) physics – without the need for experimental input. The trajectory thus generated by ab initio molecular dynamics can be analyzed in arbitrary detail, including the dynamics of the electronic structure, which hence offers deep insights into the occuring processes. As a third step, the initial state and/or the external conditions such as for instance the temperature or composition of the system can be varied, but the system can also be exposed to light, electrical current, hydrostatic pressure, or uniaxial mechanical forces. Thus, not only the investigated matter as such is virtual, but also the apparatus used to manipulate matter, for instance a laser beam or an atomic force microscope, is fully represented “in silico”. It is clear to every practitioner that this viewpoint is highly idealistic for more than one reason, but still this philosophy allows one to compute observables with predictive power and is at the same time the reason behind the broad application range and versatility of ab initio simulations. Furthermore, progress in the general availability of powerful computer hardware makes it easier as time goes on to follow the “virtual lab avenue” to achieve scientific progress.

Properties of roots

In an effort to cast the commutation relations of a semisimple Lie algebra into an eigenvalue-eigenvector format, a secular equation was constructed from the regular representation. The rank of an algebra is, among other things:

1. (i) the number of independent functions in the secular equation;

2. (ii) the number of independent roots of the secular equation;

3. (iii) the number of mutually commuting operators in the Lie algebra;

4. (iv) the number of invariant operators that commute with all elements in the Lie algebra (Casimir operators);

5. (v) the dimension of the positive-definite root space that summarizes the commutation relations.

Why bother?

Two Lie groups are isomorphic if:

1. (i) their underlying manifolds are topologically equivalent;

2. (ii) the functions defining the group composition laws are equivalent.

Two manifolds are topologically equivalent if they can be smoothly deformed into each other. This requires that all their topological indices, such as dimension, Betti numbers, connectivity properties, etc., are equal.

Two group composition laws are equivalent if there is a smooth change of variables that deforms one function into the other.

The program of Lie

Marius Sophus Lie (1842–1899) embarked on a program that is still not complete, even after a century of active work. This program attempts to use the power of the tool called group theory to solve, or at least simplify, ordinary differential equations.

Earlier in nineteenth century, Évariste Galois (1811–1832) had used group theory to solve algebraic (polynomial) equations that were quadratic, cubic, and quartic. In fact, he did more. He was able to prove that no closed form solution could be constructed for the general quintic (or any higher degree) equation using only the four standard operations of arithmetic (+, −, ×, ÷) as well as extraction of the nth roots of a complex number.

Lie initiated his program on the basis of analogy.

Galois inspired Lie. If the discrete invariance group of an algebraic equation could be exploited to generate algorithms to solve the algebraic equation “by radicals,” might it be possible that the continuous invariance group of a differential equation could be exploited to solve the differential equation “by quadratures”? Lie showed emphatically in 1874 that the answer is YES!, and work has hardly slowed down in the field that he pioneered from that time to the present.

Many years ago I wrote the book Lie Groups, Lie Algebras, and Some of Their Applications (New York: Wiley, 1974). That was a big book: long and difficult. Over the course of the years I realized that more than 90% of the most useful material in that book could be presented in less than 10% of the space. This realization was accompanied by a promise that some day I would do just that – rewrite and shrink the book to emphasize the most useful aspects in a way that was easy for students to acquire and to assimilate. The present work is the fruit of this promise.

In carrying out the revision I have created a sandwich. Lie group theory has its intellectual underpinnings in Galois theory. In fact, the original purpose of what we now call Lie group theory was to use continuous groups to solve differential (continuous) equations in the spirit that finite groups had been used to solve algebraic (finite) equations. It is rare that a book dedicated to Lie groups begins with Galois groups and includes a chapter dedicated to the applications of Lie group theory to solving differential equations. This book does just that. The first chapter describes Galois theory, and the last chapter shows how to use Lie theory to solve some ordinary differential equations. The fourteen intermediate chapters describe many of the most important aspects of Lie group theory and provide applications of this beautiful subject to several important areas of physics and geometry.

Objectives of this program

In the previous chapter we studied the commutation relations of a Lie algebra through its regular representation. This study was carried out using as a tool the Cartan–Killing inner product. As far as possible, this was the only method used. In the present chapter we introduce a second powerful tool from the theory of linear vector spaces. This is the eigenvalue decomposition. This tool is introduced in an attempt to find standard forms for the commutation relations. If a standard form is available then the properties of a Lie algebra, as well as its identification (classification), can be determined at sight.

The eigenoperator decomposition is effected by computing and studying a secular equation determined from the matrix of the regular (or any other matrix) representation of the Lie algebra.