Lie algebras and applications

This book is not a manual in the usual sense, but a compilation of facts concerning Lie algebras that continuously appear in physical problems. The material covered is the result of various seminars given by the author during many years, and synthetize the main facts that should be known to any physicist. The material is divided into 12 chapters of variable length. The first two present the main theory of semisimple Lie algebras, enumerating the key results from root theory and Dynkin-Coxeter diagrams to classify the complex simple algebras. The real forms for the classical algebras are given in table form, without going into its detailed obtainment. It should also be taken into account that for Dynkin diagrams, the author does not distinguish between positively and negatively oriented angles, thus the angles between roots in equation (2.8) are reduced to five (unoriented) angles instead of the usual eight (oriented) angles. Chapter three compiles the most important facts about Lie algebras of Lie groups, mainly focused on matrix groups. Important techniques like the exponential map and the covering of groups are nicely illustrated with the classical unitary algebra su(2) and the Lorentz group (in one dimension). I personally miss some comment on the left invariant vector fields or 1-forms (Maurer-Cartan equations), of importance in many applications to cosmology. The fourth chapter is devoted to representation theory. Although the Weyl decomposition theorem is not included, it is assumed that any representation decomposes as a direct sum of irreducible modules (valid for semisimple Lie algebras). The fundamental representations are discussed for the classical algebras (symplectic, unitary and orthogonal), and for the latter, the spinor representations are also given. The dimension formulae are given, and the tensor products (Clebsch-Gordan problem) is developed by means of Young tableaux. This is applied to the branching rules of representations with respect to some chain and the missing label problem, illustrated by examples that are typical in the interacting boson model. In chapter five, Casimir operators of Lie algebras are defined and obtained for the classical Lie algebras. Here the author uses the Perelomov-Popov approach of operators that can be identified with symmetric elements in the universal enveloping algebra. At the beginning of the chapter it is said that the number of Casimir operators equals the rank of the algebra. Again, this is only valid for semisimple Lie algebras, and generally false for arbitrary Lie algebras. The eigenvalue problem is presented using important examples, and the results resumed in a table at the end of the chapter. The previous chapter is a nice motivation for tensor operators in general, which comprise essential techniques like the coupling and recoupling coefficients, how to determine them and their symmetries (much of this material was originally developed by Racah in his Princeton lectures of 1951). This chapter is of great importance for applications. Chapters 8 and 9 are devoted to another technique of great relevance, the realizations of Lie algebras by means of creation and annihilation operators, divided into boson and fermion operators, according to commutation or anticommutation relations. Here the unitary case is exploited, and many subalgebra chains are analyzed with respect to these realizations. Of special interest are the sections concerning the L-S and j-j couplings used in spectroscopy of light nuclei and shell models, and where original examples have been used. Chapter 9 presents another possibility for realizing Lie algebras, namely by differential operators. Although a short chapter, important topics like the Casimir operators as differential operators or the Laplace-Beltrami form is presented. In chapter 10, the classical matrix realizations (in fact representations by linear operators) are briefly recalled, and the classical interpretation of the Casimir operators is recovered (without using the Schur lemma). The two last chapters deal with quite more specific topics, like dynamic symmetries, studied in both fermionic and bosonic systems, in the unitary algebras u(6) and u(4), in order to obtain mass and energy level diagrams. For the part of degeneracy algebras, the problems illustrated are the isotropic harmonic oscillator, the Coulomb problem and the Teller-Pöschl and Morse potentials. In all these problems the reader is referred to original articles to complete the information presented. The chapters of the book do not develop the theory systematically, but rather focus on a type of problem or technique which is developed using the main Lie algebras appearing (mainly) in spectroscopy, atomic, nuclear and molecular physics, as well as quantum mechanics. No proofs are given, which prevents the reader from being distracted from the main objective of the lectures. To fill the gaps, the reader is led, at many places, to consult either original references or more formal books. The book is written in an informal style, which simplifies its reading and makes it a suitable consultation work. The profusion of examples (many of them actually coming from original references) explains quite well the topics studied, and gives a concrete idea how to apply the techniques. It is a very welcomed addition to the literature that contains much topics treated for the first time in textbook form. There are few misprints and mistakes in the text, which can however confuse the reader having no previous knowledge on Lie algebras. For example, on page 7, the definition of semidirect sum is confusing and wrong. It is actually not necessary that one of the algebras is an ideal in the other, it suffices that one of them acts by derivations on the other. The definition given in the book is incompatible with example 13 on the same page. Another confusing point is subsection 1.13. Here by derivations the author means the derived series of an algebra, determining whether it is solvable or not. Derivations are linear maps satisfying the Leibniz rule, and are completely independent on the solvable character. The notation is very confusing, since the derived subalgebra (commutator ideal) is denoted in the same manner as the Lie algebra of derivations (which is actually a linear Lie algebra). Inspite of these minor details, the book will certainly be of great use for students or specialists that want to refresh their knowledge on Lie algebras applied to physics. The list of references is quite complete and provides a deeper insight into the problems where these structures appear. However, there are also some surprising absences in the references, such as the books of J. F. Cornwell or H. Lipkin, in my opinion two classicals on group theory in physics. Among the original articles, I miss for example the relevant review article by R. Slansky [Phys. Rep. 79 (1981), 1-128], although it is clear that giving a complete reference list is impossible. Resuming, the book by Iachello constitutes an excellent reference for those interested in the practical application and techniques of Lie algebras to physics, and that try to avoid the often embarrassing theoretical works. It should also be mentioned that much of the material is divided into hundreds of original articles, and therefore a unified presentation will be of great use for the physical community.