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Title: Semi-Simple Lie Algebras and Their Representations by Robert N. Cahn ISBN: 0-8053-1600-0 Publisher: Pearson Benjamin Cummings Pub. Date: June, 1984 Format: Hardcover Volumes: 1 List Price(USD): $33.75

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Average Customer Rating: 4.33 (3 reviews)

Rating: 4
Summary: A practical guide to Lie algebras and representations
Comment: The objective of this book is to provide a readable synthesis of the theory of (complex) semisimple Lie algebras and their representations which are usually needed in physics. There is no attempt to develop the theory formally, as done in usual textbooks on Lie algebras, but to present the material motivated by the rotation group SU(2), and also SU(3). The book is divided into sixteen sections. The first ten give a brief overview of the classification of semisimple algebras and their representations. For the proofs the reader is referred to the book of Jacobson [Lie algebras, Wiley 1962]. The purpose of this presentation is to introduce the concepts like Killing form, weights, root system, etc, using the examples of the two groups cited above, and then give the general description. Technical results are kept to a minimum, which causes a couple of omissions which are however used in later chapters [this is the case for the decomposition of any positive root as a sum of simple roots with integer positive coefficients]. The eleventh chapter introduces the Casimir operators of Lie algebras (more precisely the quadratic Casimir operator) and Freudenthal's formula for the dimension of weights spaces. In chapter 12 the Weyl group of a root system is discussed (but without commenting the Weyl chambers). Chapter XIII presents Weyl's formula for the dimension of irreducible representations, and illustrated with examples like sl(3) or the exceptional algebra of rank two. Chapter XIV begins with topics usually encountered in physical applications, like the decomposition of the tensor product of two irreducible representations. This and later chapters are strongly influenced by Dynkin's original work. In particular the theorem for the second highest representation is developed in detail. The last two chapters are devoted to the analysis of subalgebras of semisimple Lie algebras and the branching rules (i.e., decomposition of representations with respect to a certain subalgebra). The method based on the extension of the Dynkin diagrams is carefully developed, and the question of maximality of the subalgebra (regular or not) discussed. Here an extremely important observation is made, namely the existence of some little mistakes in the Dynkin's method (concerning the maximality of certain subalgebras in the exceptional case). This is pointed out with explicit exhibition of examples. The last chapter gives an insight into the branching rules, by the development of carefully chosen examples and the presentation of some results (without proof) due also to Dynkin.
Resuming, this book provides a quick introduction to the techniques and features of (finite dimensional) Lie algebras appearing in physical theories (e.g. the interacting boson model) without being forced to digest a formal mathematical development. Inspite of few points where the reader can get puzzled (due to the use of noncommented general properties), the text achieves its purpose and constitutes a valuable reference for physicists.

Rating: 4
Summary: A pleasant read
Comment: Not only are Lie algebras interesting and important from a mathematical standpoint, an in-depth understanding of them is essential if one is to fully comprehend the physical theories of elementary particle interactions. All of these theories, from quantum field theories to string theories, to the current research on D-branes and M-theories, are dependent on the theory of Lie groups and Lie algebras. Because of its relaxed informal style, this book would be a good choice for the physics graduate student who intends to specialize in high energy physics. Those interested in mathematical rigor would probably want to select another text. Because of space restrictions, only the first thirteen chapters will be reviewed here.

In chapter 1 the author begins the study of SU(2), the group of unitary 2 x 2 matrices of determinant 1. He does this by first considering the matrix representations of infinitesimal rotations in 3-dimenensional space. "Exponentiating" these matrices gives the finite rotational matrices. He then shows that the consideration of products of finite rotations involves knowledge of the commutators of the infinitesimal rotations. Viewing these commutators abstractly motivates the definition of a Lie algebra. He then shows that the rotation matrices form a (3-dimensional) 'representation' of the Lie algebra. Higher-dimensional representations he shows can be obtained by analogies to what is done in quantum mechanics, via the addition of angular momentum and are parametrized by spin (denoted j). The representation of smallest dimension is given by j = 1/2 and corresponds to SU(2). He is careful to point out that the rotations in 3 dimensions and SU(2) have the same Lie algebra but are not the same group.

The constructions in chapter 1, particularly the concept of "exponentiating", are central to the understanding of Lie algebras in general. This is readily apparent in the next chapter wherein he studies the Lie algebra of SU(3), the 3x3 unitary matrices of determinant 1. SU(3) has to rank as one of the most important groups in elementary particle physics. The (abstract) Lie algebra corresponding to the commutation relations of this group have various representations, the 8-dimensional, or "adjoint" representation being one of great interest. The author finds the famous 'Cartan subalgebra' of the Lie algebra, shows that it 2-dimensional and Abelian, and how eigenvectors of the adjoint operator can form a basis for the Lie algebra, as long as this operator corrresponds to an element of the Cartan subalgebra. Further, he shows that the eigenvalues of this operator depend linearly on this element, and then defines functionals on the Cartan subalgebra, called the roots, and they form the dual space to the Lie algebra. Dual spaces are familiar to physicists in the Dirac bra-ket formalism.

The geometry of Lie algebras is very well understood and is formulated in terms of the roots of the algebra and a kind of scalar product (except is not positive definite) for the Lie algebra called the 'Killing form'. The Killing form is defined on the root space, and gives a correspondence between the Cartan subalgebra and its dual. The author then shows how to use the Killing form to obtain a scalar product on the root space, and this scalar product illustrates more clearly the symmetry of the Lie algebra. The property of being semisimple is then defined abstractly by the author, namely a Lie algebra with no Abelian ideals. He states, but does not prove entirely, that the Killing form is non-degenerate if and only if the Lie algebra is semisimple.

The treatment becomes more abstract in chapter 4, wherein the author studies the structure of simple Lie algebras, since every semisimple algebra can be written as the sum of simple Lie algebras. The author shows how to obtain the Cartan subalgebra in general, motivating his procedures with what is done for SU(3). He also proves the invariance of the Lie algebra and shows that it is the only invariant bilinear form on a simple Lie algebra. After a detour on properties of representations in chapter 5, wherein he constructs some useful relations for adjoint representations, the author uses these to again study the structure of simple Lie algebras in chapters 6 and 7. This involves the notion of positive and negative roots, and simple roots, and from the latter the author constructs the 'Cartan matrix', which summarizes all of the properties of the simple Lie algebra to which it corresponds. The author shows how the contents of the Cartan matrix can be summarized in terms of 'Dynkin diagrams'.

These considerations allow an explicit characterization of the 'classical' Lie algebras: SU(n), SO(n), and Sp(2n) in chapter 8. The Dynkin diagrams of these Lie algebras are constructed. Then in chapter 9, the author considers the 'exceptional' Lie algebras, which are the last of the simple Lie algebras (5 in all). Their Dynkin diagrams are also constructed explicitly.

The author returns to representation theory in chapter 10, wherein he introduces the concept of a 'weight'. These come in sequences with successive weights differing by the roots of the Lie algebra. A finite dimensional irreducible representation has a highest weight, and each greatest weight is specified by a set of non-negative integers called 'Dynkin coefficients'. He then shows how to classify representations as 'fundamental' or 'basic', the later being ones where the Dynkin coefficients are all zero except for one entry.

In complete analogy with the theory of angular momenta in quantum mechanics, the author illustrates the role of Casimir operators in chapter 11. Freudenthal's recursion formula, which gives the dimension of the weight space, is used to derive Weyl's formula for the dimension of an irreducible representation in chapter 13. The reader can see clearly the power of the 'Weyl group' in exploiting the symmetries of representations.

Rating: 5
Summary: A nice little summary of the theory
Comment: Very well written account of the theory, with almost all the necessary proofs to get familiar with the it. It's inspired by Jacobson's book, however a lot easier to read. It's out of print, but there is an online copy.