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**Lie groups, Lie algebras, and representations. An elementary introduction.**
*(English)*
Zbl 1026.22001

Graduate Texts in Mathematics. 222. New York, NY: Springer. xiv, 351 p. (2003).

The last three or four decades have witnessed a veritable explosion of applications of Lie group and Lie algebra techniques to a wide range of physically important problems. Though there exist already several excellent text books providing the mathematical basis for all this, introductions aimed at graduate students both in mathematics and physics seem to be rare. So the guiding principle in the planning of the book by Brian Hall, a mathematician at the University of Notre Dame, was to minimize the amount of prerequisites. He avoids starting with manifold theory and left-invariant vector fields which is normally considered the correct approach to the subject. His way out of the dilemma is to restrict attention to matrix groups though it is known that not every Lie group is of this form. Moreover, the book treats the representation theory of SU(2) and SU(3) first before going to the general case. It has two major parts. Part I, called general theory, deals with the definition of matrix Lie groups and their properties (compactness, connectedness etc.), Lie algebras and the exponential mapping, the Baker-Campbell-Hausdorff formula and basic elements of unitary representation theory. Part II, called semisimple theory, deals first with the Cartan subalgebra, weights, roots and the Weyl group for SU(3), and then passes on to the general situation of semisimple Lie algebras and their representations via Verma modules. The classification of root systems is presented in terms of Dynkin diagrams. There are five appendices providing the necessary mathematical prerequisites plus some additional material (e.g. fundamental group). Physicists working in quantum mechanics, field theory or string theory might miss a detailed discussion of projective representations, homogeneous spaces, representations on Hilbert spaces, the Wigner classification of irreducible representations of the PoincarĂ© group, and more information about exceptional Lie groups (most importantly \(E_8)\). Still no doubt, students will benefit from the way the material is presented in this Introduction; for it is elementary and not intimidating, at the same time very systematic, rigorous and modern, hence a nice addition to the Springer series Graduate texts in mathematics.

Reviewer: G.Roepstorff (Aachen)

### MSC:

22-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups |

22E60 | Lie algebras of Lie groups |

17-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to nonassociative rings and algebras |