Edited by Gerald

The elementary theory of Lie1 groups has something in common with precalculus: One wants to get past it in order to get on with the beautiful and powerful theory that follows, as well as the applications. With precalculus the theory that follows is built around calculus and its fundamental theorem, extending into calculus of several variables and having profound applications through the subject of differential equations. With Lie groups the theory that follows is due to Élie Cartan (1869–1951) and Hermann Weyl (1885–1955) and concerns compact Lie groups and their representations, as well as real and complex semisimple Lie algebras and Lie groups; the Cartan–Weyl theory introduces one to the exceptional Lie algebras and their remarkable manifestations throughout mathematics, it extends via the work of Harish-Chandra (1923–1983) and others into representation theory and harmonic analysis, and it has applications in many branches of mathematics. Another thing in common for precalculus and elementary Lie theory is that the beautiful and powerful mathematics that follows—calculus in the first case and Cartan–Weyl theory in the second case—brings together several branches of mathematics, serving as a reminder that mathematics is a unifying force rather than a springboard to greater and greater specialization. In fact, because of its great beauty, its wide applicability, and its unifying effect, Cartan–Weyl theory is on my own personal list of What Every Young Mathematician Should Know. If Cartan–Weyl theory is to be accessible to every young mathematician, one has to include it earlier in the curriculum than is done at present in the United States, where it might occur as an elective for second-year graduate students. In turn, some provision has to be made to teach elementary Lie theory earlier and more rapidly than now, with fewer prerequisites. A number of people who share my sentiments have given considerable thought to how this miracle might be accomplished, and the two books under review are the authors’ contributions to this effort. Both books purport to be for “undergraduates,” and we return to them in a moment. There is another difficulty with teaching elementary Lie theory at its current normal pace. For a point of reference, consider the effect of lingering too long over precalculus. We know what happens. Large numbers of students end up with a feeling of