Book-Review Abstract

Anthony W. Knapp

Abstract of review of books by Andrew Baker and Wulf Rossmann

The review underscores the need to have elementary Lie theory be learned earlier so that Cartan-Weyl theory can be part of What Every Young Mathematician Should Know. It describes elementary Lie theory in terms of two ingredients (Lie groups and Lie algebras) and three correspondences, and it summarizes the content of Cartan-Weyl theory. Next it discusses and compares some earlier books that develop some theory for linear groups before going on to general Lie groups.

Finally it discusses the two books under review. Baker's book gets bad marks. Rossmann's book gets high praise because of its development for linear groups of all of elementary Lie theory in just 90 pages, using rather minimal prerequisites. Rossmann has found and included a way to treat analytic subgroups without the Frobenius theorem, as well as a way of passing from homomorphisms of Lie algebras to homomorphisms of linear groups without simple connectivity and without the use of local groups and local homomorphisms. The main proofs in Rossmann's book appear to be correct, apart from misprints.