Lie Algebras

Lie group theory developed by M. Sophus Lie in the nineteenth century ranks among the more important developments in modern mathematics. Lie algebras comprise a significant part of Lie group theory and are being actively studied today. This book by Professor Nathan Jacobson of Yale is the definitive treatment of the subject and can be used as a text for graduate courses.<br>Chapter 1 introduces basic concepts that are necessary for an understanding of structure theory while the following three chapters present the theory itself: solvable and nilpotent Lie algebras Cartan's criterion and its consequences and split semi-simple Lie algebras. Chapter 5 on universal enveloping algebras provides the abstract concepts underlying representation theory. The basic results on representation theory are given in three succeeding chapters: the theorem of Ado-Iwasawa classification of irreducible modules and characters of the irreducible modules. In Chapter 9 the automorphisms of semi-simple Lie algebras over an algebraically closed field of characteristic zero are determined. These results are applied in Chapter 10 to the problems of sorting out the simple Lie algebras over an arbitrary field. The reader to fully benefit from this tenth chapter should have some knowledge about the notions of Galois theory and some of the results of the Wedderburn structure theory of associative algebras.<br>Nathan Jacobson presently Henry Ford II Professor of Mathematics at Yale University is a well-known authority in the field of abstract algebra. His book <i>Lie Algebras </i> is a classic handbook both for researchers and students. Though it presupposes knowledge of linear algebra it is not overly theoretical and can be readily used for self-study.