Lie Groups

Abstract

Sophus Lie (1842–1899) and Felix Klein (1849–1925) were students together in Berlin in 1869–70 when they conceived the notion of studying mathematical systems from the perspective of the transformation groups which left these systems invariant. Thus Klein, in his famous Erlanger program, pursued the role of finite groups in the studies of regular bodies and the theory of algebraic equations, while Lie developed his notion of continuous transformation groups and their role in the theory of differential equations. Lie’s work was a tour de force of the 19th century, and today the theory of continuous groups is a fundamental tool in such diverse areas as analysis, differential geometry, number theory, differential equations, atomic structure, and high energy physics. This book is devoted to a careful exposition of the mathematical foundations of Lie groups and algebras and a sampling of their applications in differential equations, applied mathematics, and physics.