This famous book was the first treatise on Lie groups in which a modern
point of view was adopted systematically, namely, that a continuous
group can be regarded as a global object. To develop this idea to its
fullest extent, Chevalley incorporated a broad range of topics, such as
the covering spaces of topological spaces, analytic manifolds,
integration of complete systems of differential equations on a manifold,
and the calculus of exterior differential forms.
The book opens with a short description of the classical groups: unitary
groups, orthogonal groups, symplectic groups, etc. These special groups
are then used to illustrate the general properties of Lie groups, which
are considered later. The general notion of a Lie group is defined and
correlated with the algebraic notion of a Lie algebra; the subgroups,
factor groups, and homomorphisms of Lie groups are studied by making use
of the Lie algebra. The last chapter is concerned with the theory of
compact groups, culminating in Peter-Weyl's theorem on the existence of
representations. Given a compact group, it is shown how one can
construct algebraically the corresponding Lie group with complex
parameters which appears in the form of a certain algebraic variety
(associated algebraic group). This construction is intimately related to
the proof of the generalization given by Tannaka of Pontrjagin's duality
theorem for Abelian groups.
The continued importance of Lie groups in mathematics and theoretical
physics make this an indispensable volume for researchers in both
fields.