دانلود کتاب The structure of Lie groups

The organization of this book is as follows. Chapter I contains the
generalities concerning locally compact topological groups, and in particular
the existence and uniqueness of the Haar integral. It also contains a few
basic facts from general topology, such as the Stone-Weierstrass Theorem.
Chapter II contains the general theory of compact groups and their
representations. The most important results here are the Peter-Weyl
Theorem and the Tannaka Duality Theorem. Chapter III deals with a
number of basic results on the structure of locally compact groups, chiefly
abelian groups, that can be proved by elementary methods. Chapter IV
develops the topological theory of covering spaces and covering groups,
which plays a dominant role in Lie group theory. All the classical analysis
that is used later on is given in Chapter V. Chapter VI introduces analytic
manifolds. No more than the basic definitions and the main facts con¬
cerning analytic maps is given here, and no more than this is used later on.
Lie group theory proper begins with Chapter VII. The Lie algebra of
an analytic group is defined, and the main properties of the exponential
map of the Lie algebra into the group are established. Chapter VIII deals
with the closed subgroups of analytic groups, homogeneous spaces, and
factor groups. The main results are that a closed subgroup of a Lie group
is a Lie group and that the factor group of an analytic group by a closed
normal subgroup is an analytic group. Chapter IX establishes the
technique of semidirect product constructions, which is used extensively
later on. The bulk of Chapter X is the purely algebraic development of
the Campbell-Hausdorff formula which is basic for the subsequent deeper
analysis of the exponential map in the theory of analytic groups. From
this, the theory of analytic subgroups is then readily obtained at the end of
the chapter. At this stage, the road is cleared for the applications of Lie
algebra theory to the theory of analytic groups.

Chapter XI is the first instalment of pure Lie algebra theory, sufficient
for the first serious application, which is made in Chapter XII. The main
result here is that every finite-dimensional real Lie algebra is the Lie
algebra of an analytic group. The proof is based on semi-direct product
constructions, and yields topological results concerning normal analytic
subgroups of simply connected analytic groups, which play an important
role in the structure and representation theory. Chapter XIII deals with
the general structure theory of compact analytic groups. Chapter XIV
brings Lie algebra theory to the stage required for the rest of this book.
The main result concerns the existence of real forms of compact type for
the semisimple complex Lie algebras. This is applied in Chapter XV in
order to obtain the basic results concerning the maximal compact subgroups
of Lie groups with finite component groups. The principal concern of
Chapter XVI is Malcev’s criterion for an analytic subgroup of an analytic
group to be closed. This requires results on the centers of analytic groups
and on closures of analytic subgroups that are of independent interest.
Chapter XVII introduces complex analytic groups and the universal
complexifications of real analytic groups; in particular, it characterizes the
universal complexifications of the compact analytic groups in representation-
theoretical terms. Finally, Chapter XVIII gives the main results con¬
cerning the existence of faithful representations of analytic groups and the
structure of linear analytic groups.