# Book Details

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Preface | Table of Contents | Panoramic Overview | MARC record | Metadata XML | e-Book PDF (4128 KB)*Karl H. Hofmann (TU Darmstadt, Germany)*

Sidney A. Morris (University of Ballarat, Australia)

Sidney A. Morris (University of Ballarat, Australia)

#### The Lie Theory of Connected Pro-Lie Groups

A Structure Theory for Pro-Lie Algebras, Pro-Lie Groups, and Connected Locally Compact GroupsISBN print 978-3-03719-032-6, ISBN online 978-3-03719-532-1

DOI 10.4171/032

May 2007, 693 pages, hardcover, 17.0 x 24.0 cm.

88.00 Euro

Lie groups were introduced in 1870 by the Norwegian mathematician Sophus Lie. A century later Jean Dieudonné quipped that Lie groups had moved to the center of mathematics and that one cannot undertake anything without them.

If a complete topological group `G` can be approximated by Lie groups in
the sense that every identity neighborhood `U` of `G`
contains a
normal subgroup `N` such that `G/N` is a Lie group,
then it is called a *pro-Lie group*.
Every locally compact connected topological group and every
compact group is a pro-Lie group.
While the class of locally compact groups is not closed under the
formation
of arbitrary products, the class of pro-Lie groups is.

For half a century, locally compact pro-Lie groups have drifted
through the literature, yet this is the first book which
systematically treats the Lie and structure theory of pro-Lie groups
irrespective of local compactness. This study fits very well into
that current trend which addresses infinite dimensional Lie groups.
The results of this text are based on a theory of pro-Lie algebras
which parallels the structure theory of finite dimensional real Lie
algebras to an astonishing degree even though it has to overcome
greater technical obstacles.

This book exposes a Lie theory of connected pro-Lie groups (and hence of connected locally compact groups) and illuminates the manifold ways in which their structure theory reduces to that of compact groups on the one hand and of finite dimensional Lie groups on the other. It is a continuation of the authors' fundamental monograph on the structure of compact groups (1998, 2006), and is an invaluable tool for researchers in topological groups, Lie theory, harmonic analysis and representation theory. It is written to be accessible to advanced graduate students wishing to study this fascinating and important area of current research, which has so many fruitful interactions with other fields of mathematics.

*Keywords: *Pro-Lie groups, pro-Lie algebras, Lie theory of connected pro-Lie groups, exponential function, structure theory of locally compact groups, completeness, quotient groups, open mapping theorem, Levi–Malcev splitting, local Iwasawa splitting

#### Further Information

Review in Zentralblatt MATH 1153.22006