# The Lie Theory of Connected Pro-Lie Groups: A Structure Theory for Pro-Lie Algebras, Pro-Lie Groups, and Connected Locally Compact Groups

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*Karl H. Hofmann; Sidney A. Morris*

A publication of the European Mathematical Society

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 the
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 had 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.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Advanced graduate students interested in pro-Lie groups.