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

The primary focus of this book is on pro-Lie groups, a class of topological groups that encompass many prominent examples such as locally compact groups and compact groups. The aim of initiating a structure theory for these groups opens up pathways for advanced understanding in both mathematics and theoretical physics. This book meticulously elaborates on pro-Lie algebras, demonstrating their equivalence to finite-dimensional Lie algebras within this context—a revelation that simplifies the complexities inherently associated with infinite dimensions.

Across its chapters, the book offers insights into the sophisticated structure and behavioral aspects of connected pro-Lie groups. It presents innovative techniques in handling infinite-dimensional groups, extending the essence of Lie group theory beyond its traditional boundaries. The authors skillfully incorporate an array of new mathematical methods, bridging classical analysis and topology to underscore the continuity between these domains.

• A comprehensive structure theory for pro-Lie algebras and pro-Lie groups.
• Innovative techniques in dealing with infinite-dimensional groups.
• Bridges existing gaps between classical analysis, topology, and advanced Lie theory.
• Insight into the equivalence of pro-Lie algebras and finite-dimensional Lie algebras.
• Applicability of the developed theories in both mathematical and physical contexts.

"Understanding pro-Lie groups requires a rethinking of infinite dimensionality as more than a mere extension of the finite, but as a space where new meanings and structures unveil."

"The alignment of classical topological groups with the modern theories of pro-Lie groups exemplifies the fusion of old methodologies with new perspectives."