This volume studies groups equipped with topology and smooth structure.
This volume studies groups equipped with topology and smooth structure. It connects algebra, topology, and geometry, with applications in analysis and physics.
Part I. Topological Groups
Chapter 1. Topological Groups
1.1 Definitions and examples 1.2 Continuity of multiplication and inversion 1.3 Subgroups and quotient groups 1.4 Homomorphisms 1.5 Basic properties
Chapter 2. Fundamental Properties
2.1 Compactness 2.2 Connectedness 2.3 Local compactness 2.4 Separation axioms 2.5 Examples
Chapter 3. Haar Measure
3.1 Invariant measures 3.2 Existence and uniqueness 3.3 Integration on groups 3.4 Applications 3.5 Examples
Part II. Lie Groups
Chapter 4. Smooth Manifolds
4.1 Differentiable manifolds 4.2 Charts and atlases 4.3 Smooth maps 4.4 Tangent spaces 4.5 Examples
Chapter 5. Lie Groups
5.1 Definition 5.2 Matrix Lie groups 5.3 Examples 5.4 Subgroups and quotients 5.5 Basic properties
Chapter 6. Lie Algebras
6.1 Tangent space at identity 6.2 Lie bracket 6.3 Exponential map 6.4 Examples 6.5 Structure
Part III. Structure Theory
Chapter 7. Subgroups and Homomorphisms
7.1 Closed subgroups 7.2 Lie subgroups 7.3 Homomorphisms 7.4 Quotient groups 7.5 Applications
Chapter 8. Representations of Lie Groups
8.1 Linear representations 8.2 Irreducible representations 8.3 Examples 8.4 Applications 8.5 Connections
Chapter 9. Compact Lie Groups
9.1 Structure theory 9.2 Maximal tori 9.3 Weyl groups (overview) 9.4 Classification (overview) 9.5 Applications
Part IV. Advanced Lie Theory
Chapter 10. Semisimple Lie Groups
10.1 Definitions 10.2 Lie algebra decomposition 10.3 Root systems (overview) 10.4 Cartan subalgebras 10.5 Applications
Chapter 11. Representation Theory
11.1 Highest weight theory (overview) 11.2 Characters 11.3 Induced representations 11.4 Harmonic analysis 11.5 Applications
Chapter 12. Symmetric Spaces
12.1 Definitions 12.2 Examples 12.3 Classification (overview) 12.4 Geometric properties 12.5 Applications
Part V. Topological and Analytical Methods
Chapter 13. Harmonic Analysis on Groups
13.1 Fourier analysis on groups 13.2 Representations and decomposition 13.3 Applications 13.4 Examples 13.5 Connections
Chapter 14. Fiber Bundles and Connections
14.1 Principal bundles 14.2 Associated bundles 14.3 Connections and curvature 14.4 Gauge theory (overview) 14.5 Applications
Chapter 15. Lie Group Actions
15.1 Group actions on manifolds 15.2 Orbits and stabilizers 15.3 Homogeneous spaces 15.4 Applications 15.5 Examples
Part VI. Infinite-Dimensional and Generalizations
Chapter 16. Infinite-Dimensional Lie Groups
16.1 Definitions 16.2 Examples 16.3 Functional analytic aspects 16.4 Applications 16.5 Challenges
Chapter 17. Algebraic Groups (Overview)
17.1 Definitions 17.2 Examples 17.3 Structure theory 17.4 Applications 17.5 Connections
Chapter 18. Quantum Groups (Overview)
18.1 Basic concepts 18.2 Deformations 18.3 Representations 18.4 Applications 18.5 Connections
Part VII. Applications
Chapter 19. Physics Applications
19.1 Symmetry in mechanics 19.2 Quantum mechanics 19.3 Gauge theories 19.4 Relativity 19.5 Applications
Chapter 20. Geometry Applications
20.1 Transformation groups 20.2 Homogeneous geometry 20.3 Differential geometry links 20.4 Applications 20.5 Examples
Chapter 21. Computational Aspects
21.1 Numerical methods 21.2 Symbolic computation 21.3 Representation algorithms 21.4 Software tools 21.5 Applications
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Representation theory developments 22.2 Geometric analysis 22.3 Noncommutative geometry 22.4 Topological methods 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Classification challenges 23.2 Representation theory gaps 23.3 Analytical difficulties 23.4 Computational challenges 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of Lie theory 24.2 Key contributors 24.3 Evolution of ideas 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Standard Lie groups reference B. Key theorems summary C. Proof techniques checklist D. Computational tools E. Cross-reference to other MSC branches
This volume develops topological and Lie groups as the mathematics of continuous symmetry. It connects algebra, geometry, and analysis through smooth and topological structure.