This volume studies algebraic systems where associativity does not hold in general.
This volume studies algebraic systems where associativity does not hold in general. It includes Lie algebras, Jordan algebras, alternative algebras, and related structures. These systems arise naturally in geometry, physics, and symmetry theory.
Part I. Foundations
Chapter 1. Non-Associative Structures
1.1 Definitions and examples 1.2 Binary operations without associativity 1.3 Identities and laws 1.4 Homomorphisms 1.5 Subalgebras and quotients
Chapter 2. Basic Identities
2.1 Associator and commutator 2.2 Flexible and power-associative laws 2.3 Alternative identities 2.4 Jacobi identity 2.5 Examples
Chapter 3. Construction Methods
3.1 Free non-associative algebras 3.2 Quotients by identities 3.3 Universal constructions 3.4 Enveloping structures 3.5 Examples
Part II. Lie Algebras
Chapter 4. Lie Algebras
4.1 Definition and examples 4.2 Lie brackets 4.3 Subalgebras and ideals 4.4 Homomorphisms 4.5 Basic properties
Chapter 5. Structure Theory
5.1 Solvable and nilpotent algebras 5.2 Derived and lower central series 5.3 Semisimple Lie algebras 5.4 Levi decomposition 5.5 Examples
Chapter 6. Representations of Lie Algebras
6.1 Modules over Lie algebras 6.2 Irreducible representations 6.3 Highest weight theory (overview) 6.4 Universal enveloping algebra 6.5 Applications
Part III. Jordan Algebras
Chapter 7. Jordan Algebras
7.1 Definition and motivation 7.2 Commutative but non-associative structures 7.3 Special and exceptional Jordan algebras 7.4 Identities 7.5 Examples
Chapter 8. Structure Theory
8.1 Ideals and simplicity 8.2 Representations 8.3 Connections to associative algebras 8.4 Quadratic representations 8.5 Applications
Chapter 9. Applications
9.1 Quantum mechanics 9.2 Symmetric cones 9.3 Geometry connections 9.4 Optimization 9.5 Examples
Part IV. Alternative and Related Algebras
Chapter 10. Alternative Algebras
10.1 Definitions 10.2 Associativity constraints 10.3 Moufang identities 10.4 Examples 10.5 Properties
Chapter 11. Composition Algebras
11.1 Normed algebras 11.2 Real division algebras 11.3 Octonions 11.4 Classification results 11.5 Applications
Chapter 12. Other Non-Associative Systems
12.1 Quasigroups and loops 12.2 Non-associative division structures 12.3 Malcev algebras (overview) 12.4 General constructions 12.5 Examples
Part V. Representation and Cohomology
Chapter 13. Representations
13.1 Modules over non-associative algebras 13.2 Structure of representations 13.3 Decomposition 13.4 Examples 13.5 Applications
Chapter 14. Cohomology
14.1 Definitions 14.2 Extensions 14.3 Deformation theory 14.4 Applications 14.5 Examples
Chapter 15. Deformations
15.1 Formal deformations 15.2 Rigidity 15.3 Deformation quantization (overview) 15.4 Applications 15.5 Examples
Part VI. Connections and Applications
Chapter 16. Geometry and Lie Theory
16.1 Lie groups and Lie algebras 16.2 Differential geometry connections 16.3 Symmetry and transformations 16.4 Applications 16.5 Examples
Chapter 17. Physics Applications
17.1 Quantum mechanics 17.2 Particle physics symmetries 17.3 Gauge theories 17.4 String theory (overview) 17.5 Examples
Chapter 18. Algebraic and Computational Aspects
18.1 Algorithmic methods 18.2 Computation with non-associative structures 18.3 Symbolic systems 18.4 Complexity issues 18.5 Applications
Part VII. Advanced Topics
Chapter 19. Infinite-Dimensional Structures
19.1 Infinite-dimensional Lie algebras 19.2 Kac–Moody algebras (overview) 19.3 Representations 19.4 Applications 19.5 Examples
Chapter 20. Exceptional Structures
20.1 Exceptional Lie algebras 20.2 Exceptional Jordan algebras 20.3 Classification results 20.4 Connections 20.5 Applications
Chapter 21. Non-Associative Geometry
21.1 Geometric interpretations 21.2 Non-associative spaces 21.3 Connections to topology 21.4 Applications 21.5 Emerging areas
Part VIII. Research Directions
Chapter 22. Open Problems
22.1 Classification challenges 22.2 Representation theory gaps 22.3 Cohomology questions 22.4 Computational limits 22.5 Future directions
Chapter 23. Emerging Areas
23.1 Quantum algebra 23.2 Higher algebraic structures 23.3 Applications in data science 23.4 Interdisciplinary links 23.5 Trends
Chapter 24. Historical and Conceptual Notes
24.1 Development of non-associative algebra 24.2 Key contributors 24.3 Evolution of ideas 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Common identities and relations B. Standard examples reference C. Proof techniques checklist D. Algorithm templates E. Cross-reference to other MSC branches
This volume develops algebra beyond associativity. It emphasizes symmetry, structure, and applications in geometry and physics.