
Exotic Algebraic and Geometric Structures in Theoretical Physics
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Content
- Intro
- EXOTIC ALGEBRAIC ANDGEOMETRIC STRUCTURESIN THEORETICAL PHYSICS
- EXOTIC ALGEBRAIC ANDGEOMETRIC STRUCTURESIN THEORETICAL PHYSICS
- Contents
- PREFACE
- ACKNOWLEDGMENTS
- Chapter 1SEMISUPERMANIFOLDS ANDREGULARIZATION
- 1.1. Introduction
- 1.2. Patch Definition of Supermanifolds
- 1.3. Noninvertibility and Semisupermanifolds
- 1.4. n-Regularity and Semicommutative Diagrams
- 1.5. Categories and Regularization
- 1.6. Regularization of Functors and the Yang-BaxterEquation
- 1.7. Regularization of Algebras, Coalgebras,Modulesand Comodules
- 1.8. Conclusion
- Chapter 2HIGHER REGULARITY ANDOBSTRUCTED CATEGORIES
- 2.1. Introduction
- 2.2. NoninvertibleMappings and Morphisms
- 2.3. Invertibility and Regularity of Morphisms
- 2.4. Higher Analogs of Regularity
- 2.5. Semicommutative Diagrams and Obstruction
- 2.6. Monoidal Categories, Functors and Regularity
- 2.7. Higher Regular Yang-Baxter Equation
- Chapter 3OBSTRUCTED CATEGORIES ANDREGULAR GRADED ALGEBRAS
- 3.1. Introduction
- 3.2. Regular Graded Algebras
- 3.3. Regular Odd Graded Algebras
- 3.4. Regular Graded Algebra Representations
- 3.5. Regular Graded Algebra Decomposition
- 3.6. Categories and Obstructions
- 3.7. Regular Algebras and Bialgebras
- 3.8. RegularWick Algebras
- Chapter 4TOPOSES AND CATEGORIES INQUANTUM THEORY AND GRAVITY
- 4.1. Introduction
- 4.2. Toposes and Categories
- 4.3. Toposes in Quantum Theory
- 4.4. Toposes in Quantum Gravity
- 4.5. Continuum in Physical Theory
- 4.6. Alternative Conceptions of Spacetime
- 4.7. Presheaves from Topos Theory
- 4.8. Presheaves in Quantum Theory and Gravity
- 4.9. Presheaves for Topological Quantum Field Theory
- 4.10. Consistent Histories Formalism for QuantumTheory and Continuous Time
- 4.11. Presheaves of Propositions and Valuationsin Quantum Theory
- 4.12. Toposes and the Theory of Relativity
- 4.13. Regular Obstructed Categories and Toposes
- 4.14. Number Systems in Synthetic DifferentialGeometry
- Chapter 5BRAID SEMISTATISTICS AND DOUBLYREGULAR R-MATRIX
- 5.1. Introduction
- 5.2. Braid Semistatistics and the Regular Yang-BaxterEquation
- 5.3. Regular Yang-Baxter Operators
- 5.4. Bialgebras and Universal R-Matrix
- 5.5. Doubly Regular Hopf Algebras
- 5.6. Conclusion
- Chapter 6WEAK HOPF ALGEBRAS ANDYANG-BAXTER EQUATION
- 6.1. Introduction
- 6.2. Weak Quantum Algebras
- 6.3. Weak Ore Extensions
- 6.4. q = 1 Case
- 6.5. Structure of Weak Hopf Algebras
- 6.6. Group-Like Elements
- 6.7. Regular Quasi-R-Matrix
- Chapter 7COMPLETE Uq(sl2)-SYMMETRIES ONQUANTUM PLANE
- 7.1. Introduction
- 7.2. Preliminaries
- 7.3. Automorphisms of Quantum Plane
- 7.4. Structure of Uq(sl2)-Module Algebraon Quantum Plane
- 7.5. Composition Series
- 7.6. Classical Limit
- Chapter 8CODERIVATIONS ANDCODIFFERENTIAL CALCULION FINITE GROUPS
- 8.1. Introduction
- 8.2. Coalgebras and Hopf Algebras
- 8.3. Coderivations and Coalgebras
- 8.4. Coderivations and Comodules
- 8.5. Comodules and Covariant Maps Associated withFinite Groups
- 8.6. Differential Calculi on Finite Groups
- 8.7. Codifferential Calculi on Finite Groups
- 8.8. Conclusion
- Chapter 9SEMIGROUPS OF SUPERMATRICESAND SUPEROPERATORS
- 9.1. Introduction
- 9.2. Supermatrices and Even-Odd Classification
- 9.3. Odd-Reduced Supermatrix Semigroups
- 9.4. One-Parameter Supermatrix Semigroups
- 9.5. Odd-Reduced Supermatrix Operator Semigroups
- 9.6. Cauchy Problem
- 9.7. Superanalog of Resolvent for Exponential andIdempotent Superoperators
- 9.8. Properties of t-Linear Idempotent Superoperators
- 9.9. General t-Power-Type Idempotent Superoperators
- 9.10. Conclusion
- Chapter 10SUPERSYMMETRY AND NILPOTENTMECHANICS
- 10.1. Introduction
- 10.2. Classical Supersymmetry. A Toy Model
- 10.3. Lagrangian Formulation of Nilpotent Mechanics
- 10.4. Generalized Legendre Transformation
- 10.5. Hamiltonian Formulation of NilpotentMechanics
- 10.6. Conclusion
- Chapter 11CONSTRAINTLESS HAMILTONIANFORMALISM AND NEW BRACKETS
- 11.1. Introduction
- 11.2. The Legendre-Fenchel and Legendre Transforms
- 11.3. The Legendre-Clairaut Transform
- 11.4. The Mixed Legendre-Clairaut Transform
- 11.5. Hamiltonian Formulation of Singular LagrangianSystems
- 11.6. Nonabelian Gauge Theory Interpretation
- 11.7. Classification, Gauge Freedom and New Brackets
- 11.8. Singular Lagrangian Systems and Many-Time Dynamics
- 11.9. Multidimensional Clairaut Equation
- 11.10. Correspondence with the Dirac Approach
- 11.11. Conclusion
- Chapter 12GENERALIZED INTERACTION INMULTIGRAVITY
- 12.1. Introduction
- 12.2. Multigravity and the Coincidence Limit
- 12.3. Generalized Invariant Volume inMultigravity
- 12.4. Generalized Interaction Potential in Multigravity
- 12.5. Conclusion
- Chapter 13FERMIONIC LINEAL GRAVITY
- 13.1. Introduction
- 13.2. Central Extension of Poincar´e Algebra
- 13.3. Gauge Transformations
- 13.4. Lagrangian and Equations ofMotion
- 13.5. Conclusion
- Chapter 14POLYADIC ALGEBRAIC STRUCTURESAND THEIR REPRESENTATIONS
- 14.1. Introduction
- 14.2. Preliminaries
- 14.3. Special Elements and Properties ofPolyadic Systems
- 14.4. Homomorphisms of Polyadic Systems
- 14.5. Multiplace Mappings of Polyadic Systemsand Heteromorphisms
- 14.6. Associative Quivers and Heteromorphisms
- 14.7. Multiplace Representations of Polyadic Systems
- 14.8. Multiactions and G-Spaces
- 14.9. Regular Multiactions
- 14.10. Multiplace Representations of Ternary Groups
- 14.11. Matrix Representations of Ternary Groups
- 14.12. Ternary Clifford Algebra
- 14.13. Conclusion
- Chapter 15TERNARY HOPF ALGEBRASAND QUANTUM GROUPS
- 15.1. Introduction
- 15.2. Ternary Algebras and Hopf Algebras
- 15.3. Towards Ternary Quantum Groups
- 15.4. Ternary R-Matrix
- 15.5. Conclusion
- REFERENCES
- CONTRIBUTORS
- INDEX
- Blank Page
- Blank Page
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