Exotic Algebraic and Geometric Structures in Theoretical Physics

Name: Exotic Algebraic and Geometric Structures in Theoretical Physics
Brand: Nova Science Publishers Inc
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Duplij Steven(Author)

Nova Science Publishers Inc

Published on 26. November 2018

410 pages

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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
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