Associative Algebraic Geometry

website banner on leftMain subject categories: • Algebraic geometry • Associative algebras • Moduli theory • Mathematical physicsClassical Deformation Theory is used for determining the completions of local rings of an eventual moduli space. When a moduli variety exists, the main result explored in the book is that the local ring in a closed point can be explicitly computed as an algebraization of the pro-representing hull, called the local formal moduli, of the deformation functor for the corresponding closed point. The book gives explicit computational methods and includes the most necessary prerequisites for understanding associative algebraic geometry. It focuses on the meaning and the place of deformation theory, resulting in a complete theory applicable to moduli theory. It answers the question "why moduli theory", and gives examples in mathematical physics by looking at the universe as a moduli of molecules, thereby giving a meaning to most noncommutative theories. The book contains the first explicit definition of a noncommutative scheme, not necessarily covered by commutative rings. This definition does not contradict any previous abstract definitions of noncommutative algebraic geometry, but sheds interesting light on other theories, which is left for further investigation. Contents Preface About the Author Acknowledgments 1. Introduction 1.1 Associative Algebra 1.2 Deformation Theory 1.3 Affine Varieties as Moduli of Modules 1.4 Affine Associative Varieties 1.5 Associative Varieties 1.6 Associative Schemes 2. Basic Introduction to Associative Moduli 2.1 Introduction 2.2 Preliminaries 2.3 Generalized Moduli Objects 2.4 Associative Moduli and Adjoint Functors 2.5 Categorification of Deformation Theory 2.6 Geometry in Moduli Objects 2.7 A Naive Framework for Change 2.8 Concluding a Naive Framework for Change 3. Associative Algebra 3.1 Noncommutative Algebras 3.2 Artin–Wedderburn Theory 3.3 Simple Modules and the Jacobson Radical 3.4 The Classical Theorems of Burnside, Wedderburn and Malcev 3.5 Finite-Dimensional Simple Modules 3.6 Matrix Spaces over kr 3.7 Matric kr-Algebras 3.8 Quiver Algebras 3.9 GMMP Algebras 3.10 The Category of r-Pointed Artinian k-Algebras 3.11 Constructing kr-Algebras from Products 3.12 The Algebra of an n-Directed GMMP-Algebra 3.13 A Direct Example of a GMMP-Algebra 3.14 Dynamical Algebras 4. Associative Varieties I 4.1 Associative Representations of Modules 4.2 Associative Varieties 4.3 Affinity of Associative Varieties 4.4 Associative Gluing of Affine Commutative Varieties 4.5 The Structure Sheaf of an Associative Variety 4.6 The Functor Simp(−) : AlgMk→ aVarMk 5. Noncommutative Deformation Theory 5.1 Prorepresentable Functors 5.2 The Noncommutative Deformation Functor 5.3 The Tangent and the Yoneda Complex 5.4 Obstruction Theory 5.5 Computation of Prorepresenting Hulls with a Guiding Example 5.6 Generalized Matric Massey Products 5.7 The Algebra of Observables 5.8 Local Representability of the Deformation Functor 5.9 The Generalized Burnside Theorem 5.10 Generalized Obstruction Theory 5.11 Deformation of Sheaves of OX-Modules 5.12 Concluding Remarks 6. Associative Varieties II 6.1 Representable Functors, Universal Properties and Sheaves 6.2 Ordinary Varieties 6.3 Associative Varieties 6.4 Commutative Affine Schemes 6.5 Associative Affine Varieties 6.6 Associative Affine Varieties of Geometric Algebras 6.7 A First Example 6.8 Defining Associative Varieties 6.9 Deformations Due to Diagrams 6.10 The Definition of Noncommutative Schemes 6.11 Tangent Spaces of Matric Algebras 6.12 A Comment on Multi-Localization 6.13 Example 6.14 Generalized Matric Massey Products 6.15 Reconstructing Algebras from Associative Varieties 6.16 The Embedding Vark → aVark 6.17 The Embedding of Ordinary (Commutative) Varieties in the Category of Associative Varieties 7. Computational Examples of Associative Varieties 7.1 Set-Up for Noncommutative Projective Spaces 7.2 Associative Varieties of Point-Modules 7.3 Some Results from the Commutative Case 7.4 The Associative Affine Plane 7.5 The Associative Noncommutative Variety Pnass 7.6 Noncommutative Projective Varieties 7.7 The Quantum Plane 7.8 The Jordan Plane 7.9 The Quantum Polynomial Ring 7.10 A Sklyanin Algebra 7.11 To the Classification of AS Regular Algebras 7.12 Associative Projective Varieties 7.13 Noncommutative Projective Varieties 7.14 Example: The Associative Quantum Plane 7.15 The Generalized Burnside Theorem and Some Consequences 7.16 Sheaves of OX-Modules 7.17 Classifying 1-Critical Modules 7.18 Associative Schemes 8. Algebraic Invariant Theory 8.1 Basic Definitions 8.2 Fine Moduli for Orbits 8.3 Constructive Method for Noncommutative GIT 8.4 Applications of Noncommutative GIT 8.5 GL(n)-Quotients of Endk(kn) 8.6 The Setup for M3(k) 8.7 The Fine Moduli M2(k)/GL2(k) 8.8 Toric Varieties 8.9 A Toric Example 8.10 n-Lie Algebras 8.11 The Structure of 3 − Lie4 8.12 Moduli of Rank 2 Endomorphisms 9. Pre-Dynamic GIT 9.1 Generalities 9.2 Blowing Up and Desingularization 9.3 Chern Classes 9.4 The Iterated Phase Space Functor Ph∗ and the Dirac Derivation 9.5 The Generalized de Rham Complex 9.6 Excursion into the Jacobian Conjecture 10. Dynamical Algebraic Structures 10.1 Noncommutative Algebraic Geometry 10.2 Moduli of Representations 10.3 Blowing Down Subschemes 10.4 Moduli of Simple Modules 10.5 Evolution in the Moduli of Simple Modules 10.6 Dynamical Structures 10.7 Gauge Groups and Invariant Theory 10.8 The Generic Dynamical Structures Associated to a Metric 10.9 The Classical Gauge Invariance 10.10 A Generalized Yang–Mills Theory 10.11 Reuniting GR, YM and General Quantum Field Theory 10.12 Closing Remarks 10.13 Family of Representations versus Family of Metrics 10.14 Relations to Clifford Algebras Bibliography Index "Classical Deformation Theory is used for determining the completions of the local rings of an eventual moduli space. When a moduli variety exists, a main result in the book is that the local ring in a closed point can be explicitly computed as an algebraization of the pro-representing hull (therefore, called the local formal moduli) of the deformation functor for the corresponding closed point. The book gives explicit computational methods and includes the most necessary prerequisites. It focuses on the meaning and the place of deformation theory, resulting in a complete theory applicable to moduli theory. It answers the question "why moduli theory" and it give examples in mathematical physics by looking at the universe as a moduli of molecules. Thereby giving a meaning to most noncommutative theories. The book contains the first explicit definition of a noncommutative scheme, covered by not necessarily commutative rings. This definition does not contradict any of the previous abstract definitions of noncommutative algebraic geometry, but rather gives interesting relations to other theories which is left for further investigation"-- Provided by publisher