Classical 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.
Author(s): Arvid Siqveland
Publisher: World Scientific
Year: 2023
Language: English
Pages: 419
City: London
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
