Noncommutative Deformation Theory

Description

Noncommutative Deformation Theory is aimed at mathematicians and physicists studying the local structure of moduli spaces in algebraic geometry. This book introduces a general theory of noncommutative deformations, with applications to the study of moduli spaces of representations of associative algebras and to quantum theory in physics. An essential part of this theory is the study of obstructions of liftings of representations using generalised (matric) Massey products. Suitable for researchers in algebraic geometry and mathematical physics interested in the workings of noncommutative algebraic geometry, it may also be useful for advanced graduate students in these fields. Content: Cover Half Title Title Page Copyright Page Contents Introduction How to Read This Book 1 Classical Deformation Theory 1.1 General principles 1.2 Formal deformations and infinitesimal deformations 1.3 Functors of Artin rings 1.3.1 Tangent spaces 1.3.2 Obstruction calculus 1.4 Deformations of associative algebras 1.4.1 Tangent space and obstruction calculus 1.4.2 Examples 1.5 Deformations of modules 1.5.1 Tangent space and obstruction calculus 1.5.2 Examples 2 Noncommutative Algebras and Simple Modules 2.1 Noncommutative algebras 2.2 Artin-Wedderburn theory 2.3 Simple modules and the Jacobson radical2.4 The classical theorems of Burnside, Wedderburn, and Malcev 2.5 Finite dimensional simple modules 3 Noncommutative Deformation Theory 3.1 Noncommutative deformation functors 3.1.1 Flatness in Abelian categories 3.1.2 Commutative deformation functors 3.1.3 Noncommutative deformation functors 3.2 Structure of noncommutative deformation functors 3.2.1 Functors of noncommutative Artin rings 3.2.2 Algebraizations 3.2.3 Tangent spaces 3.2.4 Obstruction calculus 3.2.5 Swarms 3.2.6 Relations with commutative deformation functors 3.3 Examples of noncommutative deformation functors3.3.1 Modules 3.3.2 Modules with group action 3.4 Noncommutative deformations of sheaves and presheaves 3.4.1 Deformations of presheaves of modules 3.4.2 Deformations of quasi-coherent sheaves of modules 3.4.3 Quasi-coherent ringed schemes 3.4.4 Calculations for D-modules on elliptic curves 3.5 Matric Massey products and A-infinity structures 3.5.1 Matric Massey products on differential graded algebras 3.5.2 Matric Massey products and obstruction calculus 3.5.3 Matric A-infinity algebras 3.6 The Generalised Burnside Theorem 3.6.1 The algebra of observables3.6.2 The kernel of the miniversal morphism 3.6.3 Iterated extensions and matric Massey products 3.6.4 The Generalised Burnside Theorem 3.6.5 Properties of the algebra of observables 3.7 Iterated extension 3.7.1 Moduli of iterated extensions 3.7.2 The category of iterated extensions 4 The Noncommutative Phase Space 4.1 Introduction to noncommutative phase spaces 4.1.1 The noncommutative Kodaira-Spencer map 4.1.2 Generalised momenta 4.2 The iterated phase space functor and the Dirac derivation 4.2.1 The Dirac derivation 4.2.2 The generalised de Rham complex4.3 Differentiable structures on the moduli of representations 4.3.1 Dynamical structures 4.3.2 Representations of Ph[sup(∞)](A) 4.4 Gauge groups and invariant theory 4.5 The generic dynamical structures associated to a metric 4.5.1 The commutative case and general relativity 4.5.2 The general case 4.6 Classical gauge invariance and metric classification of representations 4.6.1 The classical gauge invariance 4.6.2 Chern characters and Chern-Simons classes 4.6.3 A generalised Yang-Mills theory 4.6.4 The classical Yang-Mills equation