Actions for Noncommutative deformation theory
Noncommutative deformation theory / Eivind Eriksen, Olav Arnfinn Laudal, Arvid Siqveland
- Author
- Eriksen, Eivind
- Published
- Boca Raton, FL : CRC Press, Taylor & Francis Group, [2017]
- Physical Description
- xvi, 242 pages : illustrations ; 27 cm.
- Additional Creators
- Laudal, Olav Arnfinn and Siqveland, Arvid, 1964-
- Series
- Contents
- Machine generated contents note: 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 radical -- 2.4.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 functors -- 3.3.1.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 observables -- 3.6.2.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 complex -- 4.3.Differentiable structures on the moduli of representations -- 4.3.1.Dynamical structures -- 4.3.2.Representations of Ph"(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 -- 4.6.5.Reuniting general relativity, Yang-Mills, and general quantum field theory -- 4.7.Entropy -- 4.7.1.The classical commutative case -- 4.7.2.The general case -- 4.8.Interactions -- 4.8.1.Interaction and noncommutative deformations -- 4.8.2.Ensembles, bialgebras, and quantum groups in our model -- 5.A Cosmological Toy Model -- 5.1.Background and some remarks on philosophy of science -- 5.2.Deformations of associative algebras -- 5.3.Spin, isospin, and supersymmetry -- 5.4.Newton's and Kepler's laws -- 5.5.The universe as a versal base space -- 5.6.Worked out formulas -- 5.6.1.Action of the gauge group g [⊗] su(2) on the tangent space -- 5.6.2.Adjoint actions of g -- 5.7.Summing up the model -- 5.7.1.The toy model -- 5.7.2.Further results -- 5.8.Elementary particles, bosons, and fermions -- 5.8.1.Spin, charge, and chirality -- 5.8.2.The weak force -- 6.Moduli of Endomorphisms of Rank 3 -- 6.1.Endomorphisms of vector spaces -- 6.2.Moduli of endomorphisms -- 6.3.Noncommutative moduli of endomorphisms of rank three -- 6.4.The computations -- 6.4.1.The orbits in Case II and Case III -- 6.4.2.The tangent space dimensions in Case I -- 6.4.3.The tangent space dimensions in Case II and III -- 6.4.4.Bases and Yoneda forms in Case I -- 6.4.5.Second-order Massey products in Case I -- 6.4.6.Computation of the first lifting -- 6.4.7.Computation of the second-order defining system -- 6.4.8.The results of the computations -- 6.4.9.The geometric picture -- 6.4.10.The isotropy groups -- 6.5.The noncommutative affine ring.
- Subject(s)
- ISBN
- 9781498796019 hardcover alkaline paper
149879601X hardcover alkaline paper - Bibliography Note
- Includes bibliographical references and index.
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