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Description: Boca Raton : CRC Press, [2017] | Series: Chapman & Hall/CRC monographs and research notes in mathematics | Includes bibliographical references and index. Identifiers: LCCN 2016054372| ISBN 9781498796019 (hardback : alk. paper) | ISBN 9781498796026 (ebook) Subjects: LCSH: Geometry, Algebraic. | Mathematical physics. | Perturbation (Mathematics) Classification: LCC QA564 .E75 2017 | DDC 516.3/5--dc23 LC record available at https://lccn.loc.gov/2016054372 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Introduction xi HowtoReadThisBook xv 1 ClassicalDeformationTheory 1 1.1 Generalprinciples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Formaldeformationsandinfinitesimaldeformations . . . . . . . . . . . . . . . . 2 1.3 FunctorsofArtinrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3.1 Tangentspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.2 Obstructioncalculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Deformationsofassociativealgebras . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.1 Tangentspaceandobstructioncalculus . . . . . . . . . . . . . . . . . . . 12 1.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5 Deformationsofmodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5.1 Tangentspaceandobstructioncalculus . . . . . . . . . . . . . . . . . . . 17 1.5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 NoncommutativeAlgebrasandSimpleModules 27 2.1 Noncommutativealgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Artin-Wedderburntheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3 SimplemodulesandtheJacobsonradical . . . . . . . . . . . . . . . . . . . . . . 30 2.4 TheclassicaltheoremsofBurnside,Wedderburn,andMalcev . . . . . . . . . . . 32 2.5 Finitedimensionalsimplemodules . . . . . . . . . . . . . . . . . . . . . . . . . 32 3 NoncommutativeDeformationTheory 35 3.1 Noncommutativedeformationfunctors . . . . . . . . . . . . . . . . . . . . . . . 35 3.1.1 FlatnessinAbeliancategories . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1.2 Commutativedeformationfunctors . . . . . . . . . . . . . . . . . . . . . 36 3.1.3 Noncommutativedeformationfunctors . . . . . . . . . . . . . . . . . . . 36 3.2 Structureofnoncommutativedeformationfunctors . . . . . . . . . . . . . . . . . 39 3.2.1 FunctorsofnoncommutativeArtinrings . . . . . . . . . . . . . . . . . . . 39 3.2.2 Algebraizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.3 Tangentspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.4 Obstructioncalculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.5 Swarms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2.6 Relationswithcommutativedeformationfunctors . . . . . . . . . . . . . . 46 3.3 Examplesofnoncommutativedeformationfunctors . . . . . . . . . . . . . . . . . 46 3.3.1 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.3.2 Moduleswithgroupaction . . . . . . . . . . . . . . . . . . . . . . . . . . 57 vii viii Contents 3.4 Noncommutativedeformationsofsheavesandpresheaves . . . . . . . . . . . . . 65 3.4.1 Deformationsofpresheavesofmodules . . . . . . . . . . . . . . . . . . . 65 3.4.2 Deformationsofquasi-coherentsheavesofmodules . . . . . . . . . . . . 70 3.4.3 Quasi-coherentringedschemes . . . . . . . . . . . . . . . . . . . . . . . 71 3.4.4 CalculationsforD-modulesonellipticcurves . . . . . . . . . . . . . . . . 73 3.5 MatricMasseyproductsandA-infinitystructures . . . . . . . . . . . . . . . . . . 76 3.5.1 MatricMasseyproductsondifferentialgradedalgebras . . . . . . . . . . . 76 3.5.2 MatricMasseyproductsandobstructioncalculus . . . . . . . . . . . . . . 78 3.5.3 MatricA-infinityalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.6 TheGeneralisedBurnsideTheorem . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.6.1 Thealgebraofobservables . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.6.2 Thekerneloftheminiversalmorphism . . . . . . . . . . . . . . . . . . . 87 3.6.3 IteratedextensionsandmatricMasseyproducts . . . . . . . . . . . . . . . 88 3.6.4 TheGeneralisedBurnsideTheorem . . . . . . . . . . . . . . . . . . . . . 90 3.6.5 Propertiesofthealgebraofobservables . . . . . . . . . . . . . . . . . . . 92 3.7 Iteratedextension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.7.1 Moduliofiteratedextensions. . . . . . . . . . . . . . . . . . . . . . . . . 93 3.7.2 Thecategoryofiteratedextensions . . . . . . . . . . . . . . . . . . . . . 95 4 TheNoncommutativePhaseSpace 97 4.1 Introductiontononcommutativephasespaces . . . . . . . . . . . . . . . . . . . . 97 4.1.1 ThenoncommutativeKodaira-Spencermap . . . . . . . . . . . . . . . . . 98 4.1.2 Generalisedmomenta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.2 TheiteratedphasespacefunctorandtheDiracderivation . . . . . . . . . . . . . . 101 4.2.1 TheDiracderivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.2.2 ThegeneraliseddeRhamcomplex . . . . . . . . . . . . . . . . . . . . . . 104 4.3 Differentiablestructuresonthemoduliofrepresentations . . . . . . . . . . . . . . 107 4.3.1 Dynamicalstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.3.2 RepresentationsofPh∞(A) . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.4 Gaugegroupsandinvarianttheory . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.5 Thegenericdynamicalstructuresassociatedtoametric . . . . . . . . . . . . . . . 119 4.5.1 Thecommutativecaseandgeneralrelativity . . . . . . . . . . . . . . . . . 120 4.5.2 Thegeneralcase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.6 Classicalgaugeinvarianceandmetricclassificationofrepresentations . . . . . . . 127 4.6.1 Theclassicalgaugeinvariance . . . . . . . . . . . . . . . . . . . . . . . . 127 4.6.2 CherncharactersandChern-Simonsclasses . . . . . . . . . . . . . . . . . 129 4.6.3 AgeneralisedYang-Millstheory . . . . . . . . . . . . . . . . . . . . . . . 130 4.6.4 TheclassicalYang-Millsequation . . . . . . . . . . . . . . . . . . . . . . 133 4.6.5 Reunitinggeneralrelativity,Yang-Mills,andgeneralquantumfieldtheory . 134 4.7 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.7.1 Theclassicalcommutativecase . . . . . . . . . . . . . . . . . . . . . . . 139 4.7.2 Thegeneralcase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.8 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.8.1 Interactionandnoncommutativedeformations. . . . . . . . . . . . . . . . 141 4.8.2 Ensembles,bialgebras,andquantumgroupsinourmodel. . . . . . . . . . 143 Contents ix 5 ACosmologicalToyModel 145 5.1 Backgroundandsomeremarksonphilosophyofscience . . . . . . . . . . . . . . 145 5.2 Deformationsofassociativealgebras . . . . . . . . . . . . . . . . . . . . . . . . 146 5.3 Spin,isospin,andsupersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.4 Newton’sandKepler’slaws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5.5 Theuniverseasaversalbasespace . . . . . . . . . . . . . . . . . . . . . . . . . 160 5.6 Workedoutformulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 5.6.1 Actionofthegaugegroupg⊕su(2)onthetangentspace . . . . . . . . . . 164 5.6.2 Adjointactionsofg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 5.7 Summingupthemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.7.1 Thetoymodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 5.7.2 Furtherresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 5.8 Elementaryparticles,bosons,andfermions . . . . . . . . . . . . . . . . . . . . . 180 5.8.1 Spin,charge,andchirality . . . . . . . . . . . . . . . . . . . . . . . . . . 180 5.8.2 Theweakforce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 6 ModuliofEndomorphismsofRank3 187 6.1 Endomorphismsofvectorspaces . . . . . . . . . . . . . . . . . . . . . . . . . . 187 6.2 Moduliofendomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 6.3 Noncommutativemoduliofendomorphismsofrankthree . . . . . . . . . . . . . 190 6.4 Thecomputations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 6.4.1 TheorbitsinCaseIIandCaseIII . . . . . . . . . . . . . . . . . . . . . . 193 6.4.2 ThetangentspacedimensionsinCaseI . . . . . . . . . . . . . . . . . . . 198 6.4.3 ThetangentspacedimensionsinCaseIIandIII . . . . . . . . . . . . . . . 203 6.4.4 BasesandYonedaformsinCaseI . . . . . . . . . . . . . . . . . . . . . . 206 6.4.5 Second-orderMasseyproductsinCaseI . . . . . . . . . . . . . . . . . . . 216 6.4.6 Computationofthefirstlifting . . . . . . . . . . . . . . . . . . . . . . . . 219 6.4.7 Computationofthesecond-orderdefiningsystem . . . . . . . . . . . . . . 221 6.4.8 Theresultsofthecomputations . . . . . . . . . . . . . . . . . . . . . . . 226 6.4.9 Thegeometricpicture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 6.4.10 Theisotropygroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 6.5 Thenoncommutativeaffinering . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 Bibliography 237 Index 241