homological methods in equations of mathematical physics-j.krasil'schchik

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1、PreprintDIPS7/98math.DG/9808130HOMOLOGICALMETHODSINEQUATIONSOFMATHEMATICALPHYSICS1JosephKRASIL′SHCHIK2IndependentUniversityofMoscowandTheDiffietyInstitute,Moscow,RussiaandAlexanderVERBOVETSKY3MoscowStateTechnicalUniversityandTheDiffietyInstitute,Moscow,RussiaarXiv:math.DG/9808130v221Dec19981Lect

2、uresgiveninAugust1998attheInternationalSummerSchoolinLevoˇca,Slovakia.ThisworkwassupportedinpartbyRFBRgrant97-01-00462andINTASgrant96-07932Correspondenceto:J.Krasil′shchik,1stTverskoy-Yamskoyper.,14,apt.45,125047Moscow,RussiaE-mail:josephk@glasnet.ru3Correspondenceto:A.Verbovetsky,Profsoyuzn

3、aya98-9-132,117485Moscow,RussiaE-mail:verbovet@mail.ecfor.rssi.ru2ContentsIntroduction41.Differentialcalculusovercommutativealgebras61.1.Lineardifferentialoperators61.2.MultiderivationsandtheDiff-Spencercomplex81.3.Jets111.4.Compatibilitycomplex131.5.DifferentialformsandthedeRhamcomplex131.6.Le

4、ftandrightdifferentialmodules161.7.TheSpencercohomology191.8.Geometricalmodules252.AlgebraicmodelforLagrangianformalism272.1.Adjointoperators272.2.Berezinianandintegration282.3.Green’sformula302.4.TheEuleroperator322.5.Conservationlaws343.Jetsandnonlineardifferentialequations.Symmetries353.1.F

5、initejets353.2.Nonlineardifferentialoperators373.3.Infinitejets393.4.Nonlinearequationsandtheirsolutions423.5.CartandistributiononJk(π)443.6.Classicalsymmetries493.7.Prolongationsofdifferentialequations533.8.Basicstructuresoninfiniteprolongations553.9.Highersymmetries624.Coveringsandnonlocalsymm

6、etries694.1.Coverings694.2.Nonlocalsymmetriesandshadows724.3.Reconstructiontheorems745.Fr¨olicher–Nijenhuisbracketsandrecursionoperators785.1.Calculusinform-valuedderivations785.2.Algebraswithflatconnectionsandcohomology835.3.Applicationstodifferentialequations:recursionoperators885.4.Passingt

7、ononlocalities966.Horizontalcohomology1016.1.C-modulesondifferentialequations1026.2.ThehorizontaldeRhamcomplex1066.3.Horizontalcompatibilitycomplex1086.4.ApplicationstocomputingtheC-cohomologygroups11036.5.Example:Evolutionequations1117.Vinogradov’s