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Unformatted text preview: arXiv:math.DG/9808130 v2 21 Dec 1998 Preprint DIPS 7/98 math.DG/9808130 HOMOLOGICAL METHODS IN EQUATIONS OF MATHEMATICAL PHYSICS 1 Joseph KRASIL ′ SHCHIK 2 Independent University of Moscow and The Diffiety Institute, Moscow, Russia and Alexander VERBOVETSKY 3 Moscow State Technical University and The Diffiety Institute, Moscow, Russia 1 Lectures given in August 1998 at the International Summer School in Levoˇ ca, Slovakia. This work was supported in part by RFBR grant 970100462 and INTAS grant 960793 2 Correspondence to: J. Krasil ′ shchik, 1st TverskoyYamskoy per., 14, apt. 45, 125047 Moscow, Russia Email : [email protected] 3 Correspondence to: A. Verbovetsky, Profsoyuznaya 989132, 117485 Moscow, Russia Email : [email protected] 2 Contents Introduction 4 1. Differential calculus over commutative algebras 6 1.1. Linear differential operators 6 1.2. Multiderivations and the DiffSpencer complex 8 1.3. Jets 11 1.4. Compatibility complex 13 1.5. Differential forms and the de Rham complex 13 1.6. Left and right differential modules 16 1.7. The Spencer cohomology 19 1.8. Geometrical modules 25 2. Algebraic model for Lagrangian formalism 27 2.1. Adjoint operators 27 2.2. Berezinian and integration 28 2.3. Green’s formula 30 2.4. The Euler operator 32 2.5. Conservation laws 34 3. Jets and nonlinear differential equations. Symmetries 35 3.1. Finite jets 35 3.2. Nonlinear differential operators 37 3.3. Infinite jets 39 3.4. Nonlinear equations and their solutions 42 3.5. Cartan distribution on J k ( π ) 44 3.6. Classical symmetries 49 3.7. Prolongations of differential equations 53 3.8. Basic structures on infinite prolongations 55 3.9. Higher symmetries 62 4. Coverings and nonlocal symmetries 69 4.1. Coverings 69 4.2. Nonlocal symmetries and shadows 72 4.3. Reconstruction theorems 74 5. Fr¨olicher–Nijenhuis brackets and recursion operators 78 5.1. Calculus in formvalued derivations 78 5.2. Algebras with flat connections and cohomology 83 5.3. Applications to differential equations: recursion operators 88 5.4. Passing to nonlocalities 96 6. Horizontal cohomology 101 6.1. Cmodules on differential equations 102 6.2. The horizontal de Rham complex 106 6.3. Horizontal compatibility complex 108 6.4. Applications to computing the Ccohomology groups 110 3 6.5. Example: Evolution equations 111 7. Vinogradov’s Cspectral sequence 113 7.1. Definition of the Vinogradov Cspectral sequence 113 7.2. The term E 1 for J ∞ ( π ) 113 7.3. The term E 1 for an equation 118 7.4. Example: Abelian pform theories 120 7.5. Conservation laws and generating functions 122 7.6. Generating functions from the antifieldBRST standpoint 125 7.7. Euler–Lagrange equations 126 7.8. The Hamiltonian formalism on J ∞ ( π ) 128 7.9. On superequations 132 Appendix: Homological algebra 135 8.1. Complexes 135 8.2. Spectral sequences 140 References 147 4 Introduction Mentioning (co)homology theory in the context of differential equations would sound a bit ridiculous some 30–40 years ago: what could be in com...
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