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Poisson structures associated with algebras of differential operators. (English. Russian original) Zbl 0859.47034
Math. Notes 58, No. 2, 850-860 (1995); translation from Mat. Zametki 58, No. 2, 256-271 (1995).
The author’s abstract: “For differential operators forming an algebra of a certain class that includes algebras of higher derivatives, a Poisson structure is introduced and the first term of the Hochschild spectral sequence is calculated”.
MSC:
47E05 General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX)
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
47L10 Algebras of operators on Banach spaces and other topological linear spaces
55T05 General theory of spectral sequences in algebraic topology
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
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References:
[1] J.-L. Brylinski, ”A differential complex for Poisson manifolds,”J. Differential Geom.,28, 93–114 (1988). · Zbl 0634.58029
[2] G. Hochschild, B. Kostant, and A. Rosenberg, ”Differential forms on regular affine algebras,”Trans. Am. Math. Soc.,102, 383–408 (1962). · Zbl 0102.27701 · doi:10.1090/S0002-9947-1962-0142598-8
[3] A. M. Vinogradov, I. S. Krasil’shchik, and V. V. Lychagin,Introduction to the Geometry of Nonlinear Differential Equations [in Russian], Nauka, Moscow (1986). · Zbl 0592.35002
[4] H. Cartan and S. Eilenberg,Homological Algebra, Princeton Univ. Press, Princeton (1956).
[5] O. V. Lychagina, ”The spectral sequence for the Hochschild homology of the algebra of higher derivations,”Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.],3, 18–22 (1993).
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