×

The homology of algebras of pseudo-differential symbols and the noncommutative residue. (English) Zbl 0646.58026

The Hochschild and cyclic homology groups of the algebra of pseudo- differential symbols \(\Psi^{\infty}(M)/\Psi^{-\infty}(M)\) on a smooth m-dimensional manifold M are calculated; the main result is that \(HH_*(\Psi^{\infty}(M)/\Psi^{-\infty}(M))=H^{2n-*}(S\quad *M\times {\mathbb{S}}^ 1,{\mathbb{C}}),\) \(HC_*(\Psi^{\infty}(M)/\Psi^{- \infty}(M))=H^{2n-*}(S^*M\times {\mathbb{S}}^ 1,{\mathbb{C}})[u].\)
Reviewer: V.Ivrij

MSC:

58J40 Pseudodifferential and Fourier integral operators on manifolds
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adler, M.: On a trace formula for formal pseudo-differential operators and the symplectic structure of the Kortweg-de Vries type equations, Invent. Math. 19 (1973), 279-330. · Zbl 0257.58008
[2] Brylinski, J. L.: A differential complex for Poisson manifolds, IHES preprint, 1986.
[3] Brylinski, J. L.: Some examples of Hochschild and cyclic cohomology, Brown preprint, 1987. · Zbl 0643.16012
[4] Connes, A.: Non-commutative differential geometry, Publ. Math. IHES 62 (1985), 41-144. · Zbl 0592.46056
[5] Flato, M. and Sternheimer, D.: Deformations of Poisson brackets, in J. Wolf et al. (eds.), Harmonic Analysis and Representations of Lie Groups, D. Reidel, Dordrecht, 1980. · Zbl 0465.53024
[6] Feigin, B. and Tsygan, P.: Cohomology of Lie algebras of generalized Jacobi matrices, Func. Anal. Appl. 17 (1983), 153-155. · Zbl 0544.17011
[7] Grothendieck, A.: On the de Rham cohomology of algebraic varieties, Publ. Math. IHES 29 (1966), 93-103. · Zbl 0145.17602
[8] Grothendieck, A. and Dieudonné, J.: E. G. A. III (Première partie), Publ. Math. IHES 11 (1961).
[9] Guillemin, V.: A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues, Adv. Math. 55 (1985), 131-160. · Zbl 0559.58025
[10] Hochschild, G., Kostant, B. and Rosenberg, A.: Differential forms on regular affine algebras, Trans. Am. Math. Soc. 102 (1962), 383-408. · Zbl 0102.27701
[11] Kashiwara, M. and Kawai, T.: On holonomic systems of microdifferential equations, III ? Systems with regular singularities, Publ. RIMS, Kyoto Univ. 17 (1981), 813-979. · Zbl 0505.58033
[12] Kashiwara, M.: Systems of Microdifferential Equations, Progress in Math., Birkhäuser, Boston, 1983. · Zbl 0521.58057
[13] Kassel, C. and Mitschi, C: Private communication.
[14] Koszul, J. L.: Crochet de Schouten-Nijenhuis et cohomologie, Astérisque hors série (1985), 257-271.
[15] Loday, J. L. and Quillen, D.: Cyclic homology and the Lie algebra homology of matrices, Comment. Math. Helv. 59 (1984), 565-591. · Zbl 0565.17006
[16] Sato, M., Kashiwara, M. and Kawai, T.: Hyperfunctions and pseudo-differential equations, in Lecture Notes in Mathematics, No. 287, Springer-Verlag, 1973, pp. 265-529. · Zbl 0277.46039
[17] Weil, A.: Sur les théorèmes de de Rham, Comment. Math. Helv. 26 (1952), 119-145. · Zbl 0047.16702
[18] De Wilde, M. and Lecomte, P. B. A.: Star-products on cotangent bundles, Lett. Math. Phys. 7 (1983), 235-241. · Zbl 0514.53031
[19] Wodzicki, M., Local invariants of spectral assymetry, Invent. Math. 75 (1984), 143-178. · Zbl 0538.58038
[20] Wodzicki, M.: Noncommutative Residue (to appear). · Zbl 0649.58033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.