Kravchenko, O. S.; Khesin, B. A. A central extension of the algebra of pseudodifferential symbols. (English. Russian original) Zbl 0729.35154 Funct. Anal. Appl. 25, No. 2, 152-154 (1991); translation from Funkts. Anal. Prilozh. 25, No. 2, 83-85 (1991). Summary: The Lie algebra of pseudodifferential symbols (\(\Psi\) \({\mathcal D}{\mathcal S})\) on the circle is a natural generalization of the algebra of differential operators (\({\mathcal D}{\mathcal O})\) on the circle, which, in turn, is a generalization of the Lie algebra of vector fields. In this note it is shown that the algebra of \(\Psi\) \({\mathcal D}{\mathcal S}\) possesses a nontrivial central extension; moreover, the restriction of the cocycle constructed here to the algebra \({\mathcal D}{\mathcal O}\) gives the cocycle found in A. O. Radul’s work [Pis’ma Zh. Eksp. Teor. Fiz. 50, 341- 343 (1989)], while its restriction to the algebra of vector fields gives the Gel’fand-Fuks cocycle. Cited in 1 ReviewCited in 25 Documents MSC: 35S99 Pseudodifferential operators and other generalizations of partial differential operators 47G99 Integral, integro-differential, and pseudodifferential operators 17B66 Lie algebras of vector fields and related (super) algebras Keywords:Lie algebra of pseudodifferential symbols; central extension; cocycle PDF BibTeX XML Cite \textit{O. S. Kravchenko} and \textit{B. A. Khesin}, Funct. Anal. Appl. 25, No. 2, 152--154 (1991; Zbl 0729.35154); translation from Funkts. Anal. Prilozh. 25, No. 2, 83--85 (1991) Full Text: DOI References: [1] A. O. Radul, Pis’ma Zh. Eksp. Teor. Fiz.,50, 341-343 (1989). [2] A. B. Zamolodchikov, Teor. Mat. Fiz.,65, 347-359 (1985). [3] S. L. Luk’yanov, Funkts. Anal. Prilozhen.,22, No. 4, 1-10 (1988). · Zbl 0667.58018 · doi:10.1007/BF01077717 [4] B. L. Feigin, Usp. Mat. Nauk,36, No. 2, 157-158 (1988). [5] E. Getzler, Proc. Am. Math. Soc.,104, 729-742 (1988). [6] A. A. Beilinson, Yu. I. Manin, in: Lecture Notes in Math., Vol. 1289, Springer, Berlin (1987), pp. 52-64. [7] J.-L. Brylinsky and E. Getzler, K-Theory,1, No. 4, 385-396. [8] M. Wodzycki, in: Lecture Notes in Math., Vol. 1289, Springer, Berlin (1987), pp. 321-392. [9] A. G. Reiman and M. A. Semenov-Tyan-Shanskii, Dokl. Akad. Nauk SSSR,251, No. 6, 1310 (1980). [10] V. G. Drinfel’d and V. V. Sokolov, in: Contemporary Problems of Mathematics,24, 81 (1984). 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.