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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.

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
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[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).
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