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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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##### References:
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