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Cyclic homology of differential operators, the Virasoro algebra and a $$q$$-analogue. (English) Zbl 0761.17020
In the algebraic setting the Virasoro-Bott cocycle, which defines the universal central extension of the Lie algebra of polynomial vector fields on the unit circle, can be obtained as the restriction of a 2-cocycle defined on the Lie algebra of algebraic differential operators.
In this paper this result is reproved using cyclic homology theory. The advantage of this method is that it can be directly generalized to a $$q$$-deformed setting, the algebra of $$q$$-difference operators on $$\mathbb C[x,x^{-1}]$$. This leads to a one-parameter deformation of the Hochschild 1-cocycle, which corresponds to the Virasoro-Bott cocycle. Moreover in the case where $$q$$ is a root of unity there arises an infinite family of linearly independent Hochschild 1-cocycles. All these cocycles are computed explicitly.
Reviewer: A. Čap (Wien)

##### MSC:
 17B55 Homological methods in Lie (super)algebras 17B68 Virasoro and related algebras 17B66 Lie algebras of vector fields and related (super) algebras
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##### References:
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