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Central extensions of some Lie algebras. (English) Zbl 0916.17017
The authors consider Lie algebras derived from the Lie algebra $$\text{Der } \mathbb{C}((t))$$ of derivations of the algebra $$\mathbb{C}((t))$$ of formal Laurent series: The Lie algebra $$\text{Der } \mathbb{C}((t))$$ itself, the Lie algebra of all differential operators on $$\mathbb{C}((t))$$ and the Lie algebra of differential operators on $$\mathbb{C}((t)) \otimes \mathbb{C}^n$$. They prove that each of these Lie algebras has an essentially unique nontrivial central extension. Up to now such results were known only for Lie algebras related to the algebra $$\mathbb{C} [t,t^{-1}]$$ of Laurent polynomials.

MSC:
 17B56 Cohomology of Lie (super)algebras 17B65 Infinite-dimensional Lie (super)algebras 17B66 Lie algebras of vector fields and related (super) algebras
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