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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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