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Deformations of Lie algebras using \(\sigma\)-derivations. (English) Zbl 1138.17012
Summary: We develop an approach to deformations of the Witt and Virasoro algebras based on \(\sigma\)-derivations. We show that \(\sigma\)-twisted Jacobi type identity holds for generators of such deformations. For the \(\sigma\)-twisted generalization of Lie algebras modeled by this construction, we develop a theory of central extensions. We show that our approach can be used to construct new deformations of Lie algebras and their central extensions, which in particular include naturally the \(q\)-deformations of the Witt and Virasoro algebras associated to \(q\)-difference operators, providing also corresponding \(q\)-deformed Jacobi identities.

MSC:
17B68 Virasoro and related algebras
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
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