The Atiyah Extension of a Lie Algebra Deformation

Ziv Ran

Abstract: We construct an analogue of the `Atiyah Class', or `Atiyah Extension' in the context of deformation theory of sheaves of Lie algebras. Given a (suitable) sheaf $\g$ of Lie algebras on a space $X$, and a $\g$-deformation $\phi$ over a formally smooth base, our Atiyah Extension is a Lie algebra extension of the algebra of base vector fields by the $\phi$-twist of $\g$. It comes equipped with a representation of the $\phi$-twist on any (admissible) $\g$-module.

Keywords: Sheaf of Lie algebras, deformation, Atiyah class, extension of Lie algebras

Classification (MSC2000): Primary 32G99, secondary 17B55, 14D14

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