Descent study of the Lie algebra of derivations of certain infinite-dimensional Lie algebras

Abstract

Letgbeafinite-dimensionalperfectLiealgebraoverafieldkofcharacteristic0. In infinite-dimensional Lie theory we encounter Lie algebras of the form g⊗k R, where R is a k-ring (usually a Laurent polynomial ring in finitely many variables over k), and étale twisted forms L of g⊗k R. Thus L is an R-Lie algebra that becomes isomorphic to the S-Lie algebra g ⊗k S after some étale cover base ring extension S/R. The interesting infinite-dimensional Lie algebras are “built” out of L by adding a centre Z and a Lie algebra of derivations D (the affine Kac-Moody Lie algebras are the simplest examples). D, which determines Z, is a Lie subalgebra of Derk(L) of L. The understanding of this last Lie algebra is crucial. While the R-Lie algebra L can be given by étale descent, the same cannot be openly said about Derk(L) since it is not an R-Lie algebra. In the present paper we give such a descent presentation within the more general framework of relative R/k-sheaves of Lie algebras that we believe is of independent interest.