Deformations of Lie algebras of vector fields arising from families of schemes

Friedrich Wagemann (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

Fialowski and Schlichenmaier constructed examples of global deformations of Lie algebras of vector fields from deforming the underlying variety. We formulate their approach in a conceptual way. Namely, we construct a stack of deformations and a morphism form the moduli stack of stable marked curves. The morphism associates to a family of marked curves the family of Lie algebras obtained by taking the Lie algebra of vertical vector fields on the family where one has extracted the marked points. We show that this morphism is almost a monomorphism by Pursell-Shanks theory.

Mots clés: global deformations of Lie algebras; moduli stack of curves; Lie algebra of vector fields on the circle; faithfully flat descent; cohomology of Lie algebras

Domaines: Géométrie algébrique [math.AG]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Friedrich Wagemann. Deformations of Lie algebras of vector fields arising from families of schemes. Journal of Geometry and Physics, 2008, 58, pp.165-178. ⟨10.1016/j.geomphys.2007.10.003⟩. ⟨hal-00165660⟩