  A Lie atom is essentially a pair of Lie algebras and its deformation theory
is that of deformations with respect to one algebra together with a
trivialization with respect to the other. Such deformations occur commonly in
Algebraic Geometry, for instance as deformations of subvarieties of a fixed
ambient variety. Here we study some basic notions related to Lie atoms,
focussing especially on their deformation theory, in particular the universal
deformation. We introduce Jacobi-Bernoulli cohomology, which yields the
deformation ring, and show that, under suitable hypotheses, infinitesimal
deformations are classified by certain Kodaira-Spencer data.(minor corrections
Feb'06/May'07; paper is to appear in GAFA)
