Lie Atoms and Their Deformations

A Lie atom is essentially a pair of Lie algebras and its deformation theory is that of a deformation with respect to the first algebra, endowed with a trivialization with respect to the second. 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. Many deformation-theoretic problems and results in algebraic and complex geometry can be profitably formulated in terms of Lie algebras, more specifically differential graded Lie algebras or dglas. These problems includes, notably, the Kodaira-Spencer theory of deformations of complex structures. Nevertheless, there are fundamental deformation problems in geometry for which no Lie theoretic formulation is known. These include, notably, the deformation theory of submanifolds in a fixed ambient manifold, i.e. the local theory of the Hilbert scheme in algebraic geometry or the Douady space in complex-analytic geometry. A principal purpose of this paper is to remedy this situation. To this end, and for what we consider its own intrinsic interest, we introduce and begin to study a notion which we call Lie atom and which generalizes that of the (shifted) quotient of a Lie algebra by a subalgebra (more precisely, a pair of Lie algebras up to bracket-preserving quasi-isomorphism)). Actually, it turns out to be preferable to work with a somewhat more general algebraic object, consisting of a pair of Lie algebras g, h, a Lie homomorphism g → h, and a g−module h ⊂ h. A special case of this is a Lie pair, where h = h. Geometrically, a Lie atom can be used to control situations where a geometric object is deformed while some aspect of the geometry ’stays the same’ (i.e. is deformed in a trivialized manner); specifically, the algebra g controls the deformation while the module h and the algebra h control the trivialization. A typical example of this situation is that of a submanifold Y in an ambient manifold X, where g is the Lie algebra of relative vector fields (infinitesimal motions of X leaving Y invariant), and h = h is the algebra and g-module of all ambient vector fields, so that the associated Lie atom is just the shifted normal bundle NY/X [−1], and the associated deformation theory is that of the Hilbert scheme or Douady space of submanifolds of X. Our point of view is that a Lie atom possesses some of the formal properties of Lie algebras. In particular, we shall see that there is a deformation theory for Lie atoms, which generalizes the case of Lie algebras and which in addition allows us to treat some classical, and disparate, deformation problems. These include, on the one hand, the Hilbert scheme, and on the other hand heat-equation deformations, introduced in the first-order case by Welters [We]. Here we will present a systematic development of some of the rudiments of the deformation theory of Lie atoms, which are closely analogous to those of (differential graded) Lie algebras. See [Rrel2] for an application of Lie atoms to the so-called Knizhnik-Zamolodchikov -Hitchin connection on the moduli space of curves. 1991 Mathematics Subject Classification. 14D15. Research supported in part by NSA grant H98230-05-1-0063; v.070104 1