Deformation of infinite dimensional differential graded Lie algebras

TL;DR

This paper introduces elliptic differential graded Lie algebras, explores their deformation theory, and establishes conditions under which formal solutions imply analytic solutions, advancing understanding of their deformation properties.

Contribution

It defines elliptic differential graded Lie algebras, constructs their deformation spaces, and links formal and analytic solutions in deformation problems.

Findings

Elliptic differential graded Lie algebras include differential forms with endomorphism values.

A complete set of deformations for elliptic algebras is constructed.

Formal solutions to deformation problems can be promoted to analytic solutions under certain conditions.

Abstract

We introduce a notion of elliptic differential graded Lie algebra. The class of elliptic algebras contains such examples as the algebra of differential forms with values in endomorphisms of a flat vector bundle over a compact manifold, etc. For elliptic differential graded algebra we construct a complete set of deformations. We show that for several deformation problems the existence of a formal power series solution guarantees the existence of an analytic solution.