Deformations of ideals in Lie algebras

TL;DR

This paper develops a deformation theory for Lie ideals, linking smooth deformations to cohomology classes, and introduces differential graded Lie algebras and $L_{}$-algebras to control and analyze these deformations and their obstructions.

Contribution

It introduces a comprehensive framework for deforming Lie ideals using cohomology, DGLAs, and $L_{}$-algebras, advancing understanding of ideal stability and obstructions.

Findings

Deformation complex enriched to a DGLA controlling ideal deformations.

Identification of an $L_{}$-algebra controlling simultaneous deformations.

Conditions for rigidity, stability, and obstructions of Lie ideals.

Abstract

This paper develops the deformation theory of Lie ideals. It shows that the smooth deformations of an ideal $\mathfrak i$ in a Lie algebra $\mathfrak g$ differentiate to cohomology classes in the cohomology of $\mathfrak g$ with values in its adjoint representation on $\operatorname{Hom}(\mathfrak i, \mathfrak g/\mathfrak i)$. The cohomology associated with the ideal $\mathfrak i$ in $\mathfrak g$ is compared with other Lie algebra cohomologies defined by $\mathfrak i$, such as the cohomology defined by $\mathfrak i$ as a Lie subalgebra of $\mathfrak g$ (Richardson, 1969), and the cohomology defined by the Lie algebra morphism $\mathfrak g \to \mathfrak g/\mathfrak i$. After a choice of complement of the ideal $\mathfrak i$ in the Lie algebra $\mathfrak g$, its deformation complex is enriched to the differential graded Lie algebra that controls its deformations, in the sense that its Maurer-Cartan elements are in one-to-one correspondence with the (small) deformations of the ideal. Furthermore, the $L_{\infty}$-algebra that simultaneously controls the deformations of $\mathfrak{i}$ and of the ambient Lie bracket is identified. Under appropriate assumptions on the low degrees of the deformation cohomology of a given Lie ideal, the (topological) rigidity and stability of ideals are studied, as well as obstructions to deformations of ideals of Lie algebras.