Deformations of ideals in Lie algebras

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.