New preprint ``Deformations of Lie algebras'', by Ilias Ermeidis and Madeleine Jotz.

The preprint ``Deformations of Lie algebras'' by Ilias Ermeidis and Madeleine Jotz, has appeared on arxiv.org.

This paper develops the deformation theory of Lie ideals. It shows that the smooth deformations  of an ideal i in a Lie algebra g differentiate to cohomology classes in the cohomology of g with values in its adjoint representation on Hom(i, g/i). The cohomology associated with the ideal i in g is compared with other Lie algebra cohomologies defined by i, such as the cohomology defined by i as a Lie subalgebra of g in (Richardson 1969), and the cohomology defined by the Lie algebra morphism g -> g/i.

After a choice of complement of the ideal i in the Lie algebra 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 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.