Correspondence, Wells and Hochschild-Serre sequences for nonabelian extensions of multiplicative Lie algebras

Abstract

For nonabelian $2^{\mathrm{nd}}$-cohomology of multiplicative Lie algebras, we properly generalize from the group case three classic results. We prove a Correspondence theorem which compares $2^{\mathrm{nd}}$-cohomology associated to a realized abstract kernel to the abelian $2^{\mathrm{nd}}$-cohomology group over the algebraic center. For arbitrary extensions, we prove a Wells's Theorem characterizing ideal-preserving automorphisms and establish the 1-dimensional Lyndon-Hochschild-Serre exact sequence. Several previously established results are recovered when restricted to extensions with group-abelian or Lie-trivial ideals.