Cohomological characterizations of non-abelian extensions of strict Lie 2-algebras

Abstract

In this paper, we study non-abelian extensions of strict Lie 2-algebras via the cohomology theory. A non-abelian extension of a strict Lie 2-algebra g by h gives rise to a strict homomorphism from g to SOut(h). Conversely, we prove that the obstruction of existence of non-abelian extensions of strict Lie 2-algebras associated to a strict Lie 2-algebra homomorphism from g to SOut(h) is given by an element in the third cohomology group. We further prove that the isomorphism classes of non-abelian extensions of strict Lie 2-algebras are classified by the second cohomology group. 0 Keyword: strict Lie 2-algebras, strict derivations, cohomology, non-abelian extensions 0 MSC: 17B99, 53D17. * Research supported by NSFC (11471139) and NSF of Jilin Province (20170101050JC).δ

Actor of a crossed module of dialgebras via tetramultipliers

“…Actions of crossed modules of different structures have also been described in terms of equations, as it can be checked in [10] for crossed modules of groups, in [8], [11] or [31] for crossed modules of Lie algebras, and in [7] for crossed modules of Leibniz algebras. Bearing those examples in mind, it is seemingly reasonable to address the problem by following the next steps: firstly, we consider a homomorphism from (M, P, η) to Act(L, D, µ) = (Tetra(D, L), Tetra(L, D, µ), ∆), and translate into equations every property satisfied by that homomorphism.…”