On Non-abelian Extensions Of Leibniz Algebras

In this paper, we mainly study the non-abelian extension of a Leib-niz algebra, and we describe it in three different ways respectively.First, we introduce the notions of non-abelian extensions of a Leib-niz algebra, its splittings and isomorphisms, which are all generaliza-tions of the corresponding concepts in Lie algebras. Given a non-abelian extension g of g by h with a splitting o, the Leibniz algebra structure on g can be transferred to g (?) h). By studying the Leibniz algebra structure on g (?) F), we find that the Leibniz algebra structure on g (?) F) is determined by three liner maps (l,r, ω), so we can study the problem of the non-abelian extensions of g by h in terms of the compatible relations with the three liner maps (l,r,ω). Based on this analysis we give the notion of a second non-abelian cohomology of g with values in h, and prove that all the non-abelian extensions of g by h are classified by the second non-abelian cohomology.Second, we study the non-abelian extensions of Leibniz algebras in terms of Leibniz 2-algebras. We find that on the vector space of left derivations and right derivations of a Leibniz algebra h, there exists a natural Leibniz structure on Der(h). Furthermore, we discover two sub-Leibniz algebras Ⅱ(h) and (?)(h). When Leibniz algebra h degen-erates into a Lie algebra, the Leibniz 2-algebra constructed by the Leibniz algebra Ⅱ(h)) turns to be a Lie 2-algebra, and it is isomorphic to the Lie 2-algebra constructed by derivations of the Lie algebra h. Thus the Leibniz 2-algebra constructed by Ⅱ (h) can be regarded as a natural generalization of the Lie 2-algebra associated to derivations of the Lie algebra h. We use Leibniz algebra (?)(h) to construct an-other Leibniz 2-algebra which is critical in studying the non-abelian extension of a Leibniz algebra on certain condition of the centre. We prove that when Z(h)= Z(g) ∩ h holds, there is a one-to-one cor-respondence between the non-abelian extensions of g by h and the morphisms (f0,f1,f2) from Leibniz 2-algebra (0,g,0, [.,.]g) to Leibniz 2-algebra (h,(?) (h), (adL, adR), l2).Finally, we study the non-abelian extensions of Leibniz algebras from the perspective of Maurer-Cartan elements in a differential grad-ed Lie algebra. According to the theory we have known, (C(g(?)h,g(?) h), [.,.]c,(?)) is a DGLA, where g (?) h is the direct sum of two Leib-niz algebras g and h, with the bracket defined by [x+α,y+β]= [x, y]g+[α,β]h, and (?) is the coboundary operator for the Leibniz algebra g (?) h with coefficients in the adjoint representation. We con-struct a sub-DGLA (C＞(g (?) h,h), of (C(g(?)h,g(?)h), [.,.]c,(?)). where Ck(g(?)h,h)= Ck＞(g(?)h,h) (?) Ck(h, h). We prove that g is a non-abelian extension of g by h if and only if l+r+ω is a Maurer-Cartan element of the DGLA (C＞(g (?) h,h)), [.,.]c,(?). We introduce the no-tion of equivalence for Maurer-Cartan elements, and prove that two non-abelian extensions of g by h are isomorphic if and only if their relevant Maurer-Cartan elements are equivalent.