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Register a new userOn non-abelian extensions of Leibniz algebras
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In this paper, first we classify non-abelian extensions of Leibniz algebras by the second non-abelian cohomology. Then, we construct Leibniz 2-algebras using derivations of Leibniz algebras, and show that under a condition on the center, a non-abelian extension of Leibniz algebras can be described by a Leibniz 2-algebra morphism. At last, we give a description of non-abelian extensions in terms of Maurer-Cartan elements in a differential graded Lie algebra.
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The present paper is devoted to the description of rigid solvable Leibniz algebras. In particular, we prove that solvable Leibniz algebras under some conditions on the nilradical are rigid and we describe four-dimensional solvable Leibniz algebras with three-dimensional rigid nilradical. We show that the Grunewald-OHallorans conjecture any $n$-dimensional nilpotent Lie algebra is a degeneration of some algebra of the same dimension holds for Lie algebras of dimensions less than six and for Leibniz algebras of dimensions less than four. The algebra of level one, which is omitted in the 1991 Gorbatsevichs paper, is indicated.
The aim of this paper is to give all quasitriangular structures on a class of semisimple Hopf algebras constructed through abelian extensions of $Bbbkmathbb{Z}_{2}$ by $Bbbk^G$ for an abelian group $G$. We first introduce the concept of symmetry of quasitriangular structures of Hopf algebras and obtain some related propositions which can be used to simplify our calculations of quasitriangular structures. Secondly, we find that quasitriangular structures of these semisimple Hopf algebras can do division-like operations. Using such operations we transform the problem of solving the quasitriangular structures into solving general solutions and giving a special solution. Then we give all general solutions and get a necessary and sufficient condition for the existence of a special solution.
We extend the classification of solvable Lie algebras with abelian nilradicals to classify solvable Leibniz algebras which are one dimensional extensions of an abelian nilradicals.
We describe infinitesimal deformations of complex naturally graded filiform Leibniz algebras. It is known that any $n$-dimensional filiform Lie algebra can be obtained by a linear integrable deformation of the naturally graded algebra $F_n^3(0)$. We establish that in the same way any $n$-dimensional filiform Leibniz algebra can be obtained by an infinitesimal deformation of the filiform Leibniz algebras $F_{n}^1,$ $F_{n}^2$ and $F_{n}^3(alpha)$. Moreover, we describe the linear integrable deformations of above-mentioned algebras with a fixed basis of $HL^2$ in the set of all $n$-dimensional Leibniz algebras. Among these deformations we found one new rigid algebra.
In this paper, first we show that under the assumption of the center of h being zero, diagonal non-abelian extensions of a regular Hom-Lie algebra g by a regular Hom-Lie algebra h are in one-to-one correspondence with Hom-Lie algebra morphisms from g to Out(h). Then for a general Hom-Lie algebra morphism from g to Out(h), we construct a cohomology class as the obstruction of existence of a non-abelian extension that induce the given Hom-Lie algebra morphism.
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