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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.
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
We describe all degenerations of three dimensional anticommutative algebras $mathfrak{Acom}_3$ and of three dimensional Leibniz algebras $mathfrak{Leib}_3$ over $mathbb{C}.$ In particular, we describe all irreducible components and rigid algebras in the corresponding varieties
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-abelia
In this paper the description of solvable Lie algebras with triangular nilradicals is extended to Leibniz algebras. It is proven that the matrices of the left and right operators on elements of Leibniz algebra have upper triangular forms. We establis
A classification exists for Lie algebras whose nilradical is the triangular Lie algebra $T(n)$. We extend this result to a classification of all solvable Leibniz algebras with nilradical $T(n)$. As an example we show the complete classification of all Leibniz algebras whose nilradical is $T(4)$.