For most complex 9-dimensional filiform Lie algebra we find another non isomorphic Lie algebra that degenerates to it. Since this is already known for nilpotent Lie algebras of rank $geq 1$, only the characteristically nilpotent ones should be considered.
In this paper we describe central extensions of some nilpotent Leibniz algebras. Namely, central extensions of the Leibniz algebra with maximal index of nilpotency are classified. Moreover, non-split central extensions of naturally graded filiform no
n-Lie Leibniz algebras are described up to isomorphism. It is shown that $k$-dimensional central extensions ($kgeq 5$) of these algebras are split.
We prove that there are no rigid complex filiform Lie algebras in the variety of (filiform) Lie algebras of dimension less than or equal to 11. More precisely we show that in any Euclidean neighborhood of a filiform Lie bracket (of low dimension), th
ere is a non-isomorphic filiform Lie bracket. This follows by constructing non trivial linear deformations in a Zariski open dense set of the variety of filiform Lie algebras of dimension 9, 10 and 11. (In lower dimensions this is well known.)
Let $p$ be a prime, $k$ be an algebraically closed field of characteristic $p$. In this paper, we provide the classification of connected Hopf algebras of dimension $p^3$, except the case when the primitive space of the Hopf algebra is two dimensiona
l and abelian. Each isomorphism class is presented by generators $x, y, z$ with relations and Hopf algebra structures. Let $mu$ be the multiplicative group of $(p^2+p-1)$-th roots of unity. When the primitive space is one-dimensional and $p$ is odd, there is an infinite family of isomorphism classes, which is naturally parameterized by $A_{k}^1/mu$.
Based on the differential graded Lie algebra controlling deformations of an $n$-Lie algebra with a representation (called an n-LieRep pair), we construct a Lie n-algebra, whose Maurer-Cartan elements characterize relative Rota-Baxter operators on n-L
ieRep pairs. The notion of an n-pre-Lie algebra is introduced, which is the underlying algebraic structure of the relative Rota-Baxter operator. We give the cohomology of relative Rota-Baxter operators and study infinitesimal deformations and extensions of order m deformations to order m+1 deformations of relative Rota-Baxter operators through the cohomology groups of relative Rota-Baxter operators. Moreover, we build the relation between the cohomology groups of relative Rota-Baxter operators on n-LieRep pairs and those on (n+1)-LieRep pairs by certain linear functions.