Infinitesimal deformations of null-filiform Leibniz superalgebras

Abstract

Abstract. In this paper we describe the infinitesimal deformations of null-filiform Leibniz superalgebras over a field of zero characteristic. It is known that up to isomorphism in each dimension there exist two such superalgebras N F n,m . One of them is a Leibniz algebra (that is m = 0) and the second one is a pure Leibniz superalgebra (that is m = 0) of maximum nilindex. We show that the closure of union of orbits of single-generated Leibniz algebras forms an irreducible component of the variety of Leibniz algebras. We prove that any single-generated Leibniz algebra is a linear integrable deformation of the algebra N F n . Similar results for the case of Leibniz superalgebras are obtained.

On Naturally Graded Lie and Leibniz Superalgebras

“…Similar to the case of null-filiform Leibniz algebras, it is easy to check that a Leibniz superalgebra is null-filiform if and only if it is single generated. Moreover, a null-filiform superalgebra has maximal supernilindex (see [27]).…”

“…Consequently, it seems to be the first case to consider the "naturally graded" structure. Note that all these Leibniz superalgebras with supernilindex (n, m) are non-Lie ones and contain in particular the only type of Leibniz superalgebra single generated or socalled null-filiform Leibniz superalgebras (for more details see [27]). It is not difficult to see that these last superalgebras, in the case of non-degenerated, are not naturally graded.…”

Complex cyclic Leibniz superalgebras

“…Note that R a is a (super) derivation if and only if A = A 0 ⊕ A 1 is a Leibniz superalgebra. In fact, the condition for being a derivation of a Leibniz superalgebra (for more details see [11]…”

“…All of the above can be extended for Leibniz superalgebras. On the other hand, in Proposition 3.6 of [11] the authors prove that an arbitray single-generated Leibniz algebra can be obtain by infinitesimal deformations of the null-filiform Leibniz algebra N F n . This result together with the classification of all the cyclic (single-generated) Leibniz algebras (Theorem 2.2) allow us to improve the expression of the irreducible component found in [11].…”

Some irreducible components of the variety of complex (n+1)-dimensional Leibniz algebras

“…In fact, we find bases of the space BL 2 and ZL 2 for these algebras. These descriptions can be applied in the study of infinitesimal deformations and extensions of mentioned algebras (see works [10,11,15,17,21,22] and references therein).…”