The classification of Leibniz superalgebras of nilindex n+m (m≠0)

AbstractIn this paper we investigate the description of the complex Leibniz superalgebras with nilindex n+m, where n and m (m≠0) are dimensions of even and odd parts, respectively. In fact, such superalgebras with characteristic sequence equal to (n1,…,nk|m1,…,ms) (where n1+⋯+nk=n, m1+⋯+ms=m) for n1⩾n−1 and (n1,…,nk|m) were classified in works by Ayupov et al. (2009) [3], Camacho et al. (2010) [4], Camacho et al. (in press) [5], Camacho et al. (in press) [6]. Here we prove that in the case of (n1,…,nk|m1,…,ms), where n1⩽n−2 and m1⩽m−1 the Leibniz superalgebras have nilindex less than n+m. Thus, we complete the classification of Leibniz superalgebras with nilindex n+m

Naturally graded quasi-filiform Leibniz algebras

AbstractThe classification of naturally graded quasi-filiform Lie algebras is known; they have the characteristic sequence (n−2,1,1) where n is the dimension of the algebra. In the present paper we deal with naturally graded quasi-filiform non-Lie–Leibniz algebras which are described by the characteristic sequence C(L)=(n−2,1,1) or C(L)=(n−2,2). The first case has been studied in [Camacho, L.M., Gómez, J.R., González, A.J., Omirov, B.A., 2006. Naturally graded 2-filiform Leibniz Algebra and its applications, preprint, MA1-04-XI06] and now, we complete the classification of naturally graded quasi-filiform Leibniz algebras. For this purpose we use the software Mathematica (the program used is explained in the last section)

Classiffication of 4-dimensional nilpotent complex Leibniz algebras

In [8] and [9] several classes of new algebras were introduced. Some of them have two generating operations and they are called dialgebras. The first motivation to introduce such algebraic structures (related with well known Lie and associative algebras) were problems in algebraic K-theory. The categories of these algebras over their operads assemble into the com- mutative diagram which re°ects the Koszul duality of those categories. The aim of the present paper is to study structural properties of one class of Lo- day's list, namely the so called Leibniz algebras. Leibniz algebras present a \non-commutative" (to be more precise, a \non- antisymmetric") analogue of Lie algebras.peerReviewe

Solvable Leibniz algebras with NFn⊕ Fm1Fm1\begin{array}{} F_{m}^{1} \end{array} nilradical

All finite-dimensional solvable Leibniz algebras L, having N = NFn⊕ Fm1$\begin{array}{} F_{m}^{1} \end{array}$ as the nilradical and the dimension of L equal to n+m+3 (the maximal dimension) are described. NFn and Fm1$\begin{array}{} F_{m}^{1} \end{array}$ are the null-filiform and naturally graded filiform Leibniz algebras of dimensions n and m, respectively. Moreover, we show that these algebras are rigid