Solvable Leibniz superalgebras whose nilradical has the characteristic sequence \((n-1, 1 |m)\) and nilindex \(n+m\) (Q6591992)

A \(\mathbb{Z}_2\)-graded vector space (or vector superspace) \(V\) is a direct sum of vector spaces \(V_{\bar{0}}\) and \(V_{\bar{1}}\), whose elements are called even and odd, respectively. For a homogeneous element (that is a non-zero element) \(v\in V_\lambda\) with \(\lambda\in\mathbb{Z}_2\), \(|v|=\lambda\) is the degree of \(v\). A (right) Leibniz superalgebra is a \(\mathbb{Z}_2\)-graded vector space \(L=L_{\bar{0}}\oplus L_{\bar{1}}\) equipped with a bilinear map \([-,-] : L \times L \to L\), satisfying the following conditions:\N\begin{itemize}\N\item[(i)] \([L_\lambda,L_\mu]\subseteq L_{\lambda+\mu}\) for every \(\lambda,\mu\in \mathbb{Z}_2\),\N\item[(ii)] \([x,[y,z]]=[[x,y],z]-(-1)^{|y||z|} [[x,z],y]\) (super Leibniz identity),\N\end{itemize}\Nfor every \(x,y,z\in L\). Note that the even part \(L_{\bar{0}}\) of a Leibniz superalgebra is a Leibniz algebra. Also, a Lie superalgebra is a Leibniz superalgebra which satisfies the graded antisymmetric identity \([x,y]=-(-1)^{|x||y|} [y,x]\). Hence Leibniz superalgebras are a generalization of both Leibniz algebras and Lie superalgebras. When a Leibniz superalgebra \(L=L_{\bar{0}}\oplus L_{\bar{1}}\) is of dimension \(n+m\) in which \(\dim L_{\bar{0}}=n\) and \(\dim L_{\bar{1}}=m\), we write \(\dim L=(n|m)\).\N\NIn this paper, it is provided a classification of solvable Leibniz superalgebras whose nilradical has nilindex (nilpotency class) \(n + m\) and characteristic sequence \((n-1, 1|m)\). Specific values are obtained for the parameters of the classes of such nilpotent superalgebras for which they have a solvable extension.