Some problems concerning to nilpotent Lie superalgebras (Q557260)

Given a nilpotent Lie superalgebra \(\mathfrak{g}=\mathfrak{g}_{\bar 0}\oplus \mathfrak{g}_{\bar 1}\), the descending central sequence is defined by \(\mathcal{C}^0(\mathfrak{g})=\mathfrak{g}\) and \(\mathcal{C}^{k+1}(\mathfrak{g})=[\mathfrak{g},\mathcal{C}^k(\mathfrak{g})]\); and the least natural number \(k\) such that \(\mathcal{C}^k(\mathfrak{g})=0\) is called the nilindex of \(\mathfrak{g}\). This nilindex is at most \(\dim\mathfrak{g}-1\). The paper under review is devoted to study the function \(f(n,m)\) given by the maximal possible nilindex among the nilpotent Lie superalgebras with \(\dim\mathfrak{g}_{\bar 0}=n\) and \(\dim\mathfrak{g}_{\bar 1}=m\). It is proven that, for \(m>0\), \(f(n,m)=n+m-1\, (=\dim\mathfrak{g} -1)\) if and only if \(n=2\) and \(m\) is odd. In this case, the maximal nilindex is attained by a unique superalgebra (up to isomorphism). Sufficient conditions for \(f(n,m)=n+m-2\) are given too, and some low dimensional examples are explored.