On a generalization of some instability results for Riccati equations via nonassociative algebras

Abstract

In [28], for any real non associative algebra of dimension \(m\geq2\), having \(k\) linearly independent nilpotent elements \(n_{1}\), \(n_{2}\), …, \(n_{k},\) \(1\leq k\leq m-1\), Mencinger and Zalar defined near idempotents and near nilpotents associated to \(n_{1}\), \(n_{2}\), …, \(n_{k}\). Assuming \(\mathcal{N}_{k}\mathcal{N}_{k}=\left\{ 0\right\}\), where \(\mathcal{N} _{k}=\operatorname*{span}\left\{ n_{1},n_{2},\ldots,n_{k}\right\} \), they showed that if there exists a near idempotent or a near nilpotent, called \(u\), associated to \(n_{1},n_{2},\ldots,n_{k}\) verifying \(n_{i}u\in\mathbb{R}n_{i},\) for \(1\leq i\leq k\), then any nilpotent element in \(\mathcal{N}_{k}\) is unstable. They also raised the question of extending their results to cases where \(\mathcal{N}_{k}\mathcal{N}_{k}\not =\left\{ 0\right\} \) with \(\mathcal{N}_{k}\mathcal{N}_{k}\subset\mathcal{N}_{k}\mathcal{\ }\)and to cases where \(\mathcal{N}_{k}\mathcal{N}_{k} \not\subset \mathcal{N}_{k}.\) In this paper, positive answers are emphasized and in some cases under the weaker conditions \(n_{i}u\in\mathcal{N}_{k}\). In addition, we characterize all such algebras in dimension 3.