Subspaces of a para-quaternionic Hermitian vector space (Q2893164)

Let \(V\) be a real vector space of dimension \(d\). A Lie subalgebra \(\tilde {Q}\subseteq\mathfrak{gl}(V)\) is called a para-quaternionic structure on \(V\) if it admits a basis \((J_{1},J_{2},J_{3})\) of anticommuting endomorphisms of \(V\) satisfying the relations: \(-J_{1}^{2}=J_{2}^{2}=J_{3}^{2}=Id\) and \(J_{1}J_{2}=J_{3}\). Such a basis is called an admissible basis for \(\tilde{Q} \). For example, the matrices NEWLINE\[NEWLINE \mathcal{I}=\left[\begin{matrix} 0 & -1\\ 1 & 0 \end{matrix} \right] ,\mathcal{J}=\left[\begin{matrix} 0 & 1\\ 1 & 0 \end{matrix} \right] ,\mathcal{K}=\left[ \begin{matrix} -1 & 0\\ 0 & 1 \end{matrix} \right] NEWLINE\]NEWLINE define an admissible basis for \(\mathfrak{sl}_{2}(\mathbb{R})\) on \(H:=\mathbb{R}^{2}\). More generally, for any \(n\geq1\) we obtain a para-quaternionic structure \(\mathfrak{sl}(H)\) on \(V\) where \(\dim_{\mathbb{R} }V=2n\) by identifying \(V\) with \(H^{2}\otimes E^{n}\) with admissible basis \(\mathcal{I\otimes}e,\mathcal{J\otimes}e,\mathcal{K\otimes}e\). Indeed, up to isomorphism, this is the only para-quaternionic vector space of this dimension. A para-quaternionic structure is called a para-quaternionic Hermitian structure with respect to a pseudo-Euclidian scalar product \(g\) if any admissible basis is Hermitian with respect to \(g\). If such a \(g\) exists then \(n\) must be even, and in this case there is a unique \(g=\omega^{H}\otimes\omega^{E}\) where \(\omega^{H}\) and \(\omega^{E}\) are symplectic forms on the two factors. The object of the present paper is to describe the para-quaternionic subspaces of \((H^{2}\otimes E^{n} ,\mathfrak{sl}(H),\omega^{H}\otimes\omega^{E})\), and more generally subspaces of the form \(H^{2}\otimes E^{\prime}\) where \(E^{\prime}\subseteq E^{n}\). The results are too complicated to be described here.