Subspaces of a para-quaternionic Hermitian vector space

TL;DR

This paper provides an explicit classification of invariant subspaces within para-quaternionic Hermitian vector spaces, using tensorial representations to describe their structure from affine and metric perspectives.

Contribution

It introduces a detailed tensorial framework to characterize invariant subspaces in para-quaternionic Hermitian spaces, enhancing understanding of their geometric structure.

Findings

Explicit description of invariant subspaces in tensorial form

Classification of subspaces based on structure group invariance

Connection between affine and metric properties of subspaces

Abstract

Let $(\tilde Q,g) $ be a para-quaternionic Hermitian structure on the real vector space $V$. By referring to the tensorial presentation $(V, \tilde{Q},g) \simeq (H^2 \otimes E^{2n}, \mathfrak{sl}(H),\omega^H \otimes \omega^E)$, we give an explicit description, from an affine and metric point of view, of main classes of subspaces of $V$ which are invariantly defined with respect to the structure group of $\tilde{Q}$ and $(\tilde{Q},g)$ respectively.