Classification of quaternionic skew-Hermitian symmetric spaces

TL;DR

This work achieves a complete classification of quaternionic skew-Hermitian symmetric spaces, i.e., simply connected symmetric spaces admitting an invariant $SO^*(2n)Sp(1)$-structure, and proves that any homogeneous qs-H manifold is necessarily symmetric. The authors develop a Lie-algebraic framework via quaternionic skew-Hermitian transvection involutive Lie algebras (qs-H tiLas) and treat both semisimple and non-semisimple cases: the semisimple case yields the known simple-type spaces $M_1, M_2, M_3$ with additional invariant geometric structures, while the non-semisimple case is captured by Cahen–Schwachhöfer parabolic constructions, parameterized by 1D subalgebras of $rak{so}^*(2n+4)$. They further show that homogeneous torsion-free qs-H structures exist and that homogeneous almost qs-H manifolds with non-zero torsion belong to the $oldsymbol{ ext X}_4$-component, illustrating the boundary between symmetric and non-symmetric homogeneous geometries. Overall, the paper unifies algebraic and geometric methods to deliver a complete classification and clarifies the role of intrinsic torsion in the qs-H setting, with implications for quaternionic pseudo-Kähler and related geometries.

Abstract

We provide a complete classification of quaternionic skew-Hermitian symmetric spaces, namely symmetric spaces that admit a torsion-free ${\rm SO}^{*}(2n){\rm Sp}(1)$-structure for arbitrary $n>1$. Moreover, we prove that any homogeneous quaternionic skew-Hermitian manifold is necessarily a symmetric space.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Corollary 2.5
• Definition 3.1
• Remark 3.2