Semisimple Quaternionic Skew-Hermitian Spaces

• Semisimple quaternionic skew-Hermitian symmetric spaces are 4n-dimensional manifolds defined by invariant torsion-free connections and quaternionic structures.
• They feature a reductive decomposition with a distinguished subalgebra and invariant tensorial data including a skew-Hermitian form and a fundamental 4-tensor.
• The classification divides these spaces into three families based on Lie group quotients and Satake diagram invariants, ensuring intrinsic zero torsion.

A semisimple quaternionic skew-Hermitian symmetric space is a $4n$-dimensional symmetric space $M = K/L$, where $K$ is a semisimple Lie group and $L$ is a closed subgroup, equipped with a $K$-invariant, torsion-free $\mathsf{SO}^*(2n)\mathsf{Sp}(1)$-structure. Such geometries, also described as quaternionic skew-Hermitian symmetric spaces, are characterized by canonical tensorial data: a skew-Hermitian form, a parallel fundamental $4$-tensor, and distinguished underlying symmetries arising from the quaternionic real form $\mathsf{SO}^*(2n)$ of $\mathsf{SO}(2n,\mathbb C)$ and the group $\mathsf{Sp}(1)$ acting as a quaternionic structure. The complete classification consists of three infinite families, each explicitly describable via Lie-theoretic, tensorial, and connection-theoretic invariants (Chrysikos et al., 2021).

1. Underlying Algebraic Structures and Symmetry

Each semisimple quaternionic skew-Hermitian symmetric space $M = K/L$ possesses the following algebraic and geometric data at the origin $o = eL$, expressed via the reductive decomposition $\mathfrak{k} = \mathfrak{l} \oplus \mathfrak{m}$, with $[\mathfrak{l}, \mathfrak{m}] \subset \mathfrak{m}$ and $[\mathfrak{m}, \mathfrak{m}] \subset \mathfrak{l}$.

Key features:

• There exists a distinguished subalgebra $\mathfrak{sp}(1) \subset \mathfrak{l}$, each mapping to a quaternionic structure on $\mathfrak{m}$ via its natural action on $\operatorname{End}(\mathfrak{m})$.
• A one-dimensional center in $\mathfrak{l}$ yields an $L$-invariant complex structure $I$ on $\mathfrak{m}$.
• The (restricted) Killing form $B_{\mathfrak{k}}$ yields a pseudo-Hermitian metric $g(\cdot, \cdot) = B_{\mathfrak{k}}(\cdot, \cdot)$ of signature $(2n, 2n)$ on $\mathfrak{m}$.

These structures combine to induce a unique $K$-invariant, torsion-free $\mathsf{SO}^*(2n)\mathsf{Sp}(1)$-geometry, and all invariants may be described at $o$ then propagated by left multiplication (Chrysikos et al., 2021).

2. Canonical Invariant Tensors and Forms

Canonical tensors arise systematically:

• Invariant $2$-form: $\omega(X, Y) = g(I X, Y)$, $L$-stable, skew-symmetric, and nondegenerate.
• Quaternionic Skew-Hermitian Form:

$$h = \omega \otimes \operatorname{Id} + \sum_{a=1}^3 g_a \otimes J_a,$$

where $J_1 = I$, and $J_2,J_3$ arise from the $\mathfrak{sp}(1) \subset \mathfrak{l}$ action, with $g_a(x, y) = g(x, J_a y)$.

• Fundamental $4$-tensor:

$$\Phi = g_1 \odot g_1 + g_2 \odot g_2 + g_3 \odot g_3.$$

Each of $h$ and $\Phi$ is $L$-invariant and extends to a parallel tensor on $M$ under the canonical symmetric-space connection.

3. Classification: The Three Families

The complete list of such symmetric spaces consists of three infinite families, distinguished by their group quotient structure and Satake/Dynkin invariants:

Family	$K$	$L$	$\dim_\mathbb{R} K/L$	Satake Data
$$\mathsf{SO}^*(2n+2)/\left(\mathsf{SO}^*(2n)\times U(1)\right)$$	$\mathsf{SO}^*(2n+2)$	$\mathsf{SO}^*(2n)\times U(1)$	$4n$	$A_{n+1}$, node $n$ black
$$SU(2+p,q)/\left(SU(2)\times SU(p,q)\times U(1)\right)$$	$SU(2+p,q)$	$SU(2)\times SU(p,q)\times U(1)$	$4n$ ($n = p+q$)	$A_{n+1}$, nodes $1,n+1$ black
$$SL(n+1,\mathbb{H})/\left(GL(1,\mathbb{H})\times SL(n,\mathbb{H})\right)$$	$SL(n+1,\mathbb{H})$	$GL(1,\mathbb{H})\times SL(n,\mathbb{H})$	$4n$	$C_{n+1}$, last two white

In each case, $\mathfrak{m} \cong \mathbb{H}^n$ (or, in the $SU(2+p,q)$ case, $\mathbb{C}^{2n}$ with adapted structures), and the various forms and tensors admit concrete saddle-point expressions in terms of Darboux bases, quaternionic units, and Killing forms, as elaborated for each family in (Chrysikos et al., 2021).

4. Geometric and Connection Properties

A defining feature is the existence of a unique, minimal, torsion-free connection compatible with all invariant structures, provided by the canonical principal $L$-connection:

$$\omega = \operatorname{pr}_{\mathfrak{l}} \circ \kappa, \quad \kappa \in \Omega^1(K, \mathfrak{k}),$$

with curvature

$$\Omega = -\frac{1}{2} [\kappa_{\mathfrak{m}} \wedge \kappa_{\mathfrak{m}} ]_{\mathfrak{l}},$$

and vanishing torsion $T = 0$. The curvature components satisfy the symmetric-space Bianchi identities and preserve the tensors $h$ and $\Phi$ by construction.

The intrinsic torsion is identically zero, as the symmetric-space condition $[\mathfrak{m}, \mathfrak{m}]\subset \mathfrak{l}$ guarantees the canonical connection is torsion-free. In root-theoretic terms, examination of the Satake diagrams confirms the absence of "torsion-modules" in the Spencer decomposition, with no obstruction arising in $H^{0,2}$ of $\mathfrak{so}^*(2n)\oplus\mathfrak{sp}(1)$ (Chrysikos et al., 2021).

5. Tensorial Realizations and Satake Diagram Data

Within each family, all invariant data can be concretely realized:

• $\mathsf{SO}^*(2n+2)$ family: At $o$, a Darboux basis for $\mathfrak{m} \cong \mathbb{H}^n$ allows explicit expressions for $g_o$, $\omega$, and the quaternionic structures $J_1, J_2, J_3$, and all associated tensors.
• $SU(2+p,q)$ family: The isotropy $U(1)\subset L$ acts by a complex structure on $\mathfrak{m}\simeq \mathbb{C}^{2n}$, complemented by quaternionic structures from $\mathfrak{sp}(1)\subset\mathfrak{l}$.
• $SL(n+1,\mathbb{H})$ family: The $$GL(1,\mathbb{H}) = \mathbb{R}^+ \times \mathsf{Sp}(1)$$ in $L$ provides both a quaternionic structure and a $U(1)$-like complex structure $I$ on $\mathfrak{m}\cong\mathbb{H}^n$.

The Satake diagrams, characterized by nodes marked for isotropy subgroups (black/white), precisely encode the algebraic conditions under which all invariant geometries are realized (Chrysikos et al., 2021).

6. Summary of Classification and Key Properties

The following properties are universal across the classification:

• Dimensionality: Each $K/L$ is $4n$-dimensional with quaternionic dimension $n$.
• Structural invariants: The forms $h$ and $\Phi$ are $L$-invariant and parallel under the canonical (symmetric-space) connection.
• Intrinsic torsion: Universally zero due to reductive symmetry.
• Canonical connection: Given by projection of the Maurer–Cartan form, its curvature and torsion fitting standard symmetric-space criteria.

No further infinite families arise; all semisimple symmetric spaces with invariant, torsion-free $\mathsf{SO}^*(2n)\mathsf{Sp}(1)$-structure appear within these three cases, and all geometric, algebraic, and connection-theoretic invariants are constructible at the origin and extend globally via the symmetry group action (Chrysikos et al., 2021).