---
title: Semisimple Quaternionic Skew-Hermitian Spaces
url: https://www.emergentmind.com/topics/semisimple-quaternionic-skew-hermitian-symmetric-spaces
type: topic
---

# Semisimple Quaternionic Skew-Hermitian Spaces

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 [2109.15253].

## 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 [2109.15253].

## 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 [2109.15253].

## 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 $\mathfrak{so}^*(2n)\oplus\mathfrak{sp}(1)$ [2109.15253].

## 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 [2109.15253].

## 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 [2109.15253].

Source: https://www.emergentmind.com/topics/semisimple-quaternionic-skew-hermitian-symmetric-spaces