[Paper Review] Weakly irreducible subgroups of $Sp(1,n+1)$

Natalia I. Bezvitnaya|arXiv (Cornell University)|Apr 2, 2008

Geometry and complex manifolds9 references3 citations

TL;DR

This paper classifies connected weakly irreducible not irreducible subgroups of $\mathrm{Sp}(1,n+1)$ up to conjugacy in $\mathrm{SO}(4,4n+4)$, under a natural condition, by analyzing their action on the boundary of quaternionic hyperbolic space via a stereographic-type map. The key result is a complete classification of such subgroups using similarity transformations on $\mathbb{H}^n$, leading to a systematic description of the corresponding holonomy algebras via types I–XI, with explicit Lie algebra structures based on the decomposition of invariant subspaces.

ABSTRACT

Connected weakly irreducible not irreducible subgroups of $Sp(1,n+1)\subset SO(4,4n+4)$ that satisfy a certain additional condition are classified. This will be used to classify connected holonomy groups of pseudo-hyper-Kählerian manifolds of index 4.

Motivation & Objective

• To classify connected weakly irreducible not irreducible subgroups of $\mathrm{Sp}(1,n+1) \subset \mathrm{SO}(4,4n+4)$ up to conjugacy.
• To extend the classification of holonomy groups of pseudo-hyper-Kählerian manifolds of index 4.
• To characterize such subgroups via their action on the boundary of quaternionic hyperbolic space and the induced action on $\mathbb{H}^n$ via a stereographic-type map.
• To identify the corresponding Lie algebras as preimages under a Lie algebra homomorphism from the algebra of similarity transformations on $\mathbb{H}^n$.

Proposed method

• Define a map $s_1: S^{4n+3} \setminus \{\text{point}\} \to \mathbb{H}^n$ resembling stereographic projection to relate the action of $G \subset \mathrm{Sp}(1,n+1)$ to $\mathrm{Sim}\, \mathbb{H}^n$.
• Construct a map $F: G \to \mathrm{Sim}\, \mathbb{H}^n$ via composition with inverse stereographic projection, so that $F(G)$ acts transitively on an affine subspace $L \subset \mathbb{R}^{4n}$.
• Analyze the structure of $F(G)$ by decomposing $L$ into orthogonal components invariant under $i$, $j$, $k$, leading to a decomposition $L = H^m \oplus \mathbb{C}^k \oplus \mathbb{R}^{n-m-k}$.
• Use the classification of transitive subgroups of $\mathrm{Sim}\, \mathbb{H}^n$ into types R, $\phi$, $\psi$ to classify the corresponding Lie algebras $\mathfrak{g} = (dF)^{-1}(\mathfrak{k})$.
• Identify the kernel of $dF$ as a 3-dimensional ideal $\mathfrak{B}$, so $\mathfrak{g} = (dF)^{-1}(\mathfrak{k})$ with $\mathfrak{k} \subset \mathfrak{L}(\mathrm{Sim}\, \mathbb{H}^n)$.
• Generalize the classification to include subspaces $L$ with multiple nontrivial inclusions in the $i,j,k$-invariant decomposition, introducing new types X and XI for the Lie algebras.

Experimental results

Research questions

• RQ1 Which connected weakly irreducible not irreducible subgroups of $\mathrm{Sp}(1,n+1)$ arise as holonomy groups of pseudo-hyper-Kählerian manifolds of index 4?
• RQ2 How can the action of such subgroups on the boundary of quaternionic hyperbolic space be used to classify their structure?
• RQ3 What is the precise structure of the Lie algebra of such subgroups, and how does it relate to the algebra of similarity transformations on $\mathbb{H}^n$?
• RQ4 How do the invariants of the affine subspace $L \subset \mathbb{R}^{4n}$ determine the possible types of the corresponding Lie algebras?

Key findings

• The classification of weakly irreducible not irreducible subgroups of $\mathrm{Sp}(1,n+1)$ is reduced to classifying transitive actions of subgroups of $\mathrm{Sim}\, \mathbb{H}^n$ on affine subspaces $L \subset \mathbb{R}^{4n}$ with $\mathrm{span}_\mathbb{H} L = \mathbb{H}^n$.
• The Lie algebra of such a subgroup is the preimage under $dF$ of a Lie subalgebra $\mathfrak{k} \subset \mathfrak{L}(\mathrm{Sim}\, \mathbb{H}^n)$, with the kernel of $dF$ being a 3-dimensional ideal $\mathfrak{B}$.
• For $L = \mathbb{H}^n$, the classification yields types I–IX of holonomy algebras, depending on the structure of $\mathfrak{k}$, with Type II, III, IV, and IX arising from specific combinations of $\mathfrak{k}$ of types R, $\phi$, $\psi$, and their generalizations.
• When $L$ is a proper subspace with nontrivial $i,j,k$-invariant components, two new types, X and XI, are introduced to account for the additional structure in the Lie algebra decomposition.
• The classification is complete up to conjugacy in $\mathrm{SO}(4,4n+4)$, and can be refined to conjugacy in $\mathrm{Sp}(1,n+1)$ by considering subspaces $L$ with at least two strict inclusions in the $i,j,k$-invariant decomposition.
• The results provide a foundation for the full classification of holonomy groups of pseudo-hyper-Kählerian manifolds of index 4, extending previous classifications for Lorentzian and pseudo-Kählerian manifolds of index 2.