The enhanced quaternionic general linear group

If we regard the quaternionic vector space ${\displaystyle \mathbb {H} ^{n}\cong \mathbb {R} ^{4n}}$ as a right ${\displaystyle \mathbb {H} }$-module, we can identify the algebra of right ${\displaystyle \mathbb {H} }$-linear maps with the algebra of ${\displaystyle n\times n}$ quaternionic matrices acting on ${\displaystyle \mathbb {H} ^{n}}$ from the left. The invertible right ${\displaystyle \mathbb {H} }$-linear maps then form a subgroup ${\displaystyle \operatorname {GL} (n,\mathbb {H} )}$ of ${\displaystyle \operatorname {GL} (4n,\mathbb {R} )}$. We can enhance this group with the group ${\displaystyle \mathbb {H} ^{\times }}$ of nonzero quaternions acting by scalar multiplication on ${\displaystyle \mathbb {H} ^{n}}$ from the right. Since this scalar multiplication is ${\displaystyle \mathbb {R} }$-linear (but not ${\displaystyle \mathbb {H} }$-linear) we have another embedding of ${\displaystyle \mathbb {H} ^{\times }}$ into ${\displaystyle \operatorname {GL} (4n,\mathbb {R} )}$. The group ${\displaystyle \operatorname {GL} (n,\mathbb {H} )\cdot \mathbb {H} ^{\times }}$ is then defined as the product of these subgroups in ${\displaystyle \operatorname {GL} (4n,\mathbb {R} )}$. Since the intersection of the subgroups ${\displaystyle \operatorname {GL} (n,\mathbb {H} )}$ and ${\displaystyle \mathbb {H} ^{\times }}$ in ${\displaystyle \operatorname {GL} (4n,\mathbb {R} )}$ is their mutual center ${\displaystyle \mathbb {R} ^{\times }}$ (the group of scalar matrices with nonzero real coefficients), we have the isomorphism

${\displaystyle \operatorname {GL} (n,\mathbb {H} )\cdot \mathbb {H} ^{\times }\cong (\operatorname {GL} (n,\mathbb {H} )\times \mathbb {H} ^{\times })/\mathbb {R} ^{\times }.}$

Almost quaternionic structure

An almost quaternionic structure on a smooth manifold ${\displaystyle M}$ is just a ${\displaystyle \operatorname {GL} (n,\mathbb {H} )\cdot \mathbb {H} ^{\times }}$-structure on ${\displaystyle M}$. Equivalently, it can be defined as a subbundle ${\displaystyle H}$ of the endomorphism bundle ${\displaystyle \operatorname {End} (TM)}$ such that each fiber ${\displaystyle H_{x}}$ is isomorphic (as a real algebra) to the quaternion algebra ${\displaystyle \mathbb {H} }$. The subbundle ${\displaystyle H}$ is called the almost quaternionic structure bundle. A manifold equipped with an almost quaternionic structure is called an almost quaternionic manifold.

The quaternion structure bundle ${\displaystyle H}$ naturally admits a bundle metric coming from the quaternionic algebra structure, and, with this metric, ${\displaystyle H}$ splits into an orthogonal direct sum of vector bundles ${\displaystyle H=L\oplus E}$ where ${\displaystyle L}$ is the trivial line bundle through the identity operator, and ${\displaystyle E}$ is a rank-3 vector bundle corresponding to the purely imaginary quaternions. Neither the bundles ${\displaystyle H}$ or ${\displaystyle E}$ are necessarily trivial.

The unit sphere bundle ${\displaystyle Z=S(E)}$ inside ${\displaystyle E}$ corresponds to the pure unit imaginary quaternions. These are endomorphisms of the tangent spaces that square to −1. The bundle ${\displaystyle Z}$ is called the twistor space of the manifold ${\displaystyle M}$, and its properties are described in more detail below. Local sections of ${\displaystyle Z}$ are (locally defined) almost complex structures. There exists a neighborhood ${\displaystyle U}$ of every point ${\displaystyle x}$ in an almost quaternionic manifold ${\displaystyle M}$ with an entire 2-sphere of almost complex structures defined on ${\displaystyle U}$. One can always find ${\displaystyle I,J,K\in \Gamma (Z|_{U})}$ such that

${\displaystyle I^{2}=J^{2}=K^{2}=IJK=-1}$

Note, however, that none of these operators may be extendable to all of ${\displaystyle M}$. That is, the bundle ${\displaystyle Z}$ may admit no global sections (e.g. this is the case with quaternionic projective space ${\displaystyle \mathbb {HP} ^{n}}$). This is in marked contrast to the situation for complex manifolds, which always have a globally defined almost complex structure.

Quaternionic structure

A quaternionic structure on a smooth manifold ${\displaystyle M}$ is an almost quaternionic structure ${\displaystyle Q}$ which admits a torsion-free affine connection ${\displaystyle \nabla }$ preserving ${\displaystyle Q}$. Such a connection is never unique, and is not considered to be part of the quaternionic structure. A quaternionic manifold is a smooth manifold ${\displaystyle M}$ together with a quaternionic structure on ${\displaystyle M}$.

Hypercomplex manifolds

A hypercomplex manifold is a quaternionic manifold with a torsion-free ${\displaystyle \operatorname {GL} (n,\mathbb {H} )}$-structure. The reduction of the structure group to ${\displaystyle \operatorname {GL} (n,\mathbb {H} )}$ is possible if and only if the almost quaternionic structure bundle ${\displaystyle H\subset \operatorname {End} (TM)}$ is trivial (i.e. isomorphic to ${\displaystyle M\times \mathbb {H} }$). An almost hypercomplex structure corresponds to a global frame of ${\displaystyle H}$, or, equivalently, triple of almost complex structures ${\displaystyle I,J}$, and ${\displaystyle K}$ such that

${\displaystyle I^{2}=J^{2}=K^{2}=IJK=-1}$

A hypercomplex structure is an almost hypercomplex structure such that each of ${\displaystyle I,J}$, and ${\displaystyle K}$ are integrable.

Quaternionic Kähler manifolds

A quaternionic Kähler manifold is a quaternionic manifold with a torsion-free ${\displaystyle \operatorname {Sp} (n)\cdot \operatorname {Sp} (1)}$-structure.

Hyperkähler manifolds

A hyperkähler manifold is a quaternionic manifold with a torsion-free ${\displaystyle \operatorname {Sp} (n)}$-structure. A hyperkähler manifold is simultaneously a hypercomplex manifold and a quaternionic Kähler manifold.