Quaternionic structure

A quaternionic structure on a real vector space $V$ is a module structure over the skew-field of quaternions $\mathbf H$, that is, a subalgebra $H$ of the algebra $\mathop{\rm End} V$ of endomorphisms of $V$ induced by two anti-commutative complex structures $J _ {1} , J _ {2}$ on $V$( cf. Complex structure). The endomorphisms $J _ {1} , J _ {2}$ are called standard generators of the quaternionic structure $H$, and the basis $\{ \mathop{\rm id} , J _ {1} , J _ {2} , J _ {3} = J _ {1} J _ {2} \}$ of $H$ defined by them is called the standard basis. A standard basis is defined up to automorphisms of $H$. The algebra $H$ is isomorphic to the algebra of quaternions (cf. Quaternion). An automorphism $A$ of the vector space $V$ is called an automorphism of the quaternionic structure if the transformation $\mathop{\rm Ad} A$ of the space of automorphisms induced by it preserves $H$, that is, if $( \mathop{\rm Ad} A ) H = A H A ^ {-1} = H$. If, moreover, the identity automorphism is induced on $H$, then $A$ is called a special automorphism of the quaternionic structure. The group of all special automorphisms of the quaternionic structure is isomorphic to the general linear group $\mathop{\rm GL} ( m , \mathbf H )$ over the skew-field $\mathbf H$, where $4 m = \mathop{\rm dim} V$. The group of all automorphisms of a quaternionic structure is isomorphic to the direct product with amalgamation of the subgroup $\mathop{\rm GL} ( m , \mathbf H )$ and the group of unit quaternions $H _ {1} \approx \mathop{\rm Sp} ( 1)$.

A quaternionic structure on a differentiable manifold is a field of quaternionic structures on the tangent spaces, that is, a subbundle $\pi : H \rightarrow M$ of the bundle $\mathop{\rm End} ( T ( M)) \rightarrow M$ of endomorphisms of tangent spaces whose fibres ${\mathcal H} _ {p} = \pi ^ {-1} ( p)$ are quaternionic structures on the tangent spaces $T _ {p} M$ for all $p \in M$. A pair of anti-commutative almost-complex structures $J _ {1} , J _ {2}$ on the manifold $M$ is called a special quaternionic structure. It induces the quaternionic structure $H$, where

$$H _ {p} = \{ {J = \lambda _ {0} \mathop{\rm id} + \lambda _ {1} J _ {1} + \lambda _ {2} J _ {2} + \lambda _ {3} J _ {1} J _ {2} } : { \lambda _ {i} \in \mathbf R } \} .$$

A quaternionic structure $H$ on a manifold $M$ is induced by a special quaternionic structure if and only if the bundle $H \rightarrow M$ is trivial. A quaternionic structure on a manifold can be regarded as a $\mathop{\rm Sp} ( 1) \cdot \mathop{\rm GL} ( m , \mathbf H )$- structure, and a special quaternionic structure as a $\mathop{\rm GL} ( m , \mathbf H )$- structure in the sense of the theory of $G$- structures (cf. $G$- structure). Hence, in order that a quaternionic structure (or a special quaternionic structure) should exist on a manifold $M$, it is necessary and sufficient that the structure group of the tangent bundle reduces to the group $\mathop{\rm Sp} ( 1) \cdot \mathop{\rm Sp} ( m)$( or $\mathop{\rm Sp} ( m)$). The first prolongation of a special quaternionic structure, regarded as a $\mathop{\rm GL} ( m , \mathbf H )$- structure, is an $e$- structure (a field of frames), which determines a canonical linear connection associated with the special quaternionic structure. The vanishing of the curvature and torsion of this connection is a necessary and sufficient condition for the special quaternionic structure to be locally equivalent to the standard flat special quaternionic structure on the vector space $\mathbf R ^ {4m}$.

A quaternionic Riemannian manifold is the analogue of a Kähler manifold for quaternionic structures. It is defined as a Riemannian manifold $M$ of dimension $4 m$ whose holonomy group $\Gamma$ is contained in the group $\mathop{\rm Sp} ( 1) \cdot \mathop{\rm Sp} ( m)$. If $\Gamma \subset \mathop{\rm Sp} ( m)$, then the quaternionic Riemannian manifold is called a special or quaternionic Kähler manifold, and it has zero Ricci curvature. A quaternionic Riemannian manifold can be characterized as a Riemannian manifold $M$ in which there exists a quaternionic structure $H$ that is invariant with respect to Levi-Civita parallel displacement. Similarly, a special quaternionic Riemannian manifold is a Riemannian manifold in which there exists a special quaternionic structure $( J _ {1} , J _ {2} )$ that is invariant with respect to Levi-Civita parallel displacement: $\nabla J _ {1} = \nabla J _ {2} = 0$, where $\nabla$ is the operator of covariant differentiation of the Levi-Civita connection.

In a quaternionic Riemannian manifold there exists a canonical parallel $4$- form that defines a number of operators in the ring $\Lambda ( M)$ of differential forms on $M$ that commute with the Laplace–Beltrami operator (exterior product operator, contraction operators). This enables one to construct an interesting theory of harmonic differential forms on quaternionic Riemannian manifolds [2] analogous to Hodge theory for Kähler manifolds, and to obtain estimates for the Betti numbers of the manifold $M$( cf. Hodge structure; Betti number). Locally Euclidean spaces account for all the homogeneous special quaternionic Riemannian manifolds. As an example of a homogeneous quaternionic Riemannian manifold that is not special one may cite the quaternionic projective space and also other Wolf symmetric spaces which are in one-to-one correspondence with simple compact Lie groups without centre (cf. Symmetric space). These account for all compact homogeneous quaternionic Riemannian manifolds. A wide class of non-compact non-symmetric homogeneous quaternionic Riemannian manifolds can be constructed by means of modules over Clifford algebras (see [5]).

References

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[2]	V.Y. Kraines, "Topology of quaternionic manifolds" Trans. Amer. Math. Soc. , 122 (1966) pp. 357–367 MR0192513 Zbl 0148.16101
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[5]	D.V. Alekseevskii, "Classification of quaternionic spaces with a transitive solvable group of motions" Math. USSR Izv. , 9 : 2 (1975) pp. 297–339 Izv. Akad. Nauk SSSR Ser. Mat. , 39 : 2 (1975) pp. 315–362 MR402649 Zbl 0324.53038
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