# 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  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 ).

How to Cite This Entry:
Quaternionic structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quaternionic_structure&oldid=48398
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article