The group Sp(n)Sp(1) - Imported Knowlpedia Content

Definition

For $n\geq1$ , let the compact symplectic group $\operatorname{Sp}(n)$ act on $\mathbb H^n$ from the left and let its unit-quaternion subgroup $\operatorname{Sp}(1)$ act from the right. The homomorphism

\operatorname{Sp}(n)\times\operatorname{Sp}(1)\longrightarrow\operatorname{SO}(4n), \qquad (A,q)\longmapsto\bigl(v\mapsto Avq^{-1}\bigr)

has kernel $\{(I,1),(-I,-1)\}$ . Its image is the group $\operatorname{Sp}(n)\operatorname{Sp}(1)$ . Hence

\operatorname{Sp}(n)\operatorname{Sp}(1) \cong \bigl(\operatorname{Sp}(n)\times\operatorname{Sp}(1)\bigr)/\{\pm(I,1)\}.

It is a compact connected Lie subgroup of $\operatorname{SO}(4n)$ , not the direct product denoted by simply placing the two factors side by side.

Action on quaternionic structures

The left $\operatorname{Sp}(n)$ -factor commutes with right quaternionic multiplication. The right $\operatorname{Sp}(1)$ -factor conjugates the three-dimensional space $\operatorname{Im}\mathbb H$ of imaginary units, inducing the standard rotation action through $\operatorname{Sp}(1)/\{\pm1\}\cong\operatorname{SO}(3)$ . Thus the quotient preserves the rank-three family of complex structures while generally rotating its local basis $I,J,K$ . This is why the group, rather than $\operatorname{Sp}(n)$ alone, is natural in quaternionic Kähler geometry Besse, §14.G.

Low-rank case and dimensions

Its Lie algebra is $\mathfrak{sp}(n)\oplus\mathfrak{sp}(1)$ , because quotienting by a finite central subgroup does not change the Lie algebra. Therefore

\dim\operatorname{Sp}(n)\operatorname{Sp}(1)=n(2n+1)+3.

For $n=1$ , the action identifies $\operatorname{Sp}(1)\operatorname{Sp}(1)$ with $\operatorname{SO}(4)$ . In quaternionic Kähler geometry the holonomy condition is normally stated for $n\geq2$ ; real dimension four is treated separately.

Relation to geometric structures

A Riemannian $4n$ -manifold whose frame bundle reduces to this subgroup has an almost quaternionic Hermitian structure. If its Levi–Civita holonomy group lies in $\operatorname{Sp}(n)\operatorname{Sp}(1)$ , it is quaternionic Kähler under the standard higher-dimensional convention. By contrast, holonomy contained in the smaller subgroup $\operatorname{Sp}(n)$ is the hyperkähler condition. The two conditions should not be conflated Salamon, pp. 143–145.

Conventions and scope

Authors variously print the middle operation as adjacency, a centered dot, or an explicit quotient. None of these denotes a direct product in this context. The expression $\{\pm(I,1)\}$ abbreviates the two diagonal central elements $(I,1)$ and $(-I,-1)$ , not four independently chosen signs.

References

Arthur L. Besse, Einstein Manifolds, Springer, 1987. DOI record. Relevant: §14.G on quaternionic Kähler manifolds and their holonomy group.
Simon Salamon, “Quaternionic Kähler Manifolds,” Inventiones Mathematicae 67 (1982), 143–171. DOI record. Relevant: pp. 143–145 on the groups $\operatorname{Sp}(n)\operatorname{Sp}(1)$ and $\operatorname{GL}(n,\mathbb H)\operatorname{Sp}(1)$ .