nLab
quaternionic unitary group

Idea

The analog of the unitary group as one passes from the complex numbers to the quaternions.

Properties

Exceptional isomorphisms

• $Sp(1) \simeq$ Spin(3) (this Prop.)

• $Sp(2) \simeq$ Spin(5) (this Prop.)

Related concepts

• A Riemannian manifold of dimension $4n$ is called a quaternion-Kähler manifold if its holonomy group is a subgroup of the quotient group Sp(n).Sp(1) of the direct product group $Sp(n) \times Sp(1)$. If it is even a subgroup of just the $Sp(n)$ factor, then it is called a hyperkähler manifold.

• Gromoll-Meyer sphere

• quaternionic projective space

References

• Quaternionic groups (pdf)