nLab quaternionic unitary group

See also compact symplectic group.

Idea

This Lie group is the analog of the unitary group as one passes from the complex numbers to the quaternions.

The quaternionic unitary group $Sp(n)$ is the group of quaternion-unitary transformations of $\mathbb{H}^n$. It is also called the compact symplectic group, since both it and the symplectic group $Sp(2n, \mathbb{R})$ are real forms of the complex Lie group $Sp(2n,\mathbb{C})$, and it is the compact form.

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

• symplectic group

• SL(2,H)

References

• Howard Georgi, §26 in: Lie Algebras In Particle Physics, Westview Press (1999), CRC Press (2019) [doi:10.1201/9780429499210]

• Quaternionic groups (pdf)