quaternionic unitary group in nLab

Contents

See also compact symplectic group.

Contents

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

Properties

Exceptional isomorphisms

• Sp(1)≃Sp(1) \simeq Spin(3) (this Prop.)

• Sp(2)≃Sp(2) \simeq Spin(5) (this Prop.)

• A Riemannian manifold of dimension 4n4n 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)×Sp(1)Sp(n) \times Sp(1). If it is even a subgroup of just the Sp(n)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)