nLab quaternionic unitary group

See also compact symplectic group.

Idea

The quaternion unitary group, often denoted $Sp(n)$ in the math literature (rarely but alternatively: $\mathrm{U}(n, \mathbb{H})$, is a Lie group which is the analog of the unitary group as one passes from the complex numbers to the quaternions, hence it is the group of quaternion-unitary transformations of the quaternionic vector space $\mathbb{H}^n$.

Beware that this group 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 $Sp(2n,\mathbb{C})$, but $Sp(n)$ is the compact form.

Definition

Definition

(1)$$Sp(n) \equiv U(n,\mathbb{H}) \;\coloneqq\; \Big\{ U \in Mat_{n \times n}(\mathbb{H}) \;\Big\vert\; U \cdot U^\dagger = I_n \Big\}$$

(cf. Zhang 1997, p. 28)

Remark

The condition in (1) is indeed sufficient, since for $B \in Mat_{n \times n}(\mathbb{H})$ we have

$$\begin{array}{cl} & B \cdot B^\dagger = I_n \\ \Leftrightarrow & B^\dagger \cdot B = I_n \end{array}$$

(cf. Zhang 1997, Prop. 4.1).

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

• Fuzhen Zhang, p. 28 of: Quaternions and matrices of quaternions, Linear Algebra and its Applications 251 (1997) 21-57 [doi:10.1016/0024-3795(95)00543-9]

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

• Nir Cohen, Stefano De Leo, Cor. 6.4 in: The quaternionic determinant, Electronic Journal of Linear Algebra 7 (2000) 100-111 [arXiv:math-ph/9907015, doi:10.13001/1081-3810.1050, eudml:121484]

See also:

• Quaternionic groups (pdf)