Sp(1) - Knowlpedia

$\operatorname{Sp}(1)$ is the compact symplectic group

\operatorname{Sp}(1)=\{A\in M_{1}(\mathbb H):A^*A=I\},

where $A^*=\overline A^{\mathsf T}$ . Matrices act on the right quaternionic module by left multiplication; the adjoint uses quaternionic conjugation.

Dimensions and structure

This group has real dimension $3$ . The tangent algebra is formed by skew-Hermitian quaternionic matrices. The parameter counts quaternionic coordinates. Thus Sp(1) is distinct from the real split group Sp(2,R). No ordinary quaternionic determinant is used in the definition.Its elements are the unit quaternions, a three-sphere under quaternion multiplication.

The scalar field of its matrix entries does not assign a complex Lie group structure to this real Lie group. In particular, a group homomorphism is not called linear merely because the group has a matrix representation.

The complex two-by-two realization of unit quaternions gives a real Lie group isomorphism. See $\operatorname{SU}(2)$ .

References

Pavel Etingof, Lie Groups and Lie Algebras (MIT lecture notes), §6.2, Proposition 6.7 and Corollary 6.8; §6.3, Lemma 6.10, pp. 40–43.