SL(1,H) - Knowlpedia

$\mathrm{SL}(1,\mathbb H)$ is the real Lie group of quaternionic-linear invertible maps of the right module $\mathbb H^{1}$ with $\det_{\mathbb C}\rho(A)=1$ , with composition as multiplication. Here a choice $\mathbb C\subset\mathbb H$ identifies $\mathbb H^{1}$ with $\mathbb C^{2}$ and gives $\rho(A)$ .

Dimensions and structure

This group has real dimension $3$ . The complex determinant of this realization is positive real for every invertible quaternionic matrix. The determinant-one equation means reduced norm one; it is not the naive Leibniz determinant of a noncommutative matrix. In size one this is the unit-quaternion group Sp(1).

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.

For a single quaternion, the determinant of its complex realization is its squared norm. The determinant-one group is precisely the unit quaternions. See $\operatorname{Sp}(1)$ .

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.