GL(1,H) - Knowlpedia

$\mathrm{GL}(1,\mathbb H)$ is the real Lie group of invertible right-quaternionic-linear maps of $\mathbb H^{1}$ , 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 $4$ . The complex determinant of this realization is positive real for every invertible quaternionic matrix. Invertibility is openness in the underlying real matrix space. Quaternionic polar decomposition gives a deformation retraction onto Sp(1). In size one this is the group of nonzero quaternions.

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.

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.