SO(1,R) - Knowlpedia

$\operatorname{SO}(1,\mathbb R)$ is the special orthogonal group

\operatorname{SO}(1,\mathbb R)=\{A\in M_{1}(\mathbb R):A^{\mathsf T}A=I,\ \det A=1\},

with matrix multiplication. The preserved form is the positive-definite real Euclidean form.

Dimensions and structure

This group has real dimension $0$ . Its tangent algebra consists of skew-symmetric matrices, $X^{\mathsf T}+X=0$ . The determinant-one subgroup is trivial.

The determinant-one subgroup includes by the identity on matrices. See $\operatorname{O}(1,\mathbb R)$ .

References

Pavel Etingof, Lie Groups and Lie Algebras (MIT lecture notes), §6.1, pp. 38–39; Proposition 6.3 and Exercise 6.5.