O(1,R) - Knowlpedia

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

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

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, $X^{\mathsf T}+X=0$ . This is the discrete two-element group $\{1,-1\}$ .

References

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