SO(2,R) - Knowlpedia

$\operatorname{SO}(2,\mathbb R)$ is the special orthogonal group

\operatorname{SO}(2,\mathbb R)=\{A\in M_{2}(\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 $1$ . Its tangent algebra consists of skew-symmetric matrices, $X^{\mathsf T}+X=0$ . It is connected.

The determinant-one subgroup includes by the identity on matrices. See $\operatorname{O}(2,\mathbb R)$ .

The map $\left(\begin{smallmatrix}a&-b\\b&a\end{smallmatrix}\right)\mapsto a+ib$ is a real Lie group isomorphism. See $\operatorname{U}(1)$ .

References