SO(2,C) - Knowlpedia

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

\operatorname{SO}(2,\mathbb C)=\{A\in M_{2}(\mathbb C):A^{\mathsf T}A=I,\ \det A=1\},

with matrix multiplication. The preserved form is the complex bilinear form $\sum_j x_jy_j$ ; transpose here does not include complex conjugation.

Dimensions and structure

This group has real dimension $2$ , complex dimension $1$ . Its tangent algebra consists of skew-symmetric matrices, $X^{\mathsf T}+X=0$ . It is connected. A Hermitian form would instead lead to a real unitary group; the complex orthogonal group uses a bilinear form.

Its complex Lie group structure uses holomorphic multiplication and inversion. Forgetting the complex structure gives the underlying real Lie group; real and holomorphic homomorphisms belong to different categories.

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

Writing $A=\left(\begin{smallmatrix}a&-b\\b&a\end{smallmatrix}\right)$ with $a^2+b^2=1$ , the map $A\mapsto a+ib$ identifies SO(2,C) with C*. Its inverse uses $a=(z+z^{-1})/2$ , $b=(z-z^{-1})/(2i)$ . See $\mathbb C^\times$ .

References

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