O(n,C) - Knowlpedia

For every integer $n\ge1$ , $\operatorname{O}(n,\mathbb C)$ is the orthogonal group

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

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 $2(n(n-1)/2)$ , complex dimension $n(n-1)/2$ . Its tangent algebra consists of skew-symmetric matrices, $X^{\mathsf T}+X=0$ . It has two connected components distinguished by determinant. 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.

References

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