I am trying to learn Lie theory. In the following I will share my thoughts.
Please, can you check my work for correctness and point out any mistakes to me?
I am trying to determine whether $O(n,\mathbb C)$ and $SL_n(\mathbb C)$ have complex Lie algebras and also whether they are complex Lie groups.
First, a counter example: $U(n)$ is not a (complex) Lie group because its defining equality $U^\ast A = I$ contains complex conjugation which is not holomorphic.
My thoughts on the Lie groups:
Unfortunately, neither of the defining equalities of $O(n,\mathbb C)$ or $SL_n(\mathbb C)$ contain complex conjugation. My thoughts are that $SL_n$ is defined in terms of the determinant and that that's a polynomial and polynomials have all the nice properties, in paritcular, are holomorphic. But that's not a mathematically rigorous argument.
For $O(n,\mathbb C)$ I have similar thoughts: the transpose seems to have nice properties and I can see no reason why it should not be holomorphic.
This leave me with the conviction that both $O(n,\mathbb C)$ and $SL_n(\mathbb C)$ are complex Lie groups but not way to verify this conjecture.
My thoughts on the Lie algebras:
The Lie algebra of $SL_n$ is the set of traceless matrices. I don't see any reason why this shouldn't be a complex Lie algebra. But this way of reasoning is highly dissatisfying to me.
The Lie algebra of $O(n,\mathbb C)$ is the set of matrices with $A^\ast + A = 0$. This contains complex conjugation which seems to suggest that this cannot be a complex Lie algebra.