What does it mean to be a real Lie-group ?
For example it is said that $SU(N)$ is a real Lie-group. While for example for $SU(2)$ the 2 dimensional matrix-representation consists of the Pauli matrices (which are complex). What does that mean for this group ? Does it imply some extra structures ?
addendum: for a Lie-group we write the group-elements as exponents:$$g=\exp\left(\sum_a\alpha_a(x)t_a\right),$$with $t_a$ the generators of the group (for the $SU(N)$-example this would be traceless anti-hermitian matrices) and $\alpha_a(x)$ the different parameters.
Does being a real Lie-group imply that the $\alpha_a(x)$ are real ?
A "real Lie group" means, simply, a Lie group, which by definition is a differentiable manifold equipped with a group operation that is differentiable and has differentiable inversion map.
A "complex Lie group" means a complex manifold equipped with a group operation that is complex differentiable and has complex differentiable inversion map.
If one were to take a group like $GL(n,\mathbb{C})$, which is a complex Lie group, and then to write a sentence like "$GL(n,\mathbb{C})$ is a real Lie group", this is an example of a forgetful functor. In it's most natural definition, $GL(n,\mathbb{C})$ is a complex Lie group. But, every complex Lie group is a real Lie group. You simply forget the imaginary unit "i" and treat a complex number $x+iy$ as a pair of real numbers $(x,y)$. Forgetting "i" lets you treat $\mathbb{C}^n$ as $\mathbb{R}^{2n}$, and it lets you treat complex differentiable maps between open subsets of $\mathbb{C}^n$ as differentiable maps, in the ordinary sense, between open subsets of $\mathbb{R}^{2n}$.
So to answer your question "Does it imply some extra structure?" --- no indeed, it implies less structure.
Where things get fun is where you ask questions like this:
The answers are "yes" and "yes", and the requisite examples can be found in textbooks on the subject.