This must be an easy question but I don't have a good argument for it and have not found a counterexample: Let $G$ be a connected semisimple algebraic group over $\mathbb{Q}$ such that the center of $G$, $Z(G)$ is trivial, i.e. $Z(G)=\{1\}$ is it true that the center of $G\_{\mathbb{R}}$, denoted $Z(G\_{\mathbb{R}})$ is also trivial?



Center of the algebraic group $G_{\mathbb{R}}$ for a centerless $G$
The short answer is that the center Z(G) of a semisimple algebraic group is a well-defined (finite) algebraic subgroup which commutes with extension of the base field, so if it's trivial over $\mathbb{Q}$, then its trivial over every field. But presumably, that is not what you meant to ask. Perhaps you mean: does $Z(\mathbb{Q})$ trivial imply $Z(\mathbb{R})$ trivial? The answer is no. You have a finite group $Z(\bar{\mathbb{Q}})$ with an action of the absolute Galois group of $\mathbb{Q}$ and you are asking: if only 1 is fixed by the full Galois group is only 1 fixed by complex conjugation? Obviously, not necessarily. It's easy to makes lots of different Galois modules as centers of semisimple groups over $\mathbb{Q}$. Or perhaps you mean: if $Z(G)$ is trivial (as an algebraic subgroup), is the center of $G(\mathbb{R})$ trivial? The answer is yes, because $G(\mathbb{R})$ is dense in $G$ for the Zariski topology. 


