
> 
> **Possible Duplicate:**  
> 
> [Complex Lie group without faithful real representations?](https://mathoverflow.net/questions/62624/complex-lie-group-without-faithful-real-representations) 
> 
> 
> 


We know that for a matrix (linear) Lie group $G$, we define it to be a closed subgroup of $GL(n,\mathbb{C})$. But Lie groups are defined as manifolds in $\mathbb{R}^n$ for some $n$, in general. The question is that, do we know any Lie group which is not a matrix Lie group? Thank you very much.



Non-linear Lie group
The traditional example is the universal cover of $SL(2,\mathbb{R})$. You can look e.g. at the wikipedia article on $SL(2,\mathbb{R})$. 


