Upper central series for the group of unitriangular automorphisms of a free associative algebra

We study some subgroups of the group of unitriangular automorphisms $U_n$ of a free associative algebra over a field of characteristic zero. We find the center of $U_n$ and describe the hypercenters of $U_2$ and $U_3$. In particular, we prove that the upper central series for $U_2$ has infinite length. As consequence, we prove that the groups $U_n$ are non-linear for all $n \geq 2$.