Rationality of the Hilbert series of Hopf-invariants of free algebras
  It is shown that the subalgebra of invariants of a free associative algebra of finite rank under a linear action of a semisimple Hopf algebra has a rational Hilbert series with respect to the usual degree function, whenever the ground field has zero characteristic. 
Vitor O. Ferreira and Lucia S. I. Murakami
math.RA