Noncommutative invariants of dihedral groups

Let $A_2(𝔐)$ and $L_2({𝔄}^2)$ be, respectively, the 2-generated free metabelian associative and Lie algebra over the field of complex numbers. In the associative case we find a finite set of generators of the algebra $A_2{(𝔐)}^{D_{2n}}$ of the dihedral group of order $2n$, $n\ge 3$. In the Lie case we find a minimal system of generators as a $ℂ{[x,y]}^{D_{2n}}$-module of the $D_{2n}$-invariants $L_2^{\prime }{({𝔄}^2)}^{D_{2n}}$ in the commutator ideal $L_2^{\prime }({𝔄}^2)$ of $L_2({𝔄}^2)$. In both cases we compute the Hilbert (or Poincaré) series of the algebras $A_2{(𝔐)}^{D_{2n}}$ and $L_2{({𝔄}^2)}^{D_{2n}}$.