Gelfand-Kirillov dimension of constants of algebraic $q$-skew derivations

Let $δ$ be an algebraic $q$-skew $σ$-derivation of an algebra $R$ over a field of arbitrary characteristic, where $σ$ is an algebraic automorphism of $R$, and let $R^δ$ be the algebra of constants of $δ$. We show that $GKdim(R) = GKdim(R^δ)$ whenever $R$ is semiprime and $R^δ$ is a PI algebra containing no infinite direct sum of nonzero ideals. The polynomial identity cannot be dropped. For every $r \geq 1$ we construct a primitive algebra $R$ with an inner derivation $d$ such that $d^{3} = 0$, $GKdim(R) \geq 2r$ and $GKdim(R^{d}) = 0$, while $R^{d}$ contains no direct sum of two nonzero ideals.