Codimension Growth of Lie algebras with a generalized action

Let $L$ be a finite dimensional Lie $F$-algebra endowed with a generalized action by an associative algebra $H$. We investigate the exponential growth rate of the sequence of $H$-graded codimensions $c_n^H(L)$ of $L$ which is a measure for the number of non-polynomial $H$-identities of $L$. More precisely, we construct the first example of an $S$-graded Lie algebra having a non-integer, even irrational, exponential growth rate $\lim_{n\rightarrow \infty} \sqrt[n]{c_n^{S}(L)}$. Hereby $S$ is a semigroup and an exact value is given. On the other hand, returning to general $H$, if $L$ is semisimple and also semisimple for the $H$-action we prove the analog of Amitsur's conjecture (i.e. $\lim_{n\rightarrow \infty} \sqrt[n]{c_n^{H}(L)} \in \mathbb{Z}$). Moreover if $H=FS$ is a semigroup algebra the semisimplicity on $L$ can be dropped which is in strong contract to the associative setting.