The generalized-action PI-exponent conjecture for H-nice Lie algebras

Let $F$ be an algebraically closed field with $\text{char}(F)=0$, and let $L$ be a finite-dimensional Lie algebra with a generalized action by an associative algebra $H$. The algebra $L$ is $H$-nice when it satisfies the structural conditions described in the source, including invariance of its nilpotent and solvable radicals and the corresponding $H$-versions of the Levi decomposition, Wedderburn–Malcev theorem, and Weyl theorem.

Generalized-action PI-exponent conjecture. If $L$ is $H$-nice, then there exist constants $C\in\mathbb{R}$, $t\in\frac{\mathbb{Z}}{2}$, and $d\in\mathbb{Z}$ such that

$$c_n^{H}(L) \simeq C n^{t} d^n.$$

This conjecture proposes that the asymptotic growth of the $H$-codimensions has the same polynomial-times-exponential form as in the semisimple case, extending the positive result for $H$-semisimple Lie algebras beyond semisimplicity. The paper presents it as a variant of the Regev–Amitsur conjecture; its resolution is not given here.