Derivations, gradings, actions of algebraic groups, and codimension growth of polynomial identities

Suppose a finite dimensional semisimple Lie algebra $\mathfrak{g}$ acts by derivations on a finite dimensional associative or Lie algebra $A$ over a field of characteristic $0$. We prove the $\mathfrak{g}$-invariant analogs of Wedderburn - Mal’cev and Levi theorems, and the analog of Amitsur’s conjecture on asymptotic behavior for codimensions of polynomial identities with derivations of $A$. It turns out that for associative algebras the differential PI-exponent coincides with the ordinary one. Also we prove the analog of Amitsur’s conjecture for finite dimensional associative algebras with an action of a reductive affine algebraic group by automorphisms and anti-automorphisms or graded by an arbitrary Abelian group. In addition, we provide criteria for $G$-, $H$- and graded simplicity in terms of codimensions.