Algebras simple with respect to a Taft algebra action

Algebras simple with respect to an action of a Taft algebra Hm2(ζ)Hm2(ζ) deliver an interesting example of H-module algebras that are H  -simple but not necessarily semisimple. We describe finite dimensional Hm2(ζ)Hm2(ζ)-simple algebras and prove the analog of Amitsur's conjecture for codimensions of their polynomial Hm2(ζ)Hm2(ζ)-identities. In particular, we show that the Hopf PI-exponent of an Hm2(ζ)Hm2(ζ)-simple algebra A   over an algebraically closed field of characteristic 0 equals dim⁡Adim⁡A. The groups of automorphisms preserving the structure of an Hm2(ζ)Hm2(ζ)-module algebra are studied as well.