Algebras simple with respect to a Sweedler's algebra action

Abstract

Algebras simple with respect to an action of Sweedler's algebra H4 deliver the easiest example of H-module algebras that are H-simple but not necessarily semisimple. We describe finite-dimensional H4-simple algebras and prove the analog of Amitsur's conjecture for codimensions of their polynomial H4-identities. In particular, we show that the Hopf PI-exponent of an H4-simple algebra A over an algebraically closed field of characteristic 0 equals dim A. The groups of automorphisms preserving the structure of an H4-module algebra are studied as well.