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Algebras simple with respect to an action of Sweedler's algebra $H_4$ deliverthe easiest example of $H$-module algebras that are $H$-simple but notnecessarily semisimple. We describe finite dimensional $H_4$-simple algebrasand prove the analog of Amitsur's conjecture for codimensions of theirpolynomial $H_4$-identities. In particular, we show that the Hopf PI-exponentof an $H_4$-simple algebra $A$ over an algebraically closed field ofcharacteristic $0$ equals $\dim A$. The groups of automorphisms preserving thestructure of an $H_4$-module algebra are studied as well.

Algebras simple with respect to a Sweedler's algebra action