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An extension of $k$-algebras $B \subset A$ is said to have depth one if thereexists a positive integer $n$ such that $ A$ is a direct summand of $ B^n$ in$_B\mtr{Mod}_B$. Depth one extensions of semisimple algebras are completelycharacterized in terms of their centers. For extensions of semisimple Hopfalgebras our results are similar to those obtained for finite group algebraextensions in \cite{BKone}.

Depth one extensions of semisimple algebras and Hopf subalgebras