Let $k$ be a field and $A$ a $k$-algebra. Are there known techniques to check that $A$ does not admit a structure of Hopf algebra? In particular if $H$ is a Hopf $k$-algebra and $M$ an $H$-bimodule, do you see any technique to check if the trivial extension algebra $H\ltimes M$ does not admit a structure of Hopf algebra?
The trivial extension $H\ltimes M$ is $H\oplus M$ as $H$-bimodules and the algebra product is defined by $(a,b)(c,d)=(ac,bc+ad)$.
As a start we can assume that $H$ is a tensor algebra and $M$ is its dual with the canonical bimodule structure.