Let $G$ be a finite group acting on an associative $k$-algebra with unity ($k$ algebraically closed of characteristic zero). The action is called ergodic if $ A^G = \{ a \in A | g \cdot a = a, \forall g \in G \} = k$. 


Anyone know an example of a non-semisimple finite dimensional algebra with an ergodic action of a finite group?



A question on group action on algebras
Let $A = k[x,y]/(x^2,xy,y^2)$ and let the generator of the two-element group act by $x \mapsto -x$, $y \mapsto -y$. 


More generally, take any finite group acting on $R = k[x\_1, \dots, x\_n]$ with invariant subring $S = k[f\_1, \dots, f\_m]$, and set $A = R/(f\_1, \dots, f\_m)$. 


