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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?

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up vote 8 down vote accepted

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 $A = R/(f_1, \dots, f_m)$.

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Another example: Let $V$ be a finite dimensional representation of G such that $V^G=0.$ Then you can take sem-direct product algebra $A=k[G]\oplus V,$ where $V^2=0$ and $g\cdot v=gv, g\in G, v\in V.$

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How do you want $G$ to act? – Mariano Suárez-Alvarez Jun 3 '10 at 19:24
You are right, my answer was incorrect. – Bedini Jun 3 '10 at 20:06

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