TITLE: Is there a countable discrete infinite group $G$ over which the group algebra $\mathbb{C} G$ is semisimple?
QUESTION [6 upvotes]: I am seeking for an Artin $k$-algebra (especially for group algebra) which is infinite-dimensional over some field $k$. It's known that any complex group algebra has trivial Jacobson radical. So I have the following question:
Is there a countable discrete infinite group $G$ over which the group algebra $\mathbb{C} G$ is semisimple?

REPLY [8 votes]: The answer is negative as any semisimple Hopf algebra is finite-dimensional. More generally, the same conclusion is true for all Artinian Hopf algebras. See e.g. Liu and Zhang - Artinian Hopf algebras are finite dimensional.