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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?

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    $\begingroup$ I.G. Connell, On the group ring, Canad. J. Math. 15 (1963), 650–685 proved the group algebra is Artinian iff the group is finite. Zelmanov generalized this to semigroups. Semisimplicity is not needed. $\endgroup$ Jun 27 at 10:30
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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.

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No: if the discrete group algebra $\mathbb C G$ is semisimple, then $G$ is finite.

Consider the trivial $G$-module $\mathbb C$ and the quotient map of $G$-modules $\mathbb C G \to \mathbb C$ which sends $g \mapsto 1$ for all $g \in G$. If $\mathbb C G$ is semisimple, then this map splits, giving a nonzero $G$-invariant element $$e = \sum_{g \in G} a_g g \in \mathbb C G$$ where $a_g \in \mathbb C$ are almost all zero and $$\sum_{g\in G} a_g = 1.$$ The element $e$ is invariant if and only if $a_g = a_1$ for all $g \in G$, hence such $e$ exists if and only if $G$ is finite.

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