Okay, so this might be a very dumb question, but I recently proved that for an associative algebra $A$, any representation $V$ of $A$ is irreducible iff every $v \in V$ is a cyclic vector, i.e $Av$ = $V$. Won't this make the regular representation of $A$ irreducible?
2026-03-27 23:23:20.1774653800
Irreducibility of regular representation
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The cyclic vectors in the regular representation are the units of the algebra, so every non-zero element of an algebra is a cyclic vector if and only if the algebra is a field. The regular representation of a field is indeed irreducible.