Semisimple and $G$-equivariant simple algebras over operads

Abstract

Let $G$ be a finite group. There is a standard theorem on the classification of $G$-equivariant finite dimensional simple commutative, associative, and Lie algebras (i.e., simple algebras of these types in the category of representations of $G$). Namely, such an algebra is of the form $A={\rm Fun}_H(G,B)$, where $H$ is a subgroup of $G$, and $B$ is a simple algebra of the corresponding type with an $H$-action. We explain that such a result holds in the generality of algebras over a linear operad. This allows one to extend Theorem 5.5 of arXiv:1506.07565 on the classification of simple commutative algebras in the Deligne category ${\rm Rep}(S_t)$ to algebras over any finitely generated linear operad.