Semisimple and G-Equivariant Simple Algebras Over Operads

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 = FunH (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 Sciarappa (arXiv: 1506.07565) on the classification of simple commutative algebras in the Deligne category Rep(St) to algebras over any finitely generated linear operad.