Semisimple and \(G\)-equivariant simple algebras over operads (Q1686691)

A family of classical theorems holds that if \(G\) is a finite group, any \(G\)-equivariant finite-dimensional simple commutative algebra, or associative algebra, or Lie algebra, is isomorphic to the space of \(H\)-invariant functions from \(G\) to \(B\), where \(B\) is a simple algebra of the good type and \(H\) a subgroup of \(G\). These theorems are extended for algebras over a a linear operad. As a consequence, this extend Sciarappa's theorem on simple commutative algebras in the Deligne category to algebras over a finitely generated linear operad.