Equivariant cohomology and deformation for algebras - an operadic approach

Let P be an operad with a multiplication, and later with a (higher) derivation. This paper introduces and develops a notion of (partial) group action and corresponding equivariant cohomology and deformation theory on P. It generalizes existing results for equivariant cohomology and deformation for associative algebras, dialgebras, and dendriform algebras, among others. At the end, we discuss compatibilities of group action and some operadic constructions.