  The purpose of this paper is to give a characterisation of divided power
algebras over a reduced operad. Such a characterisation is given in terms of
polynomial operations, following the classical example of divided power
algebras.
  We describe these polynomial operations in two different ways: one way uses
invariant elements under the action of the symmetric group, the other
coinvariant elements. Our results are then applied to the case of level
algebras, which are (non-associative) commutative algebras satisfying the
exchange law.
