On the Homotopy of Simplicial Algebras over an Operad

According to a result of H. Cartan (cf. [5]), the homotopy of a simplicial commutative algebra is equipped with divided power operations. In this paper, we provide a general approach to the construction of such operations in the context of simplicial algebras over an operad. To be precise, we work over a fixed field F, and we consider operads in the category of F-modules. An operad is an algebraic device that specifies a type of algebras. There are operads Com, As, Lie, and Pois, whose algebras are respectively commutative algebras, associative algebras, Lie algebras and Poisson algebras. In general, if P denotes an operad, then we call P-algebras the associated algebras. First, we generalize the notion of a divided power in the context of algebras over an operad. This is done as follows. Recall that the free commutative algebra is given by the formula T (Com, V ) = ⊕n(V )Sn , for V ∈ModF .