Weighted differential (q-tri)dendriform algebras.

In this paper, we first introduce a weighted derivation on algebras over an operad 풫, and prove that for the free 풫-algebra, its weighted derivation is determined by the restriction on the generators. As applications, we propose the concept of weighted differential (q-tri)dendriform algebras and study some basic properties of them. Then Novikov-(tri)dendriform algebras are initiated, which can be induced from differential (q-tri)dendriform of weight zero. Finally, the corresponding free objects are constructed, in both the commutative and noncommutative contexts. [ABSTRACT FROM AUTHOR]