Weighted differential ($q$-tri)dendriform algebras

TL;DR

This paper introduces weighted derivations on operad-based algebras, develops weighted differential ($q$-tri)dendriform structures, and constructs their free objects in commutative and noncommutative settings.

Contribution

It defines weighted derivations on operad algebras and introduces weighted differential ($q$-tri)dendriform algebras, expanding the algebraic framework and constructing free objects.

Findings

Weighted derivation determined by generators for free algebras.

Introduction of weighted differential ($q$-tri)dendriform algebras.

Construction of free objects in both commutative and noncommutative cases.

Abstract

In this paper, we first introduce a weighted derivation on algebras over an operad $\cal P$, and prove that for the free $\cal P$-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.