Simplicial Structure on Connected Multiplicative Operads

TL;DR

This paper introduces a new simplicial structure on connected multiplicative operads, along with brace algebra, bicomplex, and differential graded algebra structures, enriching the algebraic framework of operads with concrete examples.

Contribution

It develops a novel simplicial structure on connected multiplicative operads and integrates brace algebra, bicomplex, and differential graded algebra structures, providing new tools for operad theory.

Findings

Defined a new simplicial structure called connected multiplicative simplicial operad.

Constructed a bicomplex structure with coboundary and boundary operators.

Established differential graded algebra and coalgebra structures on the operad.

Abstract

In these notes, we define a new simplicial structure on a connected multiplicative operad and call it connected multiplicative simplicial operad (for short; simplicial operad). Next we introduce on this simplicial operad a brace algebra structure analogous to that of Gerstenhaber-Voronov that we call right brace algebra structure. This permits us to obtain on the operad with the above mentioned properties a bicomplex structure one of whose two differential operators is a coboundary and the other one is a boundary. Moreover we define on one hand on the above simplicial operad together with its right brace algebra structure, two distinct products up to a sign respectively called dot-product and odot-product. Then we show that the coboundary and the boundary together with the odot-product provide to this simplicial operad two distinct differential graded algebra structures. On the other hand we obtain through the Alexander-Withney map, a differential graded coalgebra structure on a simplicial operad. We end by illustrating our constructions with some examples