The lattice path operad and Hochschild cochains

TL;DR

This paper introduces new operads related to loop spaces and cochains, providing a formal construction of $E_2$-actions on Hochschild cochains of associative and Frobenius algebras.

Contribution

It presents the lattice path operad and its cyclic extension, establishing their connection to Hochschild cochains and $E_2$-actions.

Findings

Constructed the lattice path operad and cyclic extension.

Established $E_2$-action on Hochschild cochains of associative algebras.

Extended the construction to symmetric Frobenius algebras.

Abstract

We introduce two coloured operads in sets -- the lattice path operad and a cyclic extension of it -- closely related to iterated loop spaces and to universal operations on cochains. As main application we present a formal construction of an $E_2$-action (resp. framed $E_2$-action) on the Hochschild cochain complex of an associative (resp. symmetric Frobenius) algebra.