Twisting Elements in Homotopy G-algebras

TL;DR

This paper explores twisting elements within homotopy G-algebras, unifying classical deformation theory of algebras and $A()$-algebras through homotopy Gerstenhaber algebra structures.

Contribution

It introduces a framework connecting deformation theory and $A()$-algebras via twisting elements in homotopy G-algebras.

Findings

Unified perspective on deformation and $A()$-algebras

Characterization of twisting elements in homotopy G-algebras

Potential applications to algebraic deformation problems

Abstract

We study the notion of twisting elements $da=a\cup_1a$ with respect to $\cup_1$ product when it is a part of homotopy Gerstenhaber algebra structure. This allows to bring to one context the two classical concepts, the theory of deformation of algebras of M. Gerstenhaber, and $A(\infty)$-algebras of J. Stasheff.