$A_{\infty}$-structures on the additive decomposition of the Tate-Hochschild cohomology of a finite group algebra

TL;DR

This paper develops an explicit computational framework for $A_{ abla}$-structures on Tate-Hochschild cohomology of finite group algebras, providing formulas for multiplications and decompositions.

Contribution

It introduces a new computational approach for $A_{ abla}$-structures using additive decompositions and explicit formulas for multiplications.

Findings

Explicit formulas for $ abla_2$ in terms of additive decomposition.

A computational framework $ ilde{m}_n$ for Tate-Hochschild cochains.

$A_{ abla}$-multiplication formulas for Hochschild complexes and abelian groups.

Abstract

Firstly, for a finite group algebra, we provide a computational framework $\widehat{m}_n$ for the Tate-Hochschild cochain complex in terms of the additive decomposition, by decomposing each planar n-ary tree into local two children and local three children. Secondly, we give all $\widehat{m}_2$ formulas of the Tate-Hochschild cochain complex in terms of the additive decomposition. Thirdly, we give explicit $A_{\infty}$-multiplication formulas for both the Hochschild cochain complex and the Hochschild chain complex under additive decompositions. Finally, we give $A_{\infty}$-multiplication formulas in the context of abelian groups.