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

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 $A_{\mathrm{\infty }}$-multiplication formulas for both the Hochschild cochain complex and the Hochschild chain complex under additive decompositions. Finally, we give $A_{\mathrm{\infty }}$-multiplication formulas in the context of abelian groups.