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

Xiuli Bian; Longfei Li; Yuming Liu; Tianyun Wang; Zhengfang Wang; Guodong Zhou

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arXiv · 2601.00351

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

Xiuli Bian·Longfei Li·Yuming Liu·Tianyun Wang·Zhengfang Wang·Guodong Zhou

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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.

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Xiuli Bian, Longfei Li, Yuming Liu, Tianyun Wang, Zhengfang Wang, Guodong Zhou. 2026-01-01. $A_{\infty}$-structures on the additive decomposition of the Tate-Hochschild cohomology of a finite group algebra. https://arxiv.org/abs/2601.00351

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