A_∞-structures on the additive decomposition of the Tate-Hochschild cohomology of a finite group algebra

2026/01/01 by Xiuli Bian, Longfei Li, Yuming Liu +3 · 1 voice
Mathematics · #math.KT #math.RA

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Abstract

Firstly, for a finite group algebra, we provide a computational framework \widehatm_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 \widehatm₂ formulas of the Tate-Hochschild cochain complex in terms of the additive decomposition. Thirdly, we give explicit A_∞-multiplication formulas for both the Hochschild cochain complex and the Hochschild chain complex under additive decompositions. Finally, we give A_∞-multiplication formulas in the context of abelian groups.

Citations

• Algebraic Morse theory via Homological Perturbation Lemma
• The Batalin-Vilkovisky structure on the Tate-Hochschild cohomology ring of a group algebra
• Singular Hochschild cohomology and algebraic string operations
• Batalin-Vilkovisky structure over the Hochschild cohomology ring of a group algebra
• Introduction to A-infinity algebras and modules
• Deformation Quantization of Poisson Manifolds
• The Cohomology Structure of an Associative Ring
• The Higher Dimensional Cohomology Groups of Class Field Theory

Discussions

• Xiuli Bian, Longfei Li, Yuming Liu, Tianyun Wang, Zhengfang Wang, Guodong Zhou: $A_{\infty}$-structures on the additive decomposition of the Tate-Hochschild cohomology of a finite group algebra https: [bsky, 0 points, 0 comments]

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• The Batalin-Vilkovisky structure on the Tate-Hochschild cohomology ring of a group algebra
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• Singular Hochschild cohomology and algebraic string operations