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The Hochschild cohomlogy ring of a self-injective Nakayama algebra is a Batalin-Vilkoviskys algebra

Xiuli Bian, Tomohiro Itagaki, Wen Kou, Weiguo Lyu, Guodong Zhou

Abstract

Lambre, Zhou and Zimmermann showed that the Hochschild cohomology ring of a Frobenius algebra with semisimple Nakayama automorphism is a Batalin-Vilkovisky algebra. They asked whether the semisimplicity condition is necessary. In this paper, we show that for a self-injective Nakayama algebra, the Hochschild cohomology ring is always a Batalin-Vilkovisky algebra. In course of proofs, we correct some inaccuracies in the literature, hoping not to introduce new errors.

The Hochschild cohomlogy ring of a self-injective Nakayama algebra is a Batalin-Vilkoviskys algebra

Abstract

Lambre, Zhou and Zimmermann showed that the Hochschild cohomology ring of a Frobenius algebra with semisimple Nakayama automorphism is a Batalin-Vilkovisky algebra. They asked whether the semisimplicity condition is necessary. In this paper, we show that for a self-injective Nakayama algebra, the Hochschild cohomology ring is always a Batalin-Vilkovisky algebra. In course of proofs, we correct some inaccuracies in the literature, hoping not to introduce new errors.
Paper Structure (14 sections, 39 theorems, 133 equations)

This paper contains 14 sections, 39 theorems, 133 equations.

Table of Contents

  1. Preliminaries
  2. Hochschild (co)homology and Gerstenhaber algebras
  3. Hochschild cohomology and BV algebras.
  4. Minimal resolution for truncated quiver algebras and comparison morphisms
  5. The Hochschild cohomology spaces
  6. The Hochschild cohomology spaces for truncated quiver algebras
  7. The case $e=1$ for truncated basic cycle algebras $\Lambda={\mathbb K} Z_e/J^N$
  8. The case of $e\geq 2$ for truncated basic cycle algebras
  9. Cup product
  10. Hochschild cohomology ring
  11. Gerstenhaber algebra structure
  12. Gerstenhaber bracket for truncated quiver algebras
  13. Gerstenhaber brackets for truncated basic cycle algebras
  14. BV structure

Key Result

Theorem 1

Let $\Lambda={\mathbb K} Z_e/ J^N$, $N\ge 2$, be a truncated basic cycle algebra over a field ${\mathbb K}$. The the Hochschild cohomology ring $\mathrm{HH}^*(\Lambda)$ of $\Lambda$ is a BV algebra.

Theorems & Definitions (60)

  • Theorem 1: see Theorem \ref{['thm:main-result']}
  • Definition 1.1
  • Theorem 1.2: Ger63
  • Definition 1.3
  • Remark 1.4
  • Theorem 1.5: LZZ16
  • Remark 2.1
  • Lemma 3.1: Cib90, Loc99
  • Proposition 3.2: Loc99
  • Corollary 3.3: BLM00
  • ...and 50 more