Cup product and Gerstenhaber bracket on Hochschild cohomology of a family of quiver algebras

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution $\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH}^1(A)$ on $\operatorname{HH}^{\ast} (A)$.

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The cup product on Hochschild cohomology via twisting cochains and applications to Koszul rings

Journal of Pure and Applied Algebra ◽

10.1016/j.jpaa.2016.09.003 ◽

2017 ◽

Vol 221 (5) ◽

pp. 1112-1133 ◽

Cited By ~ 1

Author(s):

Cris Negron

Keyword(s):

Hochschild Cohomology ◽

Cup Product

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Comparison morphisms between two projective resolutions of monomial algebras

Revista de la Unión Matemática Argentina ◽

10.33044/revuma.v59n1a01 ◽

2017 ◽

pp. 1-31

Author(s):

María Julia Redondo ◽

Lucrecia Román

Keyword(s):

Lie Algebra ◽

Hochschild Cohomology ◽

Cohomology Groups ◽

Efficient Tool ◽

Simple Description ◽

Monomial Algebra ◽

Cup Product ◽

Lie Bracket ◽

Projective Resolutions ◽

Bar Resolution

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution $\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH}^1(A)$ on $\operatorname{HH}^{\ast} (A)$.

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$A_{\infty}$-coderivations and the Gerstenhaber bracket on Hochschild cohomology

Journal of Noncommutative Geometry ◽

10.4171/jncg/372 ◽

2020 ◽

Vol 14 (2) ◽

pp. 531-565

Author(s):

Cris Negron ◽

Yury Volkov ◽

Sarah Witherspoon

Keyword(s):

Hochschild Cohomology ◽

Gerstenhaber Bracket

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Gerstenhaber Bracket on the Hochschild Cohomology via An Arbitrary Resolution

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091518000901 ◽

2019 ◽

Vol 62 (3) ◽

pp. 817-836 ◽

Cited By ~ 1

Author(s):

Yury Volkov

Keyword(s):

Hochschild Cohomology ◽

Short Proof ◽

Hochschild Homology ◽

Full Description ◽

Algebra Structure ◽

Bimodule Resolution ◽

Symmetric Algebras ◽

Gerstenhaber Algebra ◽

Different Types ◽

Gerstenhaber Bracket

AbstractWe prove formulas of different types that allow us to calculate the Gerstenhaber bracket on the Hochschild cohomology of an algebra using some arbitrary projective bimodule resolution for it. Using one of these formulas, we give a new short proof of the derived invariance of the Gerstenhaber algebra structure on Hochschild cohomology. We also give some new formulas for the Connes differential on the Hochschild homology that lead to formulas for the Batalin–Vilkovisky (BV) differential on the Hochschild cohomology in the case of symmetric algebras. Finally, we use one of the obtained formulas to provide a full description of the BV structure and, correspondingly, the Gerstenhaber algebra structure on the Hochschild cohomology of a class of symmetric algebras.

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