Gerstenhaber algebra and Deligne’s conjecture on Tate-Hochschild cohomology

In this paper, we compute the dimension of the Hochschild cohomology groups of any [Formula: see text]-cluster tilted algebra of type [Formula: see text]. Moreover, we give conditions on the bounded quiver of an [Formula: see text]-cluster tilted algebra [Formula: see text] of type [Formula: see text] such that the Gerstenhaber algebra [Formula: see text] has nontrivial multiplicative structures. We also show that the derived class of gentle [Formula: see text]-cluster tilted algebras is not always completely determined by the dimension of the Hochschild cohomology.

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INVARIANCE OF THE GERSTENHABER ALGEBRA STRUCTURE ON TATE-HOCHSCHILD COHOMOLOGY

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748019000367 ◽

2019 ◽

pp. 1-36

Author(s):

Zhengfang Wang

Keyword(s):

Hochschild Cohomology ◽

Derived Category ◽

Algebra Structure ◽

Gerstenhaber Algebra ◽

Morita Type

Keller proved in 1999 that the Gerstenhaber algebra structure on the Hochschild cohomology of an algebra is an invariant of the derived category. In this paper, we adapt his approach to show that the Gerstenhaber algebra structure on the Tate–Hochschild cohomology of an algebra is preserved under singular equivalences of Morita type with level, a notion introduced by the author in previous work.

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The Hochschild Cohomology Ring of Temperley–Lieb Algebras Revisited

Algebra Colloquium ◽

10.1142/s1005386720000565 ◽

2020 ◽

Vol 27 (04) ◽

pp. 669-686

Author(s):

Weiguo Lyu ◽

Yuling Wu

Keyword(s):

Hochschild Cohomology ◽

Cohomology Ring ◽

Algebra Structure ◽

Hochschild Cohomology Ring ◽

Gerstenhaber Algebra

We determine the Gerstenhaber algebra structure on the Hochschild cohomology ring of Temperley–Lieb algebras in this paper.

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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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Homotopy liftings and Hochschild cohomology of some twisted tensor products

Journal of Algebra and Its Applications ◽

10.1142/s0219498822502383 ◽

2021 ◽

pp. 2250238

Author(s):

Pablo S. Ocal ◽

Tolulope Oke ◽

Sarah Witherspoon

Keyword(s):

Tensor Product ◽

Hochschild Cohomology ◽

Tensor Products ◽

Gerstenhaber Algebra ◽

Graded Tensor Product

The Hochschild cohomology of a tensor product of algebras is isomorphic to a graded tensor product of Hochschild cohomology algebras, as a Gerstenhaber algebra. A similar result holds when the tensor product is twisted by a bicharacter. We present new proofs of these isomorphisms, using Volkov’s homotopy liftings that were introduced for handling Gerstenhaber brackets expressed on arbitrary bimodule resolutions. Our results illustrate the utility of homotopy liftings for theoretical purposes.

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