Gerstenhaber brackets on Hochschild cohomology of quantum symmetric algebras and their group extensions

We examine PBW deformations of finite group extensions of quantum symmetric algebras, in particular the quantum Drinfeld orbifold algebras defined by the first author. We give a homological interpretation, in terms of Gerstenhaber brackets, of the necessary and sufficient conditions on parameter functions to define a quantum Drinfeld orbifold algebra, thus clarifying the conditions. In case the acting group is trivial, we determine conditions under which such a PBW deformation is a generalized enveloping algebra of a color Lie algebra; our PBW deformations include these algebras as a special case.

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Integrable Derivations and Stable Equivalences of Morita Type

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091517000098 ◽

2018 ◽

Vol 61 (2) ◽

pp. 343-362 ◽

Cited By ~ 1

Author(s):

Markus Linckelmann

Keyword(s):

Hochschild Cohomology ◽

Block Theory ◽

Discrete Valuation Rings ◽

Symmetric Algebras ◽

Valuation Rings ◽

Transfer Maps ◽

Stable Equivalences ◽

Morita Type

AbstractUsing that integrable derivations of symmetric algebras can be interpreted in terms of Bockstein homomorphisms in Hochschild cohomology, we show that integrable derivations are invariant under the transfer maps in Hochschild cohomology of symmetric algebras induced by stable equivalences of Morita type. With applications in block theory in mind, we allow complete discrete valuation rings of unequal characteristic.

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First Hochschild cohomology group and stable equivalence classification of Morita type of some tame symmetric algebras

Homology Homotopy and Applications ◽

10.4310/hha.2019.v21.n1.a2 ◽

2019 ◽

Vol 21 (1) ◽

pp. 19-48 ◽

Cited By ~ 1

Author(s):

Rachel Taillefer

Keyword(s):

Cohomology Group ◽

Hochschild Cohomology ◽

Stable Equivalence ◽

Symmetric Algebras ◽

Hochschild Cohomology Group ◽

Morita Type

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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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On inverse categories and transfer in cohomology

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091512000211 ◽

2012 ◽

Vol 56 (1) ◽

pp. 187-210 ◽

Cited By ~ 4

Author(s):

Markus Linckelmann

Keyword(s):

Hochschild Cohomology ◽

Abelian Groups ◽

Mackey Functor ◽

Symmetric Algebras ◽

Transfer Maps ◽

Category Algebras

AbstractIt follows from methods of B. Steinberg, extended to inverse categories, that finite inverse category algebras are isomorphic to their associated groupoid algebras; in particular, they are symmetric algebras with canonical symmetrizing forms.We deduce the existence of transfer maps in cohomology and Hochschild cohomology from certain inverse subcategories. This is in part motivated by the observation that, for certain categories $\mathcal{C}$, being a Mackey functor on $\mathcal{C}$ is equivalent to being extendible to a suitable inverse category containing $\mathcal{C}$. We further show that extensions of inverse categories by abelian groups are again inverse categories.

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