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

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

Communications in Algebra ◽

10.1080/00927872.2021.1990941 ◽

2021 ◽

pp. 1-21

Author(s):

Tolulope Oke

Keyword(s):

Hochschild Cohomology ◽

Cup Product ◽

Gerstenhaber Bracket

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Finite groups acting linearly: Hochschild cohomology and the cup product

Advances in Mathematics ◽

10.1016/j.aim.2010.09.022 ◽

2011 ◽

Vol 226 (4) ◽

pp. 2884-2910 ◽

Cited By ~ 7

Author(s):

Anne V. Shepler ◽

Sarah Witherspoon

Keyword(s):

Finite Groups ◽

Hochschild Cohomology ◽

Cup Product

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The Gerstenhaber structure on the Hochschild cohomology of a class of special biserial algebras

Journal of Algebra ◽

10.1016/j.jalgebra.2021.03.032 ◽

2021 ◽

Vol 580 ◽

pp. 264-298

Author(s):

Joanna Meinel ◽

Van C. Nguyen ◽

Bregje Pauwels ◽

María Julia Redondo ◽

Andrea Solotar

Keyword(s):

Hochschild Cohomology ◽

Special Biserial Algebras

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Twisted K-theory constructions in the case of a decomposable Dixmier-Douady class

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is014007001jkt274 ◽

2014 ◽

Vol 14 (2) ◽

pp. 247-272 ◽

Cited By ~ 4

Author(s):

Antti J. Harju ◽

Jouko Mickelsson

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Lie Groups ◽

Explicit Construction ◽

Theory Model ◽

Quantum Field ◽

Compact Lie Groups ◽

Field Theory Model ◽

Cup Product ◽

K Theory

AbstractTwisted K-theory on a manifold X, with twisting in the 3rd integral cohomology, is discussed in the case when X is a product of a circle and a manifold M. The twist is assumed to be decomposable as a cup product of the basic integral one form on and an integral class in H2(M,ℤ). This case was studied some time ago by V. Mathai, R. Melrose, and I.M. Singer. Our aim is to give an explicit construction for the twisted K-theory classes using a quantum field theory model, in the same spirit as the supersymmetric Wess-Zumino-Witten model is used for constructing (equivariant) twisted K-theory classes on compact Lie groups.

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