scholarly journals A note on (co)homologies of algebras from unpunctured surfaces

2018 ◽  
Vol 17 (05) ◽  
pp. 1850088
Author(s):  
Yadira Valdivieso-Díaz
Keyword(s):  
Hochschild Cohomology ◽  
Geometric Interpretation ◽  
Hochschild Homology ◽  
Cohomology Groups ◽  
Triangulated Surface ◽  
Triangulated Surfaces

In a previous paper, the author computed the dimension of Hochschild cohomology groups of Jacobian algebras from (unpunctured) triangulated surfaces, and gave a geometric interpretation of those numbers in terms of the number of internal triangles, the number of vertices and the existence of certain kind of boundaries. The aim of this note is to compute the cyclic (co)homology and the Hochschild homology of the same family of algebras and to give an interpretation of those dimensions through elements of the triangulated surface.

scholarly journals Comparison morphisms between two projective resolutions of monomial algebras

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 $\operatorname{HH}^{\ast} (A)$.

Hochschild cohomology of m-cluster tilted algebras of type 𝔸̃

2020 ◽  
pp. 2150018
Author(s):  
Viviana Gubitosi
Keyword(s):  
Hochschild Cohomology ◽  
Cohomology Groups ◽  
Tilted Algebra ◽  
Type Formula ◽  
Tilted Algebras ◽  
Gerstenhaber Algebra ◽  
Cluster Tilted Algebra ◽  
Multiplicative Structures

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.

Hochschild cohomology of II1 factors with Cartan maximal abelian subalgebras

2009 ◽  
Vol 52 (2) ◽  
pp. 287-295 ◽  
Author(s):  
Jan M. Cameron
Keyword(s):  
Hochschild Cohomology ◽  
Type Ii ◽  
Cohomology Groups ◽  
Maximal Abelian Subalgebra ◽  
Abelian Subalgebra ◽  
Abelian Subalgebras ◽  
Separable Predual

AbstractIn this paper we prove that, for a type-II1 factor N with a Cartan maximal abelian subalgebra, the Hochschild cohomology groups Hn(N,N)=0 for all n≥1. This generalizes the result of Sinclair and Smith, who proved this for all N having a separable predual.

scholarly journals Comparison morphisms between two projective resolutions of monomial algebras

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 $\operatorname{HH}^{\ast} (A)$.

scholarly journals On (co)homology of Frobenius Poisson algebras

2014 ◽  
Vol 14 (2) ◽  
pp. 371-386 ◽  
Author(s):  
Can Zhu ◽  
Fred Van Oystaeyen ◽  
Yinhuo Zhang
Keyword(s):  
Bilinear Form ◽  
Hochschild Cohomology ◽  
Cohomology Ring ◽  
Poisson Algebra ◽  
Hochschild Homology ◽  
Poisson Algebras ◽  
Poisson Homology ◽  
Poisson Cohomology ◽  
Frobenius Algebras

AbstractIn this paper, we study Poisson (co)homology of a Frobenius Poisson algebra. More precisely, we show that there exists a duality between Poisson homology and Poisson cohomology of Frobenius Poisson algebras, similar to that between Hochschild homology and Hochschild cohomology of Frobenius algebras. Then we use the non-degenerate bilinear form on a unimodular Frobenius Poisson algebra to construct a Batalin-Vilkovisky structure on the Poisson cohomology ring making it into a Batalin-Vilkovisky algebra.

HOCHSCHILD COHOMOLOGY OF POLYNOMIAL ALGEBRAS

2001 ◽  
Vol 03 (03) ◽  
pp. 393-402 ◽  
Author(s):  
MICHAEL PENKAVA ◽  
POL VANHAECKE
Keyword(s):  
Deformation Theory ◽  
Hochschild Cohomology ◽  
Differential Operators ◽  
Polynomial Algebra ◽  
Cohomology Groups ◽  
Polynomial Algebras ◽  
Arbitrary Polynomial

In this paper we investigate the Hochschild cohomology groups H2(A) and H3(A) for an arbitrary polynomial algebra A. We also show that the corresponding cohomology groups which are built from differential operators inject in H2(A) and H3(A) and we give an application to deformation theory.

A Cohomological Property of π-invariant Elements

2013 ◽  
Vol 56 (3) ◽  
pp. 534-543
Author(s):  
M. Filali ◽  
M. Sangani Monfared
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Orthonormal Basis ◽  
Hochschild Cohomology ◽  
Separable Hilbert Space ◽  
Continuous Representation ◽  
Cohomology Groups ◽  
Cohomological Property ◽  
Coordinate Functions ◽  
Special Case

Abstract.Let A be a Banach algebra and let be a continuous representation of A on a separable Hilbert space H with dim H = m. Let πi j be the coordinate functions of π with respect to an orthonormal basis and suppose that for each and . Under these conditions, we call an element left π-invariant if In this paper we prove a link between the existence of left π-invariant elements and the vanishing of certain Hochschild cohomology groups of A. Our results extend an earlier result by Lau on F-algebras and recent results of Kaniuth, Lau, Pym, and and the second author in the special case where π : A → C is a non-zero character on A.

scholarly journals A Physics-Based Estimation of Mean Curvature Normal Vector for Triangulated Surfaces

2019 ◽  
Vol 12 (1) ◽  
pp. 70-78
Author(s):  
Sudip Kumar Das ◽  
Mirza Cenanovic ◽  
Junfeng Zhang
Keyword(s):  
Mean Curvature ◽  
Beltrami Operator ◽  
Force Balance ◽  
Normal Vector ◽  
Triangulated Surface ◽  
Balance Principle ◽  
Alternative Expression ◽  
Triangulated Surfaces ◽  
The Mean ◽  
Mesh Structures

In this note, we derive an approximation for the mean curvature normal vector on vertices of triangulated surface meshes from the Young-Laplace equation and the force balance principle. We then demonstrate that the approximation expression from our physics-based derivation is equivalent to the discrete Laplace-Beltrami operator approach in the literature. This work, in addition to providing an alternative expression to calculate the mean curvature normal vector, can be further extended to other mesh structures, including non-triangular and heterogeneous meshes.

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