Three-bar resolution versus MTF: how different can they be anyway?

2008 ◽  
Author(s):  
John R. Rogers
Keyword(s):  
Bar Resolution

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 EXPLICIT cocycle formulas on finite abelian groups with applications to braided linear Gr-categories and Dijkgraaf–Witten invariants

2019 ◽  
Vol 150 (4) ◽  
pp. 1937-1964 ◽  
Author(s):  
Hua-Lin Huang ◽  
Zheyan Wan ◽  
Yu Ye
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Finite Abelian Groups ◽  
Bar Resolution ◽  
Chain Map ◽  
A Chain

AbstractWe provide explicit and unified formulas for the cocycles of all degrees on the normalized bar resolutions of finite abelian groups. This is achieved by constructing a chain map from the normalized bar resolution to a Koszul-like resolution for any given finite abelian group. With a help of the obtained cocycle formulas, we determine all the braided linear Gr-categories and compute the Dijkgraaf–Witten Invariants of the n-torus for all n.

scholarly journals Subring Depth, Frobenius Extensions, and Towers

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Lars Kadison
Keyword(s):  
Modular Function ◽  
Finite Representation ◽  
Finite Dimensional ◽  
Frobenius Extension ◽  
Ring Structures ◽  
Bar Resolution ◽  
Minimum Depth ◽  
Left And Right ◽  
Special Ring ◽  
Tensor Powers

The minimum depthd(B,A)of a subringB⊆Aintroduced in the work of Boltje, Danz and Külshammer (2011) is studied and compared with the tower depth of a Frobenius extension. We show thatd(B,A)< ∞ ifAis a finite-dimensional algebra andBehas finite representation type. Some conditions in terms of depth and QF property are given that ensure that the modular function of a Hopf algebra restricts to the modular function of a Hopf subalgebra. IfA⊇Bis a QF extension, minimum left and right even subring depths are shown to coincide. IfA⊇Bis a Frobenius extension with surjective Frobenius, homomorphism, its subring depth is shown to coincide with its tower depth. Formulas for the ring, module, Frobenius and Temperley-Lieb structures are noted for the tower over a Frobenius extension in its realization as tensor powers. A depth 3 QF extension is embedded in a depth 2 QF extension; in turn certain depthnextensions embed in depth 3 extensions if they are Frobenius extensions or other special ring extensions with ring structures on their relative Hochschild bar resolution groups.

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 The combinatorics of the bar resolution in group cohomology

2004 ◽  
Vol 190 (1-3) ◽  
pp. 291-327 ◽  
Author(s):  
Victor Reiner ◽  
Peter Webb
Keyword(s):  
Group Cohomology ◽  
Bar Resolution

scholarly journals Euclidean-valued group cohomology is always reduced

2018 ◽  
Vol 10 (03) ◽  
pp. 483-491
Author(s):  
Tim Austin
Keyword(s):  
Group Cohomology ◽  
Inverse Limits ◽  
Dimensional Vector ◽  
Linear Transformations ◽  
Closure Theorem ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Linear Formula ◽  
Bar Resolution ◽  
Topological Field

Let [Formula: see text] be a topological field and [Formula: see text] a countable discrete group. Then for any linear [Formula: see text]-action on a finite-dimensional vector space over [Formula: see text], the groups of coboundaries in the inhomogeneous bar resolution are closed in all degrees, and hence the cohomology is reduced in all degrees. This can be deduced from a more general automatic-closure theorem for continuous linear transformations between inverse limits of finite-dimensional vector spaces.

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