Bar Resolution Research Articles

Operads, Operadic Categories and the Blob Complex

We will show that the Morrison–Walker blob complex appearing in Topological Quantum Field Theory is an operadic bar resolution of a certain operad composed of fields and local relations. As a by-product we develop the theory of unary operadic categories and study some novel and interesting phenomena arising in this context.

On the semifree resolutions of DG algebras over the enveloping DG algebras

The goal of this paper is to construct a semifree resolution for a non-negatively graded strongly commutative DG algebra B over the enveloping DG algebra B ⊗ A B , where A ⊆ B is a DG subalgebra and B is semifree over A. Our construction of such a semifree resolution, that we denote by ( B , D ) , uses the notions of reduced bar resolution and tensor algebra of the shift of the diagonal ideal. As an application of ( B , D ) , we provide a new characterization of naïve liftability of semifree DG B-modules.

Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies

We calculate the Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies of the Taft algebra Tp for any integer p>2 which is a nonquasi-triangular Hopf algebra. We show that the bracket is indeed zero on Hopf algebra cohomology of Tp, as in all known quasi-triangular Hopf algebras. This example is the first known bracket computation for a nonquasi-triangular algebra. Also, we find a general formula for the bracket on Hopf algebra cohomology of any Hopf algebra with bijective antipode on the bar resolution that is reminiscent of Gerstenhaber's original formula for Hochschild cohomology.

Homotopía de contracción de cadenas y un método para resoluciones proyectivas relativas

The purpose of this article is to check the main results of the method that allows the construction of a relative projective resolution of an S-module N given in appendix A of the article [1], and to show an application of this method. It is achieve to show the usefulness of the method, and to put in relevance that the relative projective resolution obtained with the method is comparable to the bar resolution of an algebra because it is provided with chain contracting homotopy.

GERSTENHABER BRACKETS FOR SKEW GROUP ALGEBRAS IN POSITIVE CHARACTERISTIC

The deformation theory of an algebra is controlled by the Gerstenhaber bracket, a Lie bracket on Hochschild cohomology. We develop techniques for evaluating Gerstenhaber brackets of semidirect product algebras recording actions of finite groups over fields of positive characteristic. The Hochschild cohomology and Gerstenhaber bracket of these skew group algebras can be complicated when the characteristic of the underlying field divides the group order. We show how to investigate Gerstenhaber brackets using twisted product resolutions, which are often smaller and more convenient than the cumbersome bar resolution typically used. These resolutions provide a concrete description of the Gerstenhaber bracket suitable for exploring questions in deformation theory. We demonstrate with the prototypical example of a Drinfeld Hecke algebra (graded Hecke algebra) in positive characteristic.

Group-Twisted Alexander-Whitney and Eilenberg-Zilber Maps

We define group-twisted Alexander-Whitney and Eilenberg-Zilber maps for converting between bimodule resolutions of skew group algebras. These algebras are the natural semidirect products recording actions of finite groups by automorphisms. The group-twisted chain maps allow us to transfer information between resolutions for use in homology theories, for example, in those governing deformation theory. We show how to translate in particular from the default (but often cumbersome) reduced bar resolution to a more convenient twisted product resolution. This provides a more universal approach to some known results classifying PBW deformations.

A five-term exact sequence for Kac cohomology

We use relative group cohomologies to compute the Kac cohomology of matched pairs of finite groups. This cohomology naturally appears in the theory of abelian extensions of finite dimensional Hopf algebras. We prove that Kac cohomology can be computed using relative cohomology and relatively projective resolutions. This allows us to use other resolutions, besides the bar resolution, for computations. We compute, in terms of relative cohomology, the first two pages of a spectral sequence which converges to the Kac cohomology and its associated five-term exact sequence. Through several examples, we show the usefulness of the five-term exact sequence in computing groups of abelian extensions.

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

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.

CONSTRUCTION AND TESTING OF THE MEASUREMENT SYSTEM FOR EXCAVATOR PRODUCTIVITY

Hydraulic excavator is one of the most used machines in mining, construction and geotechnics. Monitoring its productivity can gain benefits like savings in fuel consumption and maintenance, cost optimization of working parameters and higher working efficiency. Measurement system for such purpose can be complex and expensive, requires special equipment from manufacturer and does not exist as out of the box solution that would measure working parameters such as bucket trajectory, digging energy, digging force etc. Measuring system was designed to monitor productivity and gather working parameters. It consists of three pressure transmitters for measuring cylinder pressures, three draw wire sensors for cylinder elongations, one inertial measurement unit for measurement of swing angle and one logger unit for data acquisition and storage. Logger unit, equipped with SD card module and 12-bit AD converter, is based on Atmega328 microcontroller, which allows for usage of Arduino IDE and already developed libraries. The system was tested in a laboratory to determine its accuracy, resolution and usable sample rate. Cylinder pressure can be measured in range 0-500 bar with excellent accuracy of 0.026 % FSO and 0.15 bar resolution. Cylinder elongation can be measured in range 0-2300 mm with accuracy of 0.26 % FSO and 0.58 mm resolution. Swing angle measurement accuracy is 1.5° per circle. Usable sample rate of the system is found to be 39 Hz-588 Hz, depending on the logger executing code.

Bar simplicial modules and secondary cyclic (co)homology

In this paper we study the simplicial structure of the complex C^{\bullet}((A,B,\varepsilon); M) , associated to the secondary Hochschild cohomology. The main ingredient is the simplicial object \mathcal{B}(A,B,\varepsilon) , which plays a role equivalent to that of the bar resolution associated to an algebra. We also introduce the secondary cyclic (co)homology and establish some of its properties (Theorems 4.11 and 5.11).

Euclidean-valued group cohomology is always reduced

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.

Comparison morphisms between two projective resolutions of monomial algebras

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

Comparison morphisms between two projective resolutions of monomial algebras

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

Gerstenhaber Algebra Structure on the Hochschild Cohomology of Quadratic String Algebras

We describe the Gerstenhaber algebra structure on the Hochschild cohomology HH∗(A) when A is a quadratic string algebra. First we compute the Hochschild cohomology groups using Barzdell’s resolution and we describe generators of these groups. Then we construct comparison morphisms between the bar resolution and Bardzell’s resolution in order to get formulae for the cup product and the Lie bracket. We find conditions on the bound quiver associated to string algebras in order to get non-trivial structures.

A little bit of extra functoriality for Ext and the computation of the Gerstenhaber bracket

We show that the action of the Lie algebra HH1(A) of outer derivations of an associative algebra A on the Hochschild cohomology HH•(A) of A given by the Gerstenhaber bracket can be computed in terms of an arbitrary projective resolution of A as an A-bimodule, without having recourse to comparison maps between the resolution and the bar resolution.

Hochschild (co)homology of exterior algebras using algebraic Morse theory

ABSTRACTIn [1], the authors computed the additive and multiplicative structure of HH*(A;A), where A is the n-th exterior algebra over a field. In this paper, we derive all their results using a different method (AMT) as well as calculate the additive structure of HHk(A;A) and HHk(A;A) over ℤ. We provide concise presentations of algebras HH∗(A;A) and HH*(A;A) as well as determine their generators in the Hochschild complex. Finally, we compute an explicit free resolution (spanned by multisets) of the Ae-module A and describe the homotopy equivalence to its bar resolution.

Optimisation of scintigraphic imaging of the novel therapeutic agent Ra-223 in the treatment of metastatic castration resistant prostate cancer

Introduction Imaging of radionuclide therapy agents is useful in determining biodistribution, verification of correct pharmaceutical and to rule out extravasation in clinical administrations. The aim of this study was to optimise imaging of Ra-223 dichloride (Xofigo™) for patients with metastatic castration resistant prostate cancer. Materials and methods A Jaszczak phantom and bar resolution phantom were used with 0.8 MBq of Ra-223 to evaluate image quality for a range of collimators, energy windows and matrices for planar and SPECT imaging (Siemens ECam, Erlangen). Clinical imaging was carried out twenty minutes post injection of Ra-223. Four sets of static images were obtained on two patients undergoing treatment, pre and post optimisation. Results Optimum imaging parameters were found using planar imaging with a Xenon-133 energy window (80 kV photopeak, 20% energy window), 128 × 128 matrix. A medium energy collimator resulted in higher resolution images with reduced image noise compared to a low energy general purpose collimator. These parameters allowed for clinically acceptable image quality with an imaging time acceptable for patient comfort (20 min). Clinical imaging allowed visualisation of structures post administration of Ra-223. Post optimisation images were of better quality although pre and post optimisation images were clinically useful. Conclusion Useful clinical information can be obtained from scintigraphic imaging of Ra-223. The quality of imaging improved with optimisation of imaging parameters.

The Bar Construction and Affine Stacks

We discuss the equivalence of two constructions of a unipotent group scheme attached to a differential graded algebra over a ℚ-algebra. The first construction is using the bar resolution and the second is using Toen's theory of affine stacks. We use this to establish the equivalence of two approaches to the comparison theorem in p-adic Hodge theory for the unipotent fundamental group of varieties defined over p-adic fields.

Multiplicative structures on the twisted equivariant K-theory of finite groups

Let K be a finite group and let G be a finite group acting on K by automorphisms. In this paper we study two different but intimately related subjects: on the one side we classify all possible multiplicative and associative structures with which one can endow the twisted G -equivariant K-theory of K , and on the other, we classify all possible monoidal structures with which one can endow the category of twisted and G -equivariant bundles over K . We achieve this classification by encoding the relevant information in the cochains of a sub double complex of the double bar resolution associated to the semi-direct product K \rtimes G ; we use known calculations of the cohomology of K , G and K \rtimes G to produce concrete examples of our classification. In the case in which K=G and G acts by conjugation, the multiplication map G \rtimes G \to G is a homomorphism of groups and we define a shuffle homomorphism which realizes this map at the homological level. We show that the categorical information that defines the Twisted Drinfeld Double can be realized as the dual of the shuffle homomorphism applied to any 3-cocycle of G . We use the pullback of the multiplication map in cohomology to classify the possible ring structures that the Grothendieck ring of representations of the Twisted Drinfeld Double may have, and we include concrete examples of this procedure.

The second homology of [formula omitted] of S-integers

We calculate the structure of the finitely generated groups H2(SL2(Z[1/m]),Z) when m is a multiple of 6. Furthermore, we show how to construct homology classes, represented by cycles in the bar resolution, which generate these groups and have prescribed orders. When n≥2 and m is the product of the first n primes, we combine our results with those of Jun Morita to show that the projection St(2,Z[1/m])→SL2(Z[1/m]) is the universal central extension. Our methods have wider applicability: The main result on the structure of the second homology of certain rings is valid for rings of S-integers with sufficiently many units. For a wide class of rings A, we construct explicit homology classes in H2(SL2(A),Z), functorially dependent on a pair of units, which correspond to symbols in K2(2,A).