An explicit description of the simplicial group K(A,n)K(A,n)

We give a new explicit construction for the simplicial group $K(A,n)$. We explain the topological interpretation and discuss some possible applications.Comment: 8 page

Secondary Cohomology and k-invariants

For a triple $(G,A,\kappa)$ (where $G$ is a group, $A$ is a $G$-module and $\kappa:G^3\to A$ is a 3-cocycle) and a $G$-module $B$ we introduce a new cohomology theory $_2H^n(G,A,\kappa;B)$ which we call the secondary cohomology. We give a construction that associates to a pointed topological space $(X,x_0)$ an invariant $_2\kappa^4\in_2H^4(\pi_1(X),\pi_2(X),\kappa^3;\pi_3(X))$. This construction can be seen a "3-type" generalization of the classical $k$-invariant.Comment: 8 page

Symmetric cohomology of groups in low dimension

We give an explicit characterization for group extensions that correspond to elements of the symmetric cohomology $HS^2(G,A)$. We also give conditions for the map $HS^n(G,A)\to H^n(G,A)$ to be injective

Hom-Tensor Categories and the Hom-Yang-Baxter Equation

We introduce a new type of categorical object called a \emph{hom-tensor category} and show that it provides the appropriate setting for modules over an arbitrary hom-bialgebra. Next we introduce the notion of \emph{hom-braided category} and show that this is the right setting for modules over quasitriangular hom-bialgebras. We also show how the hom-Yang-Baxter equation fits into this framework and how the category of Yetter-Drinfeld modules over a hom-bialgebra with bijective structure map can be organized as a hom-braided category. Finally we prove that, under certain conditions, one can obtain a tensor category (respectively a braided tensor category) from a hom-tensor category (respectively a hom-braided category).Comment: 36 pages, many diagram

Operations on the Secondary Hochschild Cohomology

We show that the secondary Hochschild cohomology associated to a triple $(A,B,\varepsilon)$ has several of the properties of the usual Hochschild cohomology. Among others, we prove the existence of the cup and Lie products, discuss the connection with extensions of $B$-algebras, and give a Hodge type decomposition of the secondary Hochschild cohomology

GG-Algebra Structure on the Higher Order Hochschild Cohomology HS2∗(A,A)H^*_{S^2}(A,A)

We present a deformation theory associated to the higher Hochschild cohomology $H_{S^2}^*(A,A)$. We also study a $G$-algebra structure associated to this deformation theory

Generalized (anti) Yetter-Drinfeld modules as components of a braided T-category

If H is a Hopf algebra with bijective antipode and \alpha, \beta \in Aut_{Hopf}(H), we introduce a category_H{\cal YD}^H(\alpha, \beta), generalizing both Yetter-Drinfeld and anti-Yetter-Drinfeld modules. We construct a braided T-category {\cal YD}(H) having all these categories as components, which if H is finite dimensional coincides with the representations of a certain quasitriangular T-coalgebra DT(H) that we construct. We also prove that if (\alpha, \beta) admits a so-called pair in involution, then_H{\cal YD}^H(\alpha, \beta) is isomorphic to the category of usual Yetter-Drinfeld modules_H{\cal YD}^H.Comment: 12 pages, Latex, no figure

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 3.9 and 4.11).Comment: 17 page

From 3-algebras to Δ\Delta-Groups

We introduce $\Delta$-groups and show how they fit in the context of lattice field theory. To a manifold $M$ we associate a $\Delta$-group $\Gamma(M)$. We define the symmetric cohomology $HS^n(G,A)$ of a group $G$ with coefficients in a $G$-module $A$. The $\Delta$-group $\Gamma(M)$ is determined by the action of $\pi_1(M)$ on $\pi_2(M)$ and an element of $HS^3(\pi_1(M),\pi_2(M))$.Comment: 18 page

Pseudosymmetric braidings, twines and twisted algebras

A laycle is the categorical analogue of a lazy cocycle. Twines (as introduced by Bruguieres) and strong twines (as introduced by the authors) are laycles satisfying some extra conditions. If $c$ is a braiding, the double braiding $c^2$ is always a twine; we prove that it is a strong twine if and only if $c$ satisfies a sort of modified braid relation (we call such $c$ pseudosymmetric, as any symmetric braiding satisfies this relation). It is known that symmetric Yetter-Drinfeld categories are trivial; we prove that the Yetter-Drinfeld category $_H{\cal YD}^H$ over a Hopf algebra $H$ is pseudosymmetric if and only if $H$ is commutative and cocommutative. We introduce as well the Hopf algebraic counterpart of pseudosymmetric braidings under the name pseudotriangular structures and prove that all quasitriangular structures on the $2^{n+1}$-dimensional pointed Hopf algebras E(n) are pseudotriangular. We observe that a laycle on a monoidal category induces a so-called pseudotwistor on every algebra in the category, and we obtain some general results (and give some examples) concerning pseudotwistors, inspired by properties of laycles and twines.Comment: 29 page