Related papers: Gerstenhaber-Schack Bialgebras

Graded-division algebras are building blocks in the theory of finite-dimensional associative algebras graded by a group G. If G is abelian, they can be described, using a loop construction, in terms of central simple graded-division…

We calculate the Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies of the Taft algebra $T_p$ for any integer $p>2$ which is a nonquasi-triangular Hopf algebra. We show that the bracket is indeed zero on Hopf algebra…

To every Hopf heap or quantum cotorsor of Grunspan a Hopf algebra of translations is associated. This translation Hopf algebra acts on the Hopf heap making it a Hopf-Galois co-object. Conversely, any Hopf-Galois co-object has the natural…

We give explicit formulas for maps in a long exact sequence connecting bialgebra cohomology to Hochschild cohomology. We give a sufficient condition for the connecting homomorphism to be surjective. We apply these results to compute all…

We define a "combinatorial Hopf algebra" as a Hopf algebra which is free (or cofree) and equipped with a given isomorphism to the free algebra over the indecomposables (resp. the cofree coalgebra over the primitives). The choice of such an…

Two cochain complexes are constructed for an algebra A and a coalgebra C entwined with each other via the map $\psi:C\otimes A\to A\otimes C$. One complex is associated to an A-bimodule, the other to a C-bicomodule. In the former case the…

We describe the dualization of the algebra of secondary cohomology operations in terms of generators extending the Milnor dual of the Steenrod algebra. In this way we obtain explicit formulae for the computation of the E_3-term of the Adams…

In earlier joint work with A. Connes on transverse index theory on foliations, cyclic cohomology adapted to Hopf algebras has emerged as a decisive tool in deciphering the total index class of the hypoelliptic signature operator. We have…

Let $M$ be any compact simply-connected $d$-dimensional smooth manifold and let $\mathbb{F}$ be any field. We show that the Gerstenhaber algebra structure on the Hochschild cohomology on the singular cochains of $M$, $HH^*(S^*(M);S^*(M))$,…

A $Z_3$-graded Hopf algebra structure of exterior algebra with two generators is introduced. Two covariant differential calculus on the $Z_3$-graded exterior algebra are presented. Using the generators and their partial derivatives a…

We introduce a general class of combinatorial objects, which we call \emph{multi-complexes}, which simultaneously generalizes graphs, multigraphs, hypergraphs and simplicial and delta complexes. We introduce a natural algebra of…

We show that if a finite dimensional Hopf algebra over ${\bf C}$ has a basis such that all the structure constants are non-negative, then the Hopf algebra must be given by a finite group $G$ and a factorization $G=G_+G_-$ into two…

Any finite-dimensional Hopf algebra H is Frobenius and the stable category of H-modules is triangulated monoidal. To H-comodule algebras we assign triangulated module-categories over the stable category of H-modules. These module-categories…

Given a cocommutative Hopf algebra $\mathcal{H}$ over a commutative ring $K$ and a symmetric partial action of $\mathcal{H}$ on a $K$-algebra $A,$ we obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild…

Hopf-Hecke algebras and Barbasch-Sahi algebras were defined by the first named author (2016) in order to provide a general framework for the study of Dirac cohomology. The aim of this paper is to explore new examples of these definitions…

To a finite group G one can associate a tower of wreath products S_n[G]. It is well known that the graded direct sum of the Grothendieck groups of the categories of finite dimensional complex representations of these groups can be given the…

Parallel to operated algebras built on top of planar rooted trees via the grafting operator $B^+$, we introduce and study $\vee$-algebras and more generally $\vee_\Omega$-algebras based on planar binary trees. Involving an analogy of the…

A general deformation theory of algebras which factorise into two subalgebras is studied. It is shown that the classification of deformations is related to the cohomology of a certain double complex reminiscent of the Gerstenhaber-Schack…

We construct a Frobenius algebra structure on the Hochschild cochains of a group ring k[G] that extends the known structure of a <1, 2> topological quantum field theory on HH^0(k[G]; k[G]), k a field and G a finite group. The convolution…

Quasishuffle Hopf algebras, usually defined on a commutative monoid, can be more generally defined on any associative algebra V. If V is a commutative and cocommutative bialgebra, the associated quasishuffle bialgebra QSh(V) inherits a…