Related papers: On the anti-Yetter-Drinfeld module-contramodule co…
A natural extension of the Hopf-cyclic cohomology, with coefficients, is introduced to encompass topological Hopf algebras. The topological theory allows to work with infinite dimensional Lie algebras. Furthermore, the category of…
B\"ohm and \c{S}tefan have expressed cyclic homology as an invariant that assigns homology groups $\mathrm{HC}^\chi_i(\mathrm N, \mathrm M)$ to right and left coalgebras $\mathrm N$ respectively $\mathrm M$ over a distributive law $\chi$…
The aim of this paper is to define and study Yetter-Drinfeld modules over Hom-bialgebras, a generalized version of bialgebras obtained by modifying the algebra and coalgebra structures by a homomorphism. Yetter-Drinfeld modules over a…
Generalising a result for Hopf algebras, we not only define the four possible types of Hopf modules in the bialgebroid setting but also yield the notion of two-sided two-cosided Hopf modules, also known as Hopf bimodules or tetramodules, in…
A new quantization of groupoids under the name of \times-Hopf coalgebras is introduced. We develop a Hopf cyclic theory with coefficients in stable-anti-Yetter-Drinfeld modules for \times-Hopf coalgebras. We use \times-Hopf coalgebras to…
We construct several pairings in Hopf-cyclic cohomology of (co)module (co)algebras with arbitrary coefficients. The key ideas instrumental in constructing these pairings are the derived functor interpretation of Hopf-cyclic and equivariant…
We show that all possible categories of Yetter-Drinfeld modules over a quasi-Hopf algebra $H$ are isomorphic. We prove also that the category $\yd^{\rm fd}$ of finite dimensional left Yetter-Drinfeld modules is rigid and then we compute…
Let $(R^{\vee},R)$ be a dual pair of Hopf algebras in the category of Yetter-Drinfeld modules over a Hopf algebra $H$ with bijective antipode. We show that there is a braided monoidal isomorphism between rational left Yetter-Drinfeld…
This is the first one in a series of two papers on the continuation of our study in cup products in Hopf cyclic cohomology. In this note we construct cyclic cocycles of algebras out of Hopf cyclic cocycles of algebras and coalgebras. In the…
Given a Hopf algebra A, there exist various cohomology theories for the category of Hopf bimodules over A, introduced by M. Gerstenhaber and S.D. Schack, and by C. Ospel. We prove, when A is finite dimensional, that they are equal to the…
Grothendieck-Verdier duality is a powerful and ubiquitous structure on monoidal categories, which generalises the notion of rigidity. Hopf algebroids are a generalisation of Hopf algebras, to a non-commutative base ring. Just as the…
We apply categorical machinery to the problem of defining cyclic cohomology with coefficients in two particular cases, namely quasi-Hopf algebras and Hopf algebroids. In the case of the former, no definition was thus far available in the…
We prove that both, the embedding of the category of Hopf algebras into that of bialgebras and the forgetful functor from the category of Hopf algebras to the category of algebras, have right adjoints; in other words: every bialgebra has a…
We define the exact category of contraherent cosheaves of contramodules on a locally Noetherian formal scheme, as well as the exact categories of locally contraherent cosheaves of contramodules (with respect to a given open covering). We…
We recall the notion of a Hopf (co)quasigroup defined in \cite{Kl09} and define integration and Fourier Transforms on these objects analogous to those in the theory of Hopf algebras. Using the general Hopf module theory for Hopf…
The aim of this paper is to define and study Yetter-Drinfeld modules over weak Hom-Hopf algebras. We show that the category ${}_H{\cal WYD}^H$ of Yetter-Drinfeld modules with bijective structure maps over weak Hom-Hopf algebras is a rigid…
Contramodules are module-like algebraic structures endowed with infinite summation (or, occasionally, integration) operations satisfying natural axioms. Introduced originally by Eilenberg and Moore in 1965 in the case of coalgebras over…
Covariant Hom-bimodules are introduced and the structure theory of them in the Hom-setting is studied in a detailed way. The category of bicovariant Hom-bimodules is proved to be a (pre)braided monoidal category and its structure theory is…
Monoidal product, braiding, balancing and weak duality are pieces of algebraic information that are well-known to have their origin in oriented genus zero surfaces and their mapping classes. More precisely, each of them correspond to…
We introduce the cylindrical module $A \natural \mathcal{H}$, where $\mathcal{H}$ is a Hopf algebra and $A$ is a Hopf module algebra over $\mathcal{H}$. We show that there exists an isomorphism between $\mathsf{C}_{\bullet}(A^{op} \rtimes…