Related papers: Matched pairs and Yetter-Drinfeld braces

Let $\Bbbk$ be a field, $H$ a Hopf algebra over $\Bbbk$, and $R = (_iM_j)_{1 \leq i,j \leq n}$ a generalized matrix algebra. In this work, we establish necessary and sufficient conditions for $H$ to act partially on $R$. To achieve this, we…

A generalisation of the notion of a Rota-Baxter operator is proposed. This generalisation consists of two operators acting on an associative algebra and satisfying equations similar to the Rota-Baxter equation. Rota-Baxter operators of any…

In this paper, we mainly give some equivalent characterisations of Hopf braces, show that the category $\mathcal{CB}(A)$ of Hopf braces is equivalent to the category $\mathcal{C}(A)$ of bijective 1-cocycles, and prove that the category…

For two families of Yetter-Drinfel'd Hopf algebras considered earlier by the authors, we determine which of them are isomorphic. We also determine which of their Radford biproducts are isomorphic.

Let $B$ be a bialgebra, and $A$ a left $B$-comodule algebra in a braided monoidal category $\Cc$, and assume that $A$ is also a coalgebra, with a not-necessarily associative or unital left $B$-action. Then we can define a right $A$-action…

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…

Let S be a compact connected oriented surface, whose boundary is connected or empty. A homology cylinder over the surface S is a cobordism between S and itself, homologically equivalent to the cylinder over S. The Y-filtration on the monoid…

The universal Drinfeld-Yetter algebra is an associative algebra whose co-Hochschild cohomology controls the existence of quantization functors of Lie bialgebras, such as the renowned one due to Etingof and Kazhdan. It was initially…

We classify the (filtered) Hopf actions of Hopf-Ore extensions of group algebras on path algebras of quivers, extending results in several other works from special cases to this general setting. Having done this for general Hopf-Ore…

An algebra A with a generalized H-action is a generalization of an H-module algebra where H is just an associative algebra with 1 and a relaxed compatibility condition between the multiplication in A and the H-action on A holds. At first…

If H is a finite dimensional Hopf algebra, C. Cibils and M. Rosso found an algebra X having the property that Hopf bimodules over H^* coincide with left X-modules. We find two other algebras, Y and Z, having the same property; namely, Y is…

In this paper, we begin a systematic study of modified Rota-Baxter algebras, as an associative analogue of the modified classical Yang-Baxter equation. We construct free commutative modified Rota-Baxter algebras by a variation of the…

The variety of skew braces contains several interesting subcategories as subvarieties, as for instance the varieties of radical rings, of groups and of abelian groups. In this article the methods of non-abelian homological algebra are…

A cohomology theory of the adjoint of Hopf algebras, via deformations, is presented by means of diagrammatic techniques. Explicit calculations are provided in the cases of group algebras, function algebras on groups, and the bosonization of…

We apply categorical machinery to the problem of defining anti-Yetter-Drinfeld modules for quasi-Hopf algebras. While a definition of Yetter-Drinfeld modules in this setting, extracted from their categorical interpretation as the center of…

Two important generalizations of the Hopf algebra of symmetric functions are the Hopf algebra of noncommutative symmetric functions and its graded dual the Hopf algebra of quasisymmetric functions. A common generalization of the latter is…

The notion of a coherent unit action on algebraic operads was first introduced by Loday for binary quadratic nonsymmetric operads and generalized by Holtkamp, to ensure that the free objects of the operads carry a Hopf algebra structure.…

The Drinfeld double associated to the weak multiplier Hopf ($*$-) algebra pairing $\left\langle A, B\right\rangle$ is constructed. We show that the Drinfeld double is again a weak multiplier Hopf ($*$-) algebra. If $A$ and $B$ are algebraic…

In 1999 A. Connes and D. Kreimer have discovered a Hopf algebra structure on the Feynman graphs of scalar field theory. They have found that the renormalization can be interpreted as a solving of some Riemann - Hilbert problem. In this work…

Let $(C,A,\varphi)$ be an entwining structure over $k$. In this paper, we introduce the notions of the pivotal entwined datums and ribbon entwined datums to generalize (co)pivotal Hopf algebras and (co)ribbon Hopf algebras. These notions…