Related papers: Hopf monads on monoidal categories

Let $H$ be a Hopf algebra in braided category $\cal C$. Crossed modules over $H$ are objects with both module and comodule structures satisfying some comatibility condition. Category ${\cal C}^H_H$ of crossed modules is braided and is…

This article considers cuspidal curves whose coordinate rings are numerical semigroup algebras. Using a general result about descent of Hopf algebroid structures, their rings of differential operators are shown to be cocommutative and…

We prove that the category of Hopf bimodules over any Hopf algebra has enough injectives, which enables us to extend some results on the unification of Hopf bimodule cohomologies of [T1,T2] to the infinite dimensional case. We also prove…

Let $A$ be a Hopf algebra in a braided category $\cal C$. Crossed modules over $A$ are introduced and studied as objects with both module and comodule structures satisfying a compatibility condition. The category $\DY{\cal C}^A_A$ of…

Consider a monad on an idempotent complete triangulated category with the property that its Eilenberg-Moore category of modules inherits a triangulation. We show that any other triangulated adjunction realizing this monad is 'essentially…

We introduce a notion of an integral along a bimonoid homomorphism as a simultaneous generalization of the integral and cointegral of bimonoids. The purpose of this paper is to characterize an existence of a specific integral, called a…

With the motivation of giving a more precise estimation of the quantum Brauer group of a Hopf algebra $H$ over a field $k$ we construct an exact sequence containing the quantum Brauer group of a Hopf algebra in a certain braided monoidal…

We introduce a contravariant idempotent adjunction between (i) the category of ranked monads on $\mathsf{Set}$; and (ii) the category of internal categories and internal retrofunctors in the category of locales. The left adjoint takes 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 introduce a theory of $*$-structures for bialgebroids and Hopf algebroids over a $*$-algebra, defined in such a way that the relevant category of (co)modules is a bar category. We show that if $H$ is a Hopf $*$-algebra then the action…

A subunit in a monoidal category is a subobject of the monoidal unit for which a canonical morphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules. We…

Let $H$ be a finite dimensional bialgebra. In this paper, we prove that the category of Yetter-Drinfeld-Long bimodules is isomorphic to the Yetter-Drinfeld category over the tensor product bialgebra $H\o H^*$ as monoidal category. Moreover…

Let A be a finite dimensional unital associative algebra over a field K, which is also equipped with a coassociative counital coalgebra structure (\Delta,\eps). A is called a Weak Bialgebra if the coproduct \Delta is multiplicative. We do…

We survey Hopf algebras and their generalizations. In particular, we compare and contrast three well-studied generalizations (quasi-Hopf algebras, weak Hopf algebras, and Hopf algebroids), and two newer ones (Hopf monads and hopfish…

In this paper, we study pointed rank one Hopf algebras and Hopf-Ore extensions of group algebras, over an arbitrary field $k$. It is proved that the rank of a Hopf-Ore extension of a group algebra is one or two or infinite. It is also shown…

We study Cohen-Macaulay Hopf monoids in the category of species. The goal is to apply techniques from topological combinatorics to the study of polynomial invariants arising from combinatorial Hopf algebras. Given a polynomial invariant…

In 2017 Aguiar and Ardila provided a generic way to construct polynomial invariants of combinatorial objects using the notions of Hopf monoids and characters of Hopf monoids. They show that it is possible to find a combinatorial…

We define $A_{\infty}$-structures -- algebras, coalgebras, modules, and comodules -- in an arbitrary monoidal DG category or bicategory by rewriting their definitions in terms of unbounded twisted complexes. We develop new notions of strong…

Generalized permutahedra are a family of polytopes with a rich combinatorial structure and strong connections to optimization. We prove that they are the universal family of polyhedra with a certain Hopf algebraic structure. Their antipode…

We introduce the notions of normal tensor functor and exact sequence of tensor categories. We show that exact sequences of tensor categories generalize strictly exact sequences of Hopf algebras as defined by Schneider, and in particular,…