Dual Constructions for Partial Actions of Hopf Algebras

The duality between partial actions (partial $H$-module algebras) and co-actions (partial $H$-comodule algebras) of a Hopf algebra $H$ is fully explored in this work. A connection between partial (co)actions and Hopf algebroids is established under certain commutativity conditions. Moreover, we continue this duality study, introducing also partial $H$-module coalgebras and their associated $C$-rings, partial $H$-comodule coalgebras and their associated cosmash coproducts, as well as the mutual interrelations between these structures.Comment: v3: strongly revised versio

Tangent Categories from the Coalgebras of Differential Categories

Following the pattern from linear logic, the coKleisli category of a differential category is a Cartesian differential category. What then is the coEilenberg-Moore category of a differential category? The answer is a tangent category! A key example arises from the opposite of the category of Abelian groups with the free exponential modality. The coEilenberg-Moore category, in this case, is the opposite of the category of commutative rings. That the latter is a tangent category captures a fundamental aspect of both algebraic geometry and Synthetic Differential Geometry. The general result applies when there are no negatives and thus encompasses examples arising from combinatorics and computer science

Chiral Koszul duality

We extend the theory of chiral and factorization algebras, developed for curves by Beilinson and Drinfeld in \cite{bd}, to higher-dimensional varieties. This extension entails the development of the homotopy theory of chiral and factorization structures, in a sense analogous to Quillen's homotopy theory of differential graded Lie algebras. We prove the equivalence of higher-dimensional chiral and factorization algebras by embedding factorization algebras into a larger category of chiral commutative coalgebras, then realizing this interrelation as a chiral form of Koszul duality. We apply these techniques to rederive some fundamental results of \cite{bd} on chiral enveloping algebras of $\star$-Lie algebras

Varieties of Languages in a Category

Eilenberg's variety theorem, a centerpiece of algebraic automata theory, establishes a bijective correspondence between varieties of languages and pseudovarieties of monoids. In the present paper this result is generalized to an abstract pair of algebraic categories: we introduce varieties of languages in a category C, and prove that they correspond to pseudovarieties of monoids in a closed monoidal category D, provided that C and D are dual on the level of finite objects. By suitable choices of these categories our result uniformly covers Eilenberg's theorem and three variants due to Pin, Polak and Reutenauer, respectively, and yields new Eilenberg-type correspondences

The Drinfel'd Double versus the Heisenberg Double for Hom-Hopf Algebras

Let $(A,\alpha)$ be a finite-dimensional Hom-Hopf algebra. In this paper we mainly construct the Drinfel'd double $D(A)=(A^{op}\bowtie A^{\ast},\alpha\otimes(\alpha^{-1})^{\ast})$ in the setting of Hom-Hopf algebras by two ways, one of which generalizes Majid's bicrossproduct for Hopf algebras (see \cite{M2}) and another one is to introduce the notion of dual pairs of of Hom-Hopf algebras. Then we study the relation between the Drinfel'd double $D(A)$ and Heisenberg double $H(A)=A\# A^{*}$, generalizing the main result in \cite{Lu}. Especially, the examples given in the paper are not obtained from the usual Hopf algebras