Related papers: Pointed finite tensor categories over abelian grou…
We prove that commutative algebras in braided tensor categories do not admit faithful Hopf algebra actions unless they come from group actions. We also show that a group action allows us to see the algebra as the regular algebra in the…
We prove the graded braided commutativity of the Hochschild cohomology of $A$ with trivial coefficients, where $A$ is a braided Hopf algebra in the category of Yetter-Drinfeld modules over the group algebra of an abelian group, under some…
We parameterize the finite-dimensional irreducible representations of a class of pointed Hopf algebras over an algebraically closed field of characteristic zero by dominant characters. The Hopf algebras we are considering arise in the work…
We show that if two Hopf algebras are monoidally equivalent, then their categories of bicovariant differential calculi are equivalent. We then classify, for $q \in \mathbb{C}^*$ not a root of unity, the finite dimensional bicovariant…
We present an explicit description of the Hopf 2-cocycles involved in the classification of finite dimensional pointed and copointed Hopf algebras over the symmetric group on three letters. We determine which cocycles are exponentials of…
This paper introduces group-cograded monoidal Hom-Hopf algebras, and shows that this kind of group-cograded monoidal Hom-Hopf algebras are monoidal Hom-Hopf algebras in the Turaev category $\mathcal{J}_{k}$ introduced by Canepeel and De…
This is a survey on pointed Hopf algebras with finite Gelfand-Kirillov dimension and related aspects of the theory of infinite-dimensional Hopf algebras.
We give several criteria to decide whether a given tensor category is the abelian envelope of a fixed symmetric monoidal category. As a main result we prove that the category of finite-dimensional representations of a semisimple simply…
In this paper, we compute the Clebsch-Gordan formulae and the Green rings of connected pointed tensor categories of finite type.
Davydov-Yetter cohomology classifies infinitesimal deformations of tensor categories and of tensor functors. Our first result is that Davydov-Yetter cohomology for finite tensor categories is equivalent to the cohomology of a comonad…
We show that all classes that are neither semisimple nor unipotent in finite simple Chevalley or Steinberg groups different from $PSL_n(q)$ collapse (i.e. are never the support of a finite-dimensional Nichols algebra). As a consequence, we…
We classify all finite dimensional algebras which are derived equivalent to m-cluster tilted algebras of type A.
We show that a compact rigid balanced braided monoidal category with enough compact projective objects gives rise to a system of mapping class group representations compatible with the gluing along marked intervals. A motivation to consider…
Let $p$ be a prime. We complete the classification on pointed Hopf algebras of dimension $p^2$ over an algebraically closed field $k$. When $\text{char}k \neq p$, our result is the same as the well-known result for $\text{char}k=0$. When…
In this paper, we introduce the category of brace triples in a braided monoidal setting and prove that it is isomorphic to the category of s-Hopf braces, which are a generalization of cocommutative Hopf braces. After that, we obtain a…
We introduce the notions of categorical integrals and categorical cointegrals of a finite tensor category $\mathcal{C}$ by using a certain adjunction between $\mathcal{C}$ and its Drinfeld center $\mathcal{Z}(\mathcal{C})$. These notions…
In this short note we show that representation and character varieties of discrete groups can be viewed as tensor products of suitable functors over the PROP of cocommutative Hopf algebras. Such view point has several interesting…
We show that the braided tensor product algebra $A_1\underline{\otimes}A_2$ of two module algebras $A_1, A_2$ of a quasitriangular Hopf algebra $H$ is equal to the ordinary tensor product algebra of $A_1$ with a subalgebra of…
Braided algebras are associative algebras endowed with a Yang-Baxter operator that satisfies certain compatibility conditions involving the multiplication. Along with Hochschild cohomology of algebras, there is also a notion of Yang-Baxter…
We propose the following principle to study pointed Hopf algebras, or more generally, Hopf algebras whose coradical is a Hopf subalgebra. Given such a Hopf algebra A, consider its coradical filtration and the associated graded coalgebra…