Brauer-Picard groups and pointed braided tensor categories

Tensor categories are ubiquitous in areas of mathematics involving algebraic structures. They appear, also, in other fields, such as mathematical physics (conformal field theory) and theoretical computer science (quantum computation). The study of tensor categories is, thus, a useful undertaking. Two classes of tensor categories arise naturally in this study. One consists of group-graded extensions and another of pointed tensor categories. Understanding the former involves knowledge of the Brauer-Picard group of a tensor category, while results about pointed Hopf algebras provide insights into the structure of the latter. This work consists of two main parts. In the first one we compute the Brauer-Picard group of a class of symmetric non-semisimple finite tensor categories by studying a canonical action on a vector space. In the second one we use results from the theory of Hopf algebras to prove an equivalence between the groupoid of pointed braided finite tensor categories admitting a fiber functor and a groupoid of metric quadruples

Classifying bicrossed products of two Sweedler's Hopf algebras

summary:We continue the study started recently by Agore, Bontea and Militaru in ``Classifying bicrossed products of Hopf algebras'' (2014), by describing and classifying all Hopf algebras $E$ that factorize through two Sweedler's Hopf algebras. Equivalently, we classify all bicrossed products $H_4 \bowtie H_4$. There are three steps in our approach. First, we explicitly describe the set of all matched pairs $(H_4, H_4, \triangleright , \triangleleft )$ by proving that, with the exception of the trivial pair, this set is parameterized by the ground field $k$. Then, for any $\lambda \in k$, we describe by generators and relations the associated bicrossed product, $\mathcal {H}_{16, \lambda }$. This is a $16$-dimensional, pointed, unimodular and non-semisimple Hopf algebra. A Hopf algebra $E$ factorizes through $H_4$ and $H_4$ if and only if $E \cong H_4 \otimes H_4$ or $E \cong {\mathcal H}_{16, \lambda }$. In the last step we classify these quantum groups by proving that there are only three isomorphism classes represented by: $H_4 \otimes H_4$, ${\mathcal H}_{16, 0}$ and ${\mathcal H}_{16, 1} \cong D(H_4)$, the Drinfel'd double of $H_4$. The automorphism group of these objects is also computed: in particular, we prove that ${\rm Aut}_{\rm Hopf}( D(H_4))$ is isomorphic to a semidirect product of groups, $k^{\times } \rtimes \mathbb {Z}_2$

The Classification of All Crossed Products H4#k[Cn]H_4 \# k[C_{n}]

Using the computational approach introduced in [Agore A.L., Bontea C.G., Militaru G., J. Algebra Appl. 12 (2013), 1250227, 24 pages, arXiv:1207.0411] we classify all coalgebra split extensions of $H_4$ by $k[C_n]$, where $C_n$ is the cyclic group of order $n$ and $H_4$ is Sweedler's $4$-dimensional Hopf algebra. Equivalently, we classify all crossed products of Hopf algebras $H_4 \# k[C_{n}]$ by explicitly computing two classifying objects: the cohomological 'group' ${\mathcal H}^{2} ( k[C_{n}], H_4)$ and $\text{CRP}( k[C_{n}], H_4):=$ the set of types of isomorphisms of all crossed products $H_4 \# k[C_{n}]$. More precisely, all crossed products $H_4 \# k[C_n]$ are described by generators and relations and classified: they are $4n$-dimensional quantum groups $H_{4n, \lambda, t}$, parameterized by the set of all pairs $(\lambda, t)$ consisting of an arbitrary unitary map $t : C_n \to C_2$ and an $n$-th root $\lambda$ of $\pm 1$. As an application, the group of Hopf algebra automorphisms of $H_{4n, \lambda, t}$ is explicitly described