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Register a new userQuasitriangular structures on abelian extensions of $mathbb{Z}_{2}$
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Authors
Kun Zhou
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The aim of this paper is to give all quasitriangular structures on a class of semisimple Hopf algebras constructed through abelian extensions of $Bbbkmathbb{Z}_{2}$ by $Bbbk^G$ for an abelian group $G$. We first introduce the concept of symmetry of quasitriangular structures of Hopf algebras and obtain some related propositions which can be used to simplify our calculations of quasitriangular structures. Secondly, we find that quasitriangular structures of these semisimple Hopf algebras can do division-like operations. Using such operations we transform the problem of solving the quasitriangular structures into solving general solutions and giving a special solution. Then we give all general solutions and get a necessary and sufficient condition for the existence of a special solution.
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In this paper, first we classify non-abelian extensions of Leibniz algebras by the second non-abelian cohomology. Then, we construct Leibniz 2-algebras using derivations of Leibniz algebras, and show that under a condition on the center, a non-abelian extension of Leibniz algebras can be described by a Leibniz 2-algebra morphism. At last, we give a description of non-abelian extensions in terms of Maurer-Cartan elements in a differential graded Lie algebra.
There are two very natural products of compact matrix quantum groups: the tensor product $Gtimes H$ and the free product $G*H$. We define a number of further products interpolating these two. We focus more in detail to the case where $G$ is an easy quantum group and $H=hat{mathbb{Z}}_2$, the dual of the cyclic group of order two. We study subgroups of $G*hat{mathbb{Z}}_2$ using categories of partitions with extra singletons. Closely related are many examples of non-easy bistochastic quantum groups.
Suppose that $E=A[x;sigma,delta]$ is an Ore extension with $sigma$ an automorphism. It is proved that if $A$ is twisted Calabi-Yau of dimension $d$, then $E$ is twisted Calabi-Yau of dimension $d+1$. The relation between their Nakayama automorphisms is also studied. As an application, the Nakayama automorphisms of a class of 5-dimensional Artin-Schelter regular algebras are given explicitly.
We extend cite[Theorem 4.5]{DGNO} and cite[Theorem 4.22]{LKW} to positive characteristic (i.e., to the finite, not necessarily fusion, case). Namely, we prove that if $D$ is a finite non-degenerate braided tensor category over an algebraically closed field $k$ of characteristic $p>0$, containing a Tannakian Lagrangian subcategory $Rep(G)$, where $G$ is a finite $k$-group scheme, then $D$ is braided tensor equivalent to $Rep(D^{omega}(G))$ for some $omegain H^3(G,mathbb{G}_m)$, where $D^{omega}(G)$ denotes the twisted double of $G$ cite{G2}. We then prove that the group $mathcal{M}_{{rm ext}}(Rep(G))$ of minimal extensions of $Rep(G)$ is isomorphic to the group $H^3(G,mathbb{G}_m)$. In particular, we use cite{EG2,FP} to show that $mathcal{M}_{rm ext}(Rep(mu_p))=1$, $mathcal{M}_{rm ext}(Rep(alpha_p))$ is infinite, and if $O(Gamma)^*=u(g)$ for a semisimple restricted $p$-Lie algebra $g$, then $mathcal{M}_{rm ext}(Rep(Gamma))=1$ and $mathcal{M}_{rm ext}(Rep(Gammatimes alpha_p))cong g^{*(1)}$.
A unitary fusion category is called $mathbb{Z}/2mathbb{Z}$-quadratic if it has a $mathbb{Z}/2mathbb{Z}$ group of invertible objects and one other orbit of simple objects under the action of this group. We give a complete classification of $mathbb{Z}/2mathbb{Z}$-quadratic unitary fusion categories. The main tools for this classification are skein theory, a generalization of Ostriks results on formal codegrees to analyze the induction of the group elements to the center, and a computation similar to Larsons rank-finiteness bound for $mathbb{Z}/3mathbb{Z}$-near group pseudounitary fusion categories. This last computation is contained in an appendix coauthored with attendees from the 2014 AMS MRC on Mathematics of Quantum Phases of Matter and Quantum Information.
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