Pseudosymmetric braidings, twines and twisted algebras

TL;DR

This paper introduces pseudosymmetric braidings and twines, characterizes their properties in categories and Hopf algebras, and explores their implications for algebraic structures and quasitriangular structures.

Contribution

It defines pseudosymmetric braidings and twines, characterizes pseudosymmetric Yetter-Drinfeld categories, and introduces pseudotriangular structures for Hopf algebras, expanding the understanding of symmetry in algebraic categories.

Findings

Double braiding is a strong twine iff the braiding is pseudosymmetric.

Yetter-Drinfeld category is pseudosymmetric iff the Hopf algebra is commutative and cocommutative.

All quasitriangular structures on E(n) are pseudotriangular.

Abstract

A laycle is the categorical analogue of a lazy cocycle. Twines (as introduced by Bruguieres) and strong twines (as introduced by the authors) are laycles satisfying some extra conditions. If $c$ is a braiding, the double braiding $c^2$ is always a twine; we prove that it is a strong twine if and only if $c$ satisfies a sort of modified braid relation (we call such $c$ pseudosymmetric, as any symmetric braiding satisfies this relation). It is known that symmetric Yetter-Drinfeld categories are trivial; we prove that the Yetter-Drinfeld category $_H{\cal YD}^H$ over a Hopf algebra $H$ is pseudosymmetric if and only if $H$ is commutative and cocommutative. We introduce as well the Hopf algebraic counterpart of pseudosymmetric braidings under the name pseudotriangular structures and prove that all quasitriangular structures on the $2^{n+1}$-dimensional pointed Hopf algebras E(n) are pseudotriangular. We observe that a laycle on a monoidal category induces a so-called pseudotwistor on every algebra in the category, and we obtain some general results (and give some examples) concerning pseudotwistors, inspired by properties of laycles and twines.