Quantum quasigroups and loops.
type: Item
instance of: scholarly article
title: Quantum quasigroups and loops.
zbMATH Open document ID: 1350.20051
published in: Journal of Algebra
publication date: 2016-05-20T00:00:00Z
reviewed by: Loïc Foissy
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 16T99
Mathematics Subject Classification ID: 18D10
zbMATH DE Number: 6583399
zbMATH Keywords: Hopf algebras
zbMATH Keywords: quantum groups
zbMATH Keywords: quasigroups
zbMATH Keywords: loops
zbMATH Keywords: rooted binary trees
zbMATH Keywords: skein polynomials
zbMATH Keywords: Moufang-Hopf algebras
zbMATH Keywords: bialgebras
zbMATH Keywords: Sabinin algebras
zbMATH Keywords: symmetric monoidal categories
author: J. D. H. Smith
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2016.02.014
OpenAlex ID: W2299204377
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DOI: 10.1016/J.JALGEBRA.2016.02.014
review text: A quantum quasigroup is a family \((A,\nabla,\Delta)\), where \((A,\nabla)\) is a magma in a given symmetric monoidal category, \((A,\Delta)\) is a comagma in the same category, such that the compositions \((\Delta\otimes 1_A)\circ(1_A\otimes\nabla)\) and \((1_A\otimes\Delta)(\nabla\otimes 1_A)\) are invertible. Quantum loops are quantum quasigroups with a unit and a counit. This gives a self-dual framework for nonassociative quantum groups techniques.NEWLINENEWLINE In particular, Hopf algebras are also quantum loops; sufficient conditions are given for a quantum loop to give rise to a Hopf algebra. Several examples of quantum loops are given: Moufang-Hopf algebras of Benkart and al., Hopf (co)quasigroups of Klim-Majid, coassociative \(H\)-bialgebras of Perez-Izquierdo, as well as an algebra of rooted binary trees and a Conway algebra of skein polynomials.
description: scientific article; zbMATH DE number 6583399