Quantum quasigroups and loops.
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.
scientific article; zbMATH DE number 6583399
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