On Hopf algebras in entropic Jonsson-Tarski varieties
Bull. Korean Math. Soc. 2015 Vol. 52, No. 5, 1587-1606
https://doi.org/10.4134/BKMS.2015.52.5.1587
Printed September 1, 2015
Anna B. Romanowska and Jonathan D. H. Smith
Warsaw University of Technology, Iowa State University
Abstract : Comonoid, bi-algebra, and Hopf algebra structures are studied within the universal-algebraic context of entropic varieties. Attention focuses on the behavior of setlike and primitive elements. It is shown that entropic J\'{o}nsson-Tarski varieties provide a natural universal-algebraic setting for primitive elements and group quantum couples (generalizations of the group quantum double). Here, the set of primitive elements of a Hopf algebra forms a Lie algebra, and the tensor algebra on any algebra is a bi-algebra. If the tensor algebra is a Hopf algebra, then the underlying J\'{o}nsson-Tarski monoid of the generating algebra is cancellative. The problem of determining when the J\'{o}nsson-Tarski monoid forms a group is open.
Keywords : Hopf algebra, quantum group, entropic algebra, commutative monoid, Jonsson-Tarski algebra, quantum double
MSC numbers : 08A99, 16T05
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