ON HOPF ALGEBRAS IN ENTROPIC JÓNSSON-TARSKI VARIETIES

Title & Authors
ON HOPF ALGEBRAS IN ENTROPIC JÓNSSON-TARSKI VARIETIES
ROMANOWSKA, ANNA B.; SMITH, JONATHAN D.H.;

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 $\small{J{\acute{o}}nsson}$-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 $\small{J{\acute{o}}nsson}$-Tarski monoid of the generating algebra is cancellative. The problem of determining when the $\small{J{\acute{o}}nsson}$-Tarski monoid forms a group is open.
Keywords
Hopf algebra;quantum group;entropic algebra;commutative monoid;Jonsson-Tarski algebra;quantum double;
Language
English
Cited by
1.
Quantum quasigroups and loops, Journal of Algebra, 2016, 456, 46
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