Entropic Hopf algebras

The concept of a Hopf algebra originated in topology. Classically, Hopf algebras are defined on the basis of unital modules over commutative, unital rings. The intention of the present work is to study Hopf algebra formalism (§1.2) from a universal-algebraic point of view, within the context of entropic varieties. In an entropic variety, the operations of each algebra are homomorphisms, and tensor products are well-behaved (§1.1, compare [1]). Entropic varieties include the classes of (pointed) sets, barycentric algebras [6, 7, 8], semilattices, and commutative monoids, as well as the classical example of modules over commutative rings. Hopf algebras over barycentric algebras and semilattices incorporate both coalgebraic and ordered structures. They offer an algebraic framework for the study of symmetry in these contexts. In any entropic variety, the concept of a setlike (or grouplike) element may be defined, and group (Hopf) algebras constructed (§2). In particular, group Hopf algebras within the variety of barycentric algebras consist precisely of finitely supported probability distributions on groups. For primitive elements and group quantum doubles, the natural universal-algebraic classes are entropic Jónsson-Tarski varieties (such as semilattices or commutative monoids) (§3). There, the free monoid or tensor algebra on any algebra is a bi-algebra, and the set of primitive elements of a Hopf algebra forms an abelian group. In general, we use the algebraic notations and conventions of [10]. For classical Hopf algebras (or “quantum groups”), one may refer to [5, 11].