Entropic Hopf algebras

Classically, Hopf algebras are defined on the basis of modules over commutative rings. The present study seeks to extend the Hopf algebra formalism to a more general universal-algebraic setting, entropic varieties, including (pointed) sets, barycentric algebras, semilattices, and commutative monoids. The concept of a setlike (or grouplike) element may be defined, and group algebras constructed, in any such variety. In particular, group 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). There, the tensor algebra on any algebra is a bialgebra, and the set of primitive elements of a Hopf algebra forms an abelian group. Coalgebra congruences on comonoids in entropic varieties are also studied.

Projectivity and flatness over the endomorphism ring of a finitely generated quasi-Poisson comodule

Let k be a field of characteristic 0, A a noncommutative Poisson k-algebra, U(A) the ordinary enveloping algebra of A, 𝒞 a quasi-Poisson A-coring that is projective as a left A-module, *𝒞 the left dual ring of 𝒞 (it is a right U(A)-module algebra) and Λ a right quasi-Poisson 𝒞-comodule that is finitely generated as a right U(A)#*𝒞-module. The vector space End 𝒫,𝒞(Λ) of right quasi-Poisson 𝒞-colinear maps from Λ to Λ is a ring. We give necessary and sufficient conditions for projectivity and flatness of a module over End 𝒫,𝒞(Λ). If 𝒞 contains a fixed quasi-Poisson grouplike element, we can replace Λ with A.

FULLY BOUNDED NOETHERIAN RINGS AND FROBENIUS EXTENSIONS

Let i: A → R be a ring morphism, and χ: R → A a right R-linear map with χ(χ(r)s) = χ(rs) and χ(1R) = 1A. If R is a Frobenius A-ring, then we can define a trace map tr: A → AR. If there exists an element of trace 1 in A, then A is right FBN if and only if AR is right FBN and A is right noetherian. The result can be generalized to the case where R is an I-Frobenius A-ring. We recover results of García and del Río, and Dǎscǎlescu, Kelarev and Torrecillas on actions of group and Hopf algebras on FBN rings as special cases. We also obtain applications to extensions of Frobenius algebras, and to Frobenius corings with a grouplike element.