scholarly journals The structure of corings with a grouplike element

2003 ◽  
Author(s):  
Tomasz Brzeziński
Keyword(s):  
Grouplike Element

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

2014 ◽  
Vol 13 (08) ◽  
pp. 1450060
Author(s):  
T. Guédénon
Keyword(s):  
Vector Space ◽  
Endomorphism Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Module Algebra ◽  
Finitely Generated ◽  
Enveloping Algebra ◽  
Grouplike Element ◽  
Necessary And Sufficient ◽  
Dual Ring

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.

scholarly journals Entropic Hopf algebras

10.29007/39rd ◽  
2018 ◽  
Author(s):  
Anna Romanowska ◽  
Jonathan Smith
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Probability Distributions ◽  
Commutative Rings ◽  
Group Algebras ◽  
Tensor Algebra ◽  
Primitive Elements ◽  
Commutative Monoids ◽  
Algebraic Setting ◽  
Grouplike Element

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.

scholarly journals FULLY BOUNDED NOETHERIAN RINGS AND FROBENIUS EXTENSIONS

2007 ◽  
Vol 06 (02) ◽  
pp. 189-206 ◽  
Author(s):  
S. CAENEPEEL ◽  
T. GUÉDÉNON
Keyword(s):  
Hopf Algebras ◽  
Noetherian Rings ◽  
Linear Map ◽  
Trace Map ◽  
Frobenius Algebras ◽  
Grouplike Element ◽  
Special Cases ◽  
Del Rio

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.

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