These lecture notes were written for a short course to be delivered in March 2017 at the Atlantic Algebra Centre of the Memorial University of Newfoundland, Canada. Folklore says that (Hopf) bialgebras are distinguished algebras whose representation category admits a (closed) monoidal structure. Here we discuss generalizations of (Hopf) bialgebras based on this principle. • The first lecture is...

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 ten...

We reduce certain proofs in [16, 11, 12] to depth two quasibases from one side only, a minimalistic approach which leads to a characterization of Galois extensions for finite projective bialgebroids without the Frobenius extension property. We prove that a proper algebra extension is a left T -Galois extension for some right finite projective left bialgebroid over some algebra R if and only if ...

We study indecomposable codes over the well-known family of Radford Hopf algebras. We use properties of Hopf algebras to show that tensors of ideal codes are ideal codes, extending the corresponding result given in [4] and showing that in this case, semisimplicity is lost.

Let L|K be a Galois extension of fields with finite Galois group G. Greither and Pareigis [GP87] showed that there is a bijection between Hopf Galois structures on L|K and regular subgroups of Perm(G) normalized by G, and Byott [By96] translated the problem into that of finding equivalence classes of embeddings of G in the holomorph of groups N of the same cardinality as G. In [CCo06] we showed...

William M. Singer’s theory of extensions of connected Hopf algebras is used to give a complete list of the cocommutative connected Hopf algebras over a field of positive characteristic p which have vector space dimension less than or equal to p3. The theory shows that there are exactly two noncommutative non-primitively generated Hopf algebras on the list, one of which is the Hopf algebra corre...

We give a Decomposition Theorem for a family of Hopf algebras containing the well-know family of Taft Hopf algebras. Therefore, those indecomposable codes over this family of algebras (cf. [4]) is an indecomposable code over the studied case. We use properties of Hopf algebras to show that dual (in the module sense) of an ideal code is again an ideal code.