Equivalences of (co)module algebra structures over Hopf algebras

Abstract

We introduce the notion of support equivalence for (co)module algebras (over Hopf algebras), which generalizes in a natural way (weak) equivalence of gradings. We show that for each equivalence class of (co)module algebra structures on a given algebra A, there exists a unique universal Hopf algebra H together with an H-(co)module structure on A such that any other equivalent (co)module algebra structure on A factors through the action of H. We study support equivalence and the universal Hopf algebras mentioned above for group gradings, Hopf-Galois extensions, actions of algebraic groups and cocommutative Hopf algebras. We show how the notion of support equivalence can be used to reduce the classification problem of Hopf algebra (co)actions. In particular, we apply support equivalence to study the asymptotic behaviour of codimensions of H-identities of a certain class of H-module algebras. This result proves the analogue (formulated by Yu. A. Bahturin) of Amitsur's conjecture which was originally concerned with ordinary polynomial identities.

V-universal Hopf algebras (co)acting on Ω-algebras

“…(The limit lim T is still taken in C.) In the case C = Vect , where is a field, supp ρ corresponds to taking the intersection of all such subcoalgebras that the map can be factored through them. Therefore supp ρ coincides with the support defined in[3]. (See also Examples 4.10 (2).)…”

“…However, the universal group of a grading allows us to recover all groups that realize a concrete grading. The corresponding notions of equivalence and universal Hopf algebras of (co)module structures on algebras were introduced in [3], generalizing the aforementioned universal group of a grading. Furthermore, a unifying theory for universal Hopf algebras of (co)module structures and universal (co)acting bi/Hopf algebras of Sweedler -Manin -Tambara ([21, 17, 22]) was introduced in [4], by considering V -universal (co)acting bi/Hopf algebras where V is a unital subalgebra of End (A) and is the base field.…”

“…The classification up to equivalence may seem coarser, however the notion of the universal group of the grading makes it possible to recover all groups that realize a concrete grading. Inspired by this, the notions of equivalence and universal Hopf algebras of (co)module structures on algebras were introduced in [3] as a natural generalization of the aforementioned universal group of a grading. This construction was unified in [4] with the universal (co)acting bi/Hopf algebras of Sweedler -Manin -Tambara, introducing the V -universal (co)acting bi/Hopf algebras for a given algebra A, where V is a unital subalgebra of End (A) and is the base field.…”

Functors between representation categories. Universal modules

“…a Poisson Hopf algebra structure as in [2]), seem to play a prominent role in solving other seemingly unrelated problems such as the classification of gradings on various kinds of algebras ( [4,12]), the description of the automorphisms group of certain algebraic structures ( [4]) and even in quantum Galois theory ([6]). Another related universal (co)acting construction was considered in [3] as the Hopf algebraic analogue of the universal group of a grading and its connections to the problem of classifying Hopf algebra coactions have been highlighted.…”