Split extension classifiers in the category of cocommutative Hopf algebras

Gran, Marino; Kadjo, Gabriel; Vercruysse, Joost

doi:10.48550/arxiv.1702.05595

Cited by 2 publications

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“…Accordingly, for any Hopf K-algebra H, an element h in H is said to be a group-like element if ∆(h) = h ⊗ h for the comultiplication ∆ of H. Identifying the category of cocommutative Hopf K-algebras with the category Grp(CCoAlg K ) of internal groups in the category of cocommutative K-coalgebras (see [29] for instance), and identifying the category of Hopf K-algebras of the form K[G] (for all groups G) with the category of groups, yields now the group-like element functor Grp(CCoAlg K ) → Grp. As shown in [16] (see also [17]), when K is a field of characteristic zero, Grp(CCoAlg K ) is semi-abelian and this functor is a localization.…”

Section: Our Factorization Of This Reflection Ismentioning

confidence: 81%

Some remarks on protolocalizations and protoadditive reflections

2018

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Abstract. We investigate additional properties of protolocalizations, introduced and studied by F. Borceux, M. M. Clementino, M. Gran, and L. Sousa, and of protoadditive reflections, introduced and studied by T. Everaert and M. Gran. Among other things we show that there are no non-trivial (protolocalizations and) protoadditive reflections of the category of groups, and establish a connection between protolocalizations and Kurosh-Amitsur radicals of groups with multiple operators whose semisimple classes form subvarieties.

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Section: Our Factorization Of This Reflection Ismentioning

confidence: 81%

Some remarks on protolocalizations and protoadditive reflections

2018

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“…and there exists a unique Hopf algebra homomorphism ϕ 1 : H 1 → R(ζ(H)) such that the following diagram commutes: In the case of an algebraically closed field of characteristic 0, the Cartier-Gabriel-Kostant theorem (see e.g. [27, Corollary 5.6.4 and Theorem 5.6.5]) allows us to give a concrete description of the universal cocommutative Hopf algebra, using a similar technique as in [23].…”

Section: Module Structures On Algebrasmentioning

confidence: 99%

Equivalences of (co)module algebra structures over Hopf algebras

Agore¹,

Gordienko²,

Vercruysse³

2018

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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.

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Split extension classifiers in the category of cocommutative Hopf algebras

Cited by 2 publications

References 0 publications

Some remarks on protolocalizations and protoadditive reflections

Some remarks on protolocalizations and protoadditive reflections

Equivalences of (co)module algebra structures over Hopf algebras

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