Hopf measuring comonoids and enrichment

Hyland, Martin; Franco, Ignacio López; Vasilakopoulou, Christina

doi:10.1112/plms.12064

Cited by 21 publications

(29 citation statements)

References 32 publications

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“…This universal object may be viewed as a generalization of Sweedler's finite dual B • (taking A = k). In the recent literature, one finds other generalizations [7,13], the former of which shares some overlap with our work (see the remark following Proposition 2.2). In Theorem 2.1, we adapt Sweedler's proof to construct universal partial coverings.…”

Section: 2supporting

confidence: 80%

“…Acknowledgements. We would like to thank the referee for a number of useful comments, in particular for pointing us to the references [7], [13] and for explaining a more categorical way of approaching some of the constructions discussed in the paper: see the discussion at the end of Section 1, the paragraph preceding Theorem 2.1, the remark following Proposition 2.2, and the comment preceding Question 2.…”

Section: Introductionmentioning

confidence: 99%

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Bialgebra coverings and transfer of structure

Lauve¹,

Mastnak²

2019

Tensor Categories and Hopf Algebras

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We introduce the bicategory of bialgebras with coverings (which can be thought of as coalgebra-indexed families of morphisms), and provide a motivating application to the transfer of formulas for primitives and antipode. Additionally, we study properties of this bicategory and various sub-bicategories, and describe some universal constructions. Finally, we generalize Nichols' result on bialgebra quotients of Hopf algebra, which gives conditions on when the resulting bialgebra quotient is a Hopf algebra.2010 Mathematics Subject Classification. 18D05, 16T05, 05E05, 16T30.

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Section: 2supporting

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Bialgebra coverings and transfer of structure

Lauve¹,

Mastnak²

2019

Tensor Categories and Hopf Algebras

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“…Since the categories Alg R and Ring R are locally finitely presentable (trivially), while the categories Coalg R and Coring R are locally presentable (see [15], [14]) (in fact locally countably presentable by [24]), we deduce: 10 Remark Note that, when C is symmetric, Proposition 8 above is a special instance of the more general result, that in this case all so-called universal measuring comonoids exist and, thus, MonC is enriched over ComonC (see [5], [9]).…”

Section: Semi-dualization In Monoidal Closed Categoriesmentioning

confidence: 99%

Generalizations of the Sweedler Dual

Porst

Street

2016

Appl Categor Struct

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As left adjoint to the dual algebra functor, Sweedler's finite dual construction is an important tool in the theory of Hopf algebras over a field. We show in this note that the left adjoint to the dual algebra functor, which exists over arbitrary rings, shares a number of properties with the finite dual. Nonetheless the requirement that it should map Hopf algebras to Hopf algebras needs the extra assumption that this left adjoint should map an algebra into its linear dual. We identify a condition guaranteeing that Sweedler's construction works when generalized to noetherian commutative rings. We establish the following two apparently previously unnoticed dual adjunctions: For every commutative ring R the left adjoint of the dual algebra functor on the category of R-bialgebras has a right adjoint. This dual adjunction can be restricted to a dual adjunction on the category of Hopf R-algebras, provided that R is noetherian and absolutely flat.

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“…Another way to motivate this is by viewing measuring coalgebras P (A, B) as a quantization [5] of the set of maps between two k-algebras A and B. This provides an enrichment of the category Alg of k-algebras over the category Coalg; see [80,39]. If k is algebraically closed and A is a commutative affine algebra, then the Nullstellensatz yields a bijection…”

Section: The Finite Dual As a Quantized Maximal Spectrummentioning

confidence: 99%

The finite dual coalgebra as a quantization of the maximal spectrum

Reyes¹

2021

Preprint

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In pursuit of a noncommutative spectrum functor, we argue that the Heyneman-Sweedler finite dual coalgebra can be viewed as a quantization of the maximal spectrum of a commutative affine algebra, integrating prior perspectives of Takeuchi, Batchelor, Kontsevich-Soibelman, and Le Bruyn. We introduce fully residually finite-dimensional algebras A as those with enough finite-dimensional representations to let A • act as an appropriate depiction of the noncommutative maximal spectrum of A; importantly, this class includes affine noetherian PI algebras. We investigate cases where the finite dual coalgebra of a twisted tensor product is a crossed product coalgebra of the respective finite duals. This is achieved by interpreting the finite dual as a topological dual. Sufficient conditions for this result to be applied to Ore extensions, smash product algebras, and crossed product bialgebras are described. In the case of prime affine algebras that are module-finite over their center, we describe how the Azumaya locus is represented in the finite dual. Finally, we implement these techniques for quantum planes at roots of unity as an endeavor to visualize the noncommutative space on which these algebras act as functions.

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Hopf measuring comonoids and enrichment

Cited by 21 publications

References 32 publications

Bialgebra coverings and transfer of structure

Bialgebra coverings and transfer of structure

Generalizations of the Sweedler Dual

The finite dual coalgebra as a quantization of the maximal spectrum

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