Hopf Algebras—Variant Notions and Reconstruction Theorems

Vercruysse, Joost; Heunen, Chris; Sadrzadeh, Mehrnoosh; Grefenstette, Edward

doi:10.1093/acprof:oso/9780199646296.003.0005

Cited by 10 publications

(8 citation statements)

References 39 publications

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“…It is known [22] that a (unital) algebra is a sesqui-unital sesqui-algebra if and only if its category of modules is closed monoidal (without assumptions on the existence of a monoidal fibre functor). In [21], a notion of antipode for sesqui-unital sesqui-algebras was proposed, which was however rather rigid.…”

Section: The Dilations Of a Partial Smash Productmentioning

confidence: 99%

Dilations of partial representations of Hopf algebras

Alves¹,

Batista

Vercruysse

2019

J. London Math. Soc.

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We introduce the notion of a dilation for a partial representation (that is, a partial module) of a Hopf algebra, which in case the partial representation origins from a partial action (that is, a partial module algebra) coincides with the enveloping action (or globalization). This construction leads to categorical equivalences between the category of partial H-modules, a category of (global) H-modules endowed with a projection satisfying a suitable commutation relation and the category of modules over a (global) smash product constructed upon H, from which we deduce the structure of a Hopfish algebra on this smash product. These equivalences are used to study the interactions between partial and global representation theory.

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Section: The Dilations Of a Partial Smash Productmentioning

confidence: 99%

Dilations of partial representations of Hopf algebras

Alves¹,

Batista

Vercruysse

2019

J. London Math. Soc.

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“…Indeed: consider a base {e i } for X, then x = a i e i , ∆(x) = a ij e i ⊗ e j . Now apply condition (19) for the dual base elements f = e * i and g = e * j , then one finds that a ij = a i + a j . The cocommutativity now follows.…”

Section: Examples 32mentioning

confidence: 99%

Lie monads and dualities

Goyvaerts¹,

Vercruysse²

2014

Journal of Algebra

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We study dualities between Lie algebras and Lie coalgebras, and their respective (co)representations. To allow a study of dualities in an infinite-dimensional setting, we introduce the notions of Lie monads and Lie comonads, as special cases of YB-Lie algebras and YB-Lie coalgebras in additive monoidal categories. We show that (strong) dualities between Lie algebras and Lie coalgebras are closely related to (iso)morphisms between associated Lie monads and Lie comonads. In the case of a duality between two Hopf algebras -in the sense of Takeuchi-we recover a duality between a Lie algebra and a Lie coalgebra -in the sense defined in this note-by computing the primitive and the indecomposables elements, respectively.

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“…This is a version of Tannaka duality for Hopf algebras (e.g. as in [Sch92,Ver12] and the many references therein): The CQG algebra constructed in [Wor88, 1.3] given a concrete monoidal W *category C is what in [Sch92] would be called the coendomorphism Hopf algebra of the functor C → Hilb that is implicitly part of Woronowicz's definition. Now, if one starts with the category of unitary comodules of a CQG algebra A and performs the above construction, the resulting CQG algebra is again A.…”

Section: Propositionmentioning

confidence: 99%

Categorical Aspects of Compact Quantum Groups

Chirvăsitu

2013

Appl Categor Struct

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We show that either of the two reasonable choices for the category of compact quantum groups is nice enough to allow for a plethora of universal constructions, all obtained "by abstract nonsense" via the adjoint functor theorem. This approach both recovers constructions which have appeared in the literature, such as the quantum Bohr compactification of a locally compact semigroup, and provides new ones, such as the coproduct of a family of compact quantum groups, and the compact quantum group freely generated by a locally compact quantum space. In addition, we characterize epimorphisms and monomorphisms in the category of compact quantum groups.

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Hopf Algebras—Variant Notions and Reconstruction Theorems

Cited by 10 publications

References 39 publications

Dilations of partial representations of Hopf algebras

Dilations of partial representations of Hopf algebras

Lie monads and dualities

Categorical Aspects of Compact Quantum Groups

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