The Drinfel’d double versus the Heisenberg double for Hom-Hopf algebras

Lu, Daowei; Wang, Shuanhong

doi:10.1142/s0219498816500596

Cited by 12 publications

(17 citation statements)

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“…A notable example is that the braided Heisenberg double is a comodule algebra over the braided Drinfeld double Drin H pC, Bq (Example 3.12). This case recovers a result of [Lau15] generalizing [Lu94].…”

Section: Introductionsupporting

confidence: 84%

“…In this case, Theorem 2.3 recovers the result of [Lau15] (proved there for quasi-Hopf algebras) that the tensor product V b W of a Yetter-Drinfeld module V with a Hopf module W is again a Hopf module over B. We will remark in Examples 3.13 and 4.19 how this tensor product generalizes the main result of [Lu94].…”

Section: A Tensor Product Action By Yetter-drinfeld Modulessupporting

confidence: 72%

“…In particular, the regular comodule algebra gives an algebra D q pgq containing the quantum Weyl algebra of [Jos95] . This is a generalization of an earlier result of [Lu94]. An example is that the q-Weyl algebra D q pgq is a 2-cocycle twist of U q pgq.…”

Section: Introductionsupporting

confidence: 67%

“…Example 2.11). In the special case where A " B with the regular coaction, the result gives an action of B B YD on Hopf modules over B (Example 2.10) similar to [Lu94,Lau15]. If B " H is a commutative Hopf algebra, using A " H with the adjoint coaction, the functor Ź corresponds, under equivalence, to the tensor product of H H YD (see Example 2.15).…”

Section: Introductionmentioning

confidence: 99%

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Comodule algebras and 2-cocycles over the (Braided) Drinfeld double

Laugwitz

2019

Commun. Contemp. Math.

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We show that for dually paired bialgebras, every comodule algebra over one of the paired bialgebras gives a comodule algebra over their Drinfeld double via a crossed product construction. These constructions generalize to working with bialgebra objects in a braided monoidal category of modules over a quasitriangular Hopf algebra. Hence two ways to provide comodule algebras over the braided Drinfeld double (the double bosonization) are provided. Furthermore, a map of second Hopf algebra cohomology spaces is constructed. It takes a pair of 2-cocycles over dually paired Hopf algebras and produces a 2-cocycle over their Drinfeld double. This construction also has an analogue for braided Drinfeld doubles.

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Section: Introductionsupporting

confidence: 84%

Section: A Tensor Product Action By Yetter-drinfeld Modulessupporting

confidence: 72%

Section: Introductionsupporting

confidence: 67%

Section: Introductionmentioning

confidence: 99%

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Comodule algebras and 2-cocycles over the (Braided) Drinfeld double

Laugwitz

2019

Commun. Contemp. Math.

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“…The result of a categorical action of the monoidal (or Drinfeld center) on the Hopf center is hence a special case of Theorem 4.1. Such a result was first proved in [Lu94] in the case of finite-dimensional Hopf algebras over a field k.…”

Section: 1mentioning

confidence: 88%

The relative monoidal center and tensor products of monoidal categories

Laugwitz

2019

Commun. Contemp. Math.

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This paper develops a theory of monoidal categories relative to a braided monoidal category, called augmented monoidal categories. For such categories, balanced bimodules are defined using the formalism of balanced functors. The two main constructions are a relative tensor product of monoidal categories as well as a relative version of the monoidal center, which are Morita dual constructions. A general existence statement for a relative tensor products is derived from the existence of pseudo-colimits.In examples, a category of locally finite weight modules over a quantized enveloping algebra is equivalent to the relative monoidal center of modules over its Borel part. A similar result holds for small quantum groups, without restricting to locally finite weight modules. More generally, for modules over braided bialgebras, the relative center is shown to be equivalent to the category of braided Yetter-Drinfeld modules (or crossed modules). This category corresponds to modules over the braided Drinfeld double (or double bosonization) which are locally finite for the action of the dual.1.3. Summary of Results. In the spirit of moving from the representation theory of algebras to modules of categories (in this case, monoidal categories), this paper studies the representation theory of the relative monoidal center in the framework of [EGNO15]. After establishing the categorical setup and recalling generalities on categorical modules (Sections 2.1 & 3.1), the relative tensor product of categorical bimodules is reviewed in the generality of finitely cocomplete k-linear categories (Section 3.2 & Appendix A).The concept of the relative monoidal center from [Lau15] is refined using the language of Bbalanced functors to allow a better study of its categorical modules. For this, B-augmented monoidal categories are introduced in Section 3.3. These can be thought of as categorical analogues of Caugmented C-algebras R over a commutative k-algebra C. Over a B-augmented monoidal category, we can study B-balanced bimodules (Section 3.4). This construction is a categorical analogue of R b C R op -modules, over a C-algebra R. In other words, R-bimodules for which the left and right C-action coincide. Based on this concept, we present two constructions for a B-augmented monoidal category C:(1) the relative monoidal center Z B pCq, which is defined as Z B pCq " Hom B C-C pC, Cq, i.e. the category of endofunctors of B-balanced bimodule functors of the regular C-bimodule (Section 3.5);(2) the relative tensor product C B C op (Theorem 3.21) which is a monoidal category, generalizing the tensor product of categories of Kelly [Kel05]. The monoidal category C B C op has the universal property that its categorical modules give all B-balanced bimodules over C (Theorem 3.27). We show that there is a natural construction to turn a C-bimodule into a B-balanced C-bimodule.Theorem 3.42. Restriction along the monoidal functor C C op Ñ C B C op has a left 2-adjointThis construction is a categorical analogue of restricting a R-R-bimodule to the subspa...

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