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

Laugwitz, Robert

doi:10.1142/s0219199718500451

Cited by 5 publications

(13 citation statements)

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“…In addition to Davydov's work [11], [12] and the first author's work on comodule algebras over braided Drinfeld doubles [29], our results have connections to several other articles in the literature. See several works on braided commutative algebras in Yetter-Drinfeld categories, including [6,9,10].…”

Section: Introductionsupporting

confidence: 63%

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Braided Commutative Algebras Over Quantized Enveloping Algebras

Laugwitz

Walton

2020

Transformation Groups

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We produce braided commutative algebras in braided monoidal categories by generalizing Davydov’s full center construction of commutative algebras in centers of monoidal categories. Namely, we build braided commutative algebras in relative monoidal centers $$ {\mathcal{Z}}_{\mathrm{\mathcal{B}}}\left(\mathcal{C}\right) $$ Z ℬ C from algebras in ℬ-central monoidal categories $$ \mathcal{C} $$ C , where ℬ is an arbitrary braided monoidal category; Davydov’s (and previous works of others) take place in the special case when ℬ is the category of vector spaces $$ {\mathbf{Vect}}_{\mathbbm{K}} $$ Vect K over a field $$ \mathbbm{K} $$ K . Since key examples of relative monoidal centers are suitable representation categories of quantized enveloping algebras, we supply braided commutative module algebras over such quantum groups. One application of our work is that we produce Morita invariants for algebras in ℬ-central monoidal categories. Moreover, for a large class of ℬ-central monoidal categories, our braided commutative algebras arise as a braided version of centralizer algebras. This generalizes the fact that centers of algebras in $$ {\mathbf{Vect}}_{\mathbbm{K}} $$ Vect K serve as Morita invariants. Many examples are provided throughout.

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Section: Introductionsupporting

confidence: 63%

“…We use the graphical calculus as in [29], similar to that used in [34], for computations in (braided) monoidal categories D. The braiding in D is denoted by Ψ = .…”

Section: Notation and Conventionsmentioning

confidence: 99%

Braided Commutative Algebras Over Quantized Enveloping Algebras

Laugwitz

Walton

2020

Transformation Groups

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“…Further, the results of this paper relate to the constructions of comodule algebras over the braided Drinfeld double in [Lau18]. Let A be a left B-comodule algebra in B " H-Mod.…”

Section: 2mentioning

confidence: 94%

“…Graphical calculus, as in [Lau18], which is inspired by that used in [Maj94], is helpful for computations in B-ModpBq. For example, the tensor product of modules is depictured as…”

Section: Augmented Monoidal Categoriesmentioning

confidence: 99%

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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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An Invitation to Noncommutative Algebra

Walton

2019

Association for Women in Mathematics Series

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

Cited by 5 publications

References 39 publications

Braided Commutative Algebras Over Quantized Enveloping Algebras

Braided Commutative Algebras Over Quantized Enveloping Algebras

The relative monoidal center and tensor products of monoidal categories

An Invitation to Noncommutative Algebra

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