Braided Drinfeld and Heisenberg doubles

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...

Section: Balanced Bimodules and The Relative Monoidal Centermentioning

confidence: 99%

The relative monoidal center and tensor products of monoidal categories

2019

Commun. Contemp. Math.

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“…We need two braided Hopf algebras which are only required to be dually paired considered as braided Hopf algebra in the category of modules (rather than YDmodules). That is, the requirement that is weakened compared to the definition of a braided Drinfeld double (as in [24] or [21]) is that the comodule structures do not need to be dually paired. We refer to this generalization as the asymmetric braided Drinfeld double.…”

Section: 2]mentioning

confidence: 99%

“…This is a common feature of quantum groups and rational Cherednik algebras, but more generally shared by all braided Drinfeld or Heisenberg doubles (cf. [21,Section 3]). Here, we are using the definitions introduced in [12] to study such algebras with triangular decomposition (so-called braided doubles).…”

Section: Algebras With Triangular Decomposition (Free Case)mentioning

confidence: 99%

“…We can visualize conditions (26) and (27) using graphical calculus (with the conventions from [21]), see Fig. 1.…”

Section: Asymmetric Braided Drinfeld Doublesmentioning

confidence: 99%

“…With these structures, the relation (36) becomes Note that a partial converse statement also holds: Given an asymmetric braided Drinfeld double that is symmetric, then it can be displayed as a braided Drinfeld double in the sense of [21,24], but unless H is quasitriangular (and the coactions induced by the R-matrix), we need to view it over the base Hopf algebra Drin(H ) instead. If the positive and negative part are perfectly paired, then we can give a formal power series describing the R-matrix and an appropriate subcategory (corresponding to the Drinfeld center) is braided.…”

Section: Symmetric Triangular Decompositionsmentioning

confidence: 99%

See 2 more Smart Citations

Pointed Hopf Algebras with Triangular Decomposition

2016

Algebr Represent Theor

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In this paper, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable type over a group, we obtain a class of pointed Hopf algebras which can be viewed as natural generalizations of multiparameter deformations of universal enveloping algebras of Lie algebras. These Hopf algebras are instances of a new version of braided Drinfeld doubles, which we call asymmetric braided Drinfeld doubles. This is a generalization of an earlier result by Benkart and Witherspoon (Algebr. Represent. Theory 7(3) 2014) who showed that two-parameter quantum groups are Drinfeld doubles. It is possible to recover a Lie algebra from these doubles in the case where the group is free abelian and the parameters are generic. The Lie algebras arising are generated by Lie subalgebras isomorphic to sl 2 .

Braided Commutative Algebras Over Quantized Enveloping Algebras

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.