Braided Commutative Algebras Over Quantized Enveloping Algebras

Abstract

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.

Constructing non-semisimple modular categories with local modules

“…Next, in Section 5, we build on previous work by the authors on the construction of nonsemisimple modular categories with relative monoidal centers [LW21a,LW21b]. In Section 5.2, we review facts about the relative monoidal center, Z B pCq, of a finite tensor category C with respect to a braided finite tensor category B.…”

“…In Section 5.2, we review facts about the relative monoidal center, Z B pCq, of a finite tensor category C with respect to a braided finite tensor category B. Such categories are useful as they naturally include representation categories of quantum groups [Lau20,LW21a], and when B is the category of k-vector spaces, Vect k , we recover the usual monoidal center ZpCq. We recall conditions from [LW21b] that imply when Z B pCq is a modular tensor category [Theorem 5.10].…”

“…In this section, we summarize the setup required for relative monoidal centers from [LW21a,LW21b]. Let B be a braided finite tensor category, with mirror B, throughout this section.…”