Braided commutative algebras over quantized enveloping algebras

TL;DR

This paper generalizes the construction of braided commutative algebras in monoidal categories, producing new algebraic structures over quantum groups and establishing Morita invariants, with numerous examples.

Contribution

It extends Davydov's full center construction to relative monoidal centers over arbitrary braided categories, enabling the creation of braided commutative algebras over quantum groups.

Findings

Constructed braided commutative algebras in relative monoidal centers.

Produced Morita invariants for algebras in braided categories.

Generalized centers as braided centralizer 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 $\mathcal{Z}_{\mathcal{B}}(\mathcal{C})$ from algebras in $\mathcal{B}$-central monoidal categories $\mathcal{C}$, where $\mathcal{B}$ is an arbitrary braided monoidal category; Davydov's (and previous works of others) take place in the special case when $\mathcal{B}$ is the category of vector spaces $\mathbf{Vect}$ over a field $\Bbbk$. 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 $\mathcal{B}$-central monoidal categories. Moreover, for a large class of $\mathcal{B}$-central monoidal categories, our braided commutative algebras arise as a braided version of centralizer algebras. This generalizes the fact that centers of algebras in $\mathbf{Vect}$ serve as Morita invariants. Many examples are provided throughout.

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