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On braided commutative algebras

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I will discuss in this talk joint work with Robert Laugwitz on a new mechanism for producing braided commutative algebras in braided monoidal categories. Namely, we construct braided commutative algebras in relative monoidal centers (in the sense of Laugwitz), which generalizes work of Davydov. Similar to how monoidal centers include representation categories of Drinfel'd doubles of Hopf algebras, an example of a relative monoidal center is a suitable category of modules over a quantized enveloping algebra. We also show that our braided commutative algebras arise as certain centralizer algebras, for a large class of braided monoidal categories, which leads to new Morita invariants for algebras in monoidal categories. Despite all of this categorical language, I aim to make this talk down-to-earth.