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Session S16 - Quantum symmetries

Friday, July 16, 11:35 ~ 12:00 UTC-3

The adjoint algebra for 2-categories


Universidad Nacional de San Luis, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In 2017, Kenichi Shimizu introduced the notion of the adjoint algebra $\mathcal{A}_{\mathcal{C}}$ and the space of class functions CF($\mathcal{C}$) for a finite tensor category. The adjoint algebra is defined as the end $\int_{X \in \mathcal{C}} X \otimes X^*$. In fact, $\mathcal{A}_\mathcal{C}$ is an algebra in the Drinfeld center $\mathcal{Z}(\mathcal{C})$. Both, the adjoint algebra and the space of class functions generalize the well know adjoint representation and character algebra for a finite group. Later, in 2019, the author generalized these concepts for any left $\mathcal{C}$-module category $\mathcal{M}$.

In this talk, I will present a generalization of the adjoint algebra to the realm of 2-categories. For any 0-cell B in a 2-category $\mathcal{B}$, I will define the adjoint algebra$\mathcal{Ad}_B$ wich is an algebra in the center of $\mathcal{B}$. For $\mathcal{C}$ a finite tensor category, we will see that when we apply this notion to the 2-category ${}_\mathcal{C} \mathcal{Mod}$ of $\mathcal{C}$-module categories, it coincides with the one introduced by Shimizu, and how it behaves under 2-equivalences.


K. Shimizu. "The monoidal center and the character algebra". J. Pure Appl. Algebra 221, No. 9, 2338–2371 (2017).

K. Shimizu, "Further results on the structure of (Co)ends in fintite tensor categories", Appl. Categor. Struct. (2019).

N. Bortolussi, M. Mombelli. "The adjoint algebra for 2-categories". Kyoto Journal of Mathematics, to appear. arXiv:2005.05271.

Joint work with Martín Mombelli (Universidad Nacional de Córdoba, Argentina).

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