Let $\mathcal{C}$ be a spherical tensor category. It is known that the Drinfeld center of $\mathcal{C}$ is modular (and therefore also spherical), see for example, Corollary 8.20.14 in [1]. Recall the notion of dimension in spherical tensor categories obtained by taking the quantum trace of the identity.

Question: For a given object $Z = (X,\gamma)$ in the Drinfeld center is there a simple way to express the dimension $d(Z)$ in terms of $X$ and $\gamma$?

[1] Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor, Tensor categories, Mathematical Surveys and Monographs 205. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2024-6/hbk). xvi, 343 p. (2015). ZBL1365.18001..

  • $\begingroup$ Isn't the spherical structure you put on the Drinfeld center "equal" to that of $\mathcal{C}$? I.e. $d(Z)=d(X)$? $\endgroup$
    – JeCl
    Nov 23, 2020 at 16:49
  • $\begingroup$ You are quite correct, this can be seen by doing Exercise 7.13.6. in the reference I provided. I feel a bit silly asking just a trivial question! $\endgroup$
    – Arthur
    Nov 25, 2020 at 0:01


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