4
$\begingroup$

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..

Improve this question
$\endgroup$
2
  • $\begingroup$ 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

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.