On the Drinfeld center of the Verlinde category $\Ver_p$

Abstract

We provide some information about the Drinfeld center $\Z(\Ver_p)$ of the Verlinde category $\Ver_p$. We compute explicitly the Cartan matrix, bound quiver and cohomology of $\Z(\Ver_p^+)$, and prove that the category $\mathscr{Z}(\Ver_p^+)$ is wild for every $p\ge 7$. In the special case $p=5$, we show that $\Z(\Ver_5^+)$ has exactly $10$ non-isomorphic indecomposable objects, classify them and describe the Green ring of $\Z(\Ver_5^+)$, and compute the semisimplification of $\Z(\Ver_5^+)$.