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Let $\mathscr{C}$ be a strict monoidal category. I will denote the product of $\mathscr{C}$ by $\otimes$. The Drinfeld center $\mathscr{Z(C)}$ of $\mathscr{C}$ is the category with object $(X,\phi)$ where $X$ is an object of $\mathscr{C}$ and $\phi$ is a natural isomorphism from $X \otimes -$ to $ - \otimes X$. Morhphisms from $(X,\phi)$ to $(Y,\psi)$ in $\mathscr{Z(C)}$ are elements all $f \in \mathrm{hom_\mathscr{C}(X,Y)}$ such that $(\mathrm{id}_W \otimes f) \circ \phi_W = \psi_W \circ (f \otimes \mathrm{id}_W)$ for all $W \in \mathscr{C}$.

My question is the following. There must be a problem in the follwoing reasoning, but I cannot find it. I wonder if anybody can point out the mistake. Fix any field $k$ of characteristic $0$. If $G$ is a finite group, then $\mathrm{Vec}_G$ the category of $G$-graded vector spaces is strict monoidal and moreover it is semisimple. Its simple objects are one dimensional vector spaces $V_g$ with grading given by an element $g \in G$. Morphisms are grading-preserving linear maps. In particular $V_g$ and $V_h$ are isomorphic if and only if $g = h$. The monoidal structure of $\mathrm{Vec}_G$ is given on simple object by multiplication of elements in $G$: $V_g \otimes V_h = V_{gh}$. Now assume that $G$ has a trivial center. Then for any $g \in G$ there is no natural isomorphism from $V_g \otimes - $ to $ - \otimes V_g$, since $V_{gh}$ and $V_{hg}$ are not isomorphic for some $h \in G$. Hence, its Drinfeld center is trivial.

Note that the conclusion cannot be true, since $\mathscr{Z}(\mathrm{Vec}_G)$ is the representation category of the Drinfeld double $k[G] \ltimes \mathrm{Fun}(G)$ of $\mathrm{Fun}(G)$, where $k[G]$ is the group ring of $G$, $\mathrm{Fun}(G)$ are the $k$-valued functions on $G$ and $\ltimes$ denotes the crossed product with respect to the natural action.

Thank you very much for your help.

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How is $\mathrm{Vec}_G$ strict monoidal? Is $\mathrm{Vec}$ strict? –  Mariano Suárez-Alvarez Mar 10 '11 at 22:26
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up vote 4 down vote accepted

Your observation only means that there is no object $(X,\phi)$ in the Drinfeld center which has $X$ simple and non-isomorphic to $V_e$ ($e$ being the identity element of $G$)

But let $C\subseteq G$ be a conjugacy class, and let $V_C=\bigoplus_{g\in C}V_g$. Then you should be able to find a natural isomorphism $\phi_C:V_C\otimes(\mathord-)\to(\mathord-)\otimes V_C$, so that $(V_C,\phi_C)$ is a non-trivial element in $\mathcal{Z}(\mathrm{Vec}_G)$.

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Thank you very much. I was a little too fixed to simple objects. –  Sven Raum Mar 10 '11 at 22:39
    
With a little work, you can surely describe all indecomposable objects in the center of that category. –  Mariano Suárez-Alvarez Mar 10 '11 at 22:40
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