I'm currently working on understanding the Drinfeld centre construction. My professor gave me the exercises to understand an example:
Take $G$ a group and $Vec_G$ the category of $G$-graded vector spaces. My professor gave my the clue that the centre looks kind of like this $$ Z(Vec_G)=\{\text{G-graded vector space + representation of G, s.th. }\rho(h)(V_g) \subset V_{hgh^{-1}} \} $$
I know that the elements in the Drinfeld centre look like $(V, c_{-,V})$ with maps $c_{X, V}: X \otimes V \rightarrow V \otimes X$ for each $X$ in the given category, such that a few conditions are fulfilled. The first thing I want to try, is to find such maps with help of his clue.
My tensor product in $Vec_G$ is given on simple elements by $V_g \otimes V_h = V_{gh}$. Lets look at $(V_h, c_{-, V_h})$, my idea was to use $\rho$ intuitively like this: $$c_{V_g, V_h}:V_g \otimes V_h \rightarrow \rho_h(V_{gh}) \subset V_{hghh^{-1}}=V_hg= V_h \otimes V_g$$
Does this make any sense at all? The notion is probably pretty bad, but I'm honestly very lost and just trying to find a starting point.
Furthermore, with this method I'll only get that the set my professor gave me is subset of the centre. Any tipps on how I proof that the center is all of the above?
Thanks in advance!
First, let's clarify some notation. Generally, one reserves $V_g$ for the $g$-component of the $G$-graded vector space $V$. That is, $V\in \text{Vec}_G$ decomposes canonically as
$$V=\bigoplus_{g\in G}V_g.$$
Now, given $V,W\in\text{Vec}_G$ we do indeed define $V\otimes W$ on $g$-components as
$$(V\otimes W)_g=\bigoplus_{h\cdot k=g}V_h\otimes W_k.$$
Now, we want to study $Z(\text{Vec}_G)$. Suppose we are given a representation $\rho:G\to \text{Aut}(V)$ of $G$ on a $G$-graded vector space, such that $\rho_h(V_g)\subseteq V_{h^{-1}gh}$. Then, we can define a half-braiding $c_{V,-}$ as follows:
$$c_{V,W}:(V\otimes W)_g=\bigoplus_{h\cdot k=g}V_h\otimes W_k\to \bigoplus_{h\cdot k =g}W_k\otimes V_h\to \bigoplus_{h\cdot k=g}W_k\otimes \rho_k(V_g)\subseteq \bigoplus_{h\cdot k=g}W_k\otimes V_{k^{-1}gk}=(W\otimes V)_g.$$
The key point is that you aren't allowed to simply switch $V_h$ and $W_k$, since $h\cdot k=g$ does not imply $k\cdot h=g$. The if and only if statemet is that $h\cdot k=g$ implies $k\cdot (k^{-1}h k)=g$. Hence, we have to have a coherent way of sending $V_h$ to $V_{k^{-1}hk}$. That is, we need a nice representation $\rho$.
We now check the converse, that every half braiding must come from something of this form. I'll assume the ground field is $\mathbb{C}$ out of an abundance of caution, but this should work more generally. Applying the half braiding to get a map $V\otimes \mathbb{C}_h\to \mathbb{C}_h\otimes V$. Seeing as there is a canonical isomorphism $V\otimes \mathbb{C}\cong \mathbb{C}\otimes V\cong V$, we deduce from the definition of tensor product a map
$$V\otimes \mathbb{C}_h=\bigoplus_{g\in G}V_{gh^{-1}}\to \bigoplus_{g\in G}V_{h^-1 g}=\mathbb{C}_h\otimes V.$$
Since this map is graded, we get morphisms $V_{gh^{-1}}\to V_{h^{-1}g}$ for every $h,g$. Collecting and using the right naturality conditions, we conclude that this does indeed stitch together a representation $G\to \text{Aut}(V)$, as desired. It is then clear that going back and forth between these processes gives a bijection between our two sets, and moreover will carry through whatever algebraic structures you desire.
I don't have a reference for this stuff (I would love it if someone else did), and this is mostly just from memory. What I do have is detailed notes about topological quantum computing, in which there is a section on Drinfeld centers of $G$-graded vector spaces. You can find it on my GitHub page here: https://github.com/Milo-Moses/Toric-Code-Introduction. This one of the propositions at the end of what is currently labeled as section 6.