2
$\begingroup$

$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Ind{Ind}\DeclareMathOperator\End{End}$In Ostrik - Module categories, weak Hopf algebras and modular invariants, it is proven in Theorem 2 that all indecomposable module categories over $\Rep(G)$ are equivalent to $\Rep^1(H,\omega)$ for some subgroup $H$ and $\omega\in H^2(H,k^*)$. I have a few questions about this proof:

They begin with that it is easy to see that for any $V\in\Rep^1(\tilde H)$ we have $\underline{\Hom}(V,V)=\Ind^G _H \End(V)$, but I do not see this. We need a bijection $\Hom_G(X,\Ind^G _H\End(V))\to\Hom_{\tilde H}(X\otimes V,V)$. I would at least be able to construct this map if $\Ind$ were a right-adjoint instead of a left-adjoint, but it is not.

Second, I don't understand their use of the minimal central idempotents. When $A$ is a direct sum of matrix algebras, are these the elements corresponding to the matrices $E_{ii}$? Why can I not use primitive orthogonal idempotents here? The action $G$ is also transitive on them, would they not suffice?

Third, they define $H$ to be the stabilizer of the idempotent $e$. Wouldn't this make the action of $H$ on the subalgebra $eAe$ trivial?

Lastly, I don't see how they have recovered the extension $\tilde H$, as they only define $H$ in the proof.

Improve this question
$\endgroup$
3
  • $\begingroup$ What is $\tilde H$? $\endgroup$
    – LSpice
    Commented Jun 23, 2023 at 19:53
  • $\begingroup$ @LSpice Elements $\omega\in H^2(H,k^*)$ are one-to-one with central extensions $1\to k^*\to \tilde H\to H\to1$. With $\operatorname{Rep}^1(H,\omega)=\operatorname{Rep}(\tilde H)$ we denote representations of $\tilde H$ such that $k^*$ acts by the trivial character. $\endgroup$
    – shin chan
    Commented Jun 23, 2023 at 20:33
  • $\begingroup$ This might not be as satisfying, but it is easier to show that indecomposable module categories for $\text{Vec}_G$ are classified by such pairs $(H,\omega)$. From there you can tensor with the invertible $\big(\text{Rep}(G),\text{Vec}_G\big)$-bimodule category $\text{Vec}$ (invertible=Morita equivalence) to find that all indecomposable module categories for $\text{Rep}(G)$ are classified by the same pairs. This is a more modern way of arriving at this classification, but it's certainly less explicit. $\endgroup$
    – Sean Sanford
    Commented Sep 20, 2023 at 16:05

0

Reset to default

You must log in to answer this question.