Timeline for subfactor of finite rank but infinite index: is this possible?
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
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| Mar 9, 2016 at 12:40 | comment | added | André Henriques | I'd like an irreducible object $x$ in a semisimple category $\mathcal C$ such that $\not \exists y\in \mathcal C$ with $1< x\otimes y$. The category should have finitely many simples, and ind-$\mathcal C$ should be a tensor category: given two simples in $\mathcal C$ their product is allowed to be an ind-object. A variant question is when one insists that $\mathcal C$ itself be a tensor category (and not just ind-$\mathcal C$). Such catrgories certainly exists, and my question is whether they can be realised inside bimodules on a vN algebra. | |
| Mar 9, 2016 at 4:11 | comment | added | Marcel Bischoff | Do you want it irreducible? Because the object $\rho_\oplus$ has no dual. But I see the problem that is does not live in a semisimple category but is rather an ind object. So semisimple means that you have to assume that the relative commutant is finite?! | |
| Mar 8, 2016 at 23:07 | comment | added | André Henriques | The example you are providing is not really what I was asking for. What I'd like is a tensor category of bimodules (or of endomorphisms of a $III_1$ factor) which is semi-simple, with finitely many isomorhpism classes of simple object, but such that there exists an object with no dual: $\exists x\in \mathcal C$ such that $\not\exists y\in\mathcal C$ with $1_{\mathcal C}< x\otimes y$. | |
| Mar 8, 2016 at 17:38 | history | answered | Marcel Bischoff | CC BY-SA 3.0 |