| Title:
|
Unitary dual functors for unitary multitensor categories (English) |
| Author:
|
Penneys, David |
| Language:
|
English |
| Journal:
|
Higher Structures |
| ISSN:
|
2209-0606 |
| Volume:
|
4 |
| Issue:
|
2 |
| Year:
|
2020 |
| Pages:
|
22-56 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We classify which dual functors on a unitary multitensor category are compatible with the dagger structure in terms of groupoid homomorphisms from the universal grading groupoid to $\Bbb R_{>0}$ where the latter is considered as a groupoid with one object. We then prove that all unitary dual functors induce unitarily equivalent bi-involutive structures. As an application, we provide the unitary version of the folklore correspondence between shaded planar C$^*$ algebras with finite dimensional box spaces and unitary multitensor categories with a chosen unitary dual functor and chosen generator. We make connection with the recent work of Giorgetti-Longo to determine when the loop parameters in these planar algebras are scalars. Finally, we show that we can correct for many non-spherical choices of dual functor by adding the data of a spherical state on End$_C(1c)$, similar to the spherical state for a graph planar algebra. This is the published version of http://arxiv.org/pdf/1808.00323. (English) |
| Keyword:
|
tensor category |
| Keyword:
|
planar algebra |
| Keyword:
|
pivotal structure |
| Keyword:
|
unitary dual functor |
| MSC:
|
18D05 |
| MSC:
|
18D10 |
| MSC:
|
46L37 |
| idZBL:
|
Zbl 1457.18019 |
| idMR:
|
MR4133163 |
| DOI:
|
10.21136/HS.2020.09 |
| . |
| Date available:
|
2026-03-12T11:00:39Z |
| Last updated:
|
2026-03-12 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153425 |
| . |
| Reference:
|
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| . |
