Timeline for Choice of a ground ring for cluster algebras
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Oct 8, 2019 at 8:04 | vote | accept | amator2357 | ||
| Oct 4, 2019 at 17:54 | answer | added | John Machacek | timeline score: 1 | |
| Oct 4, 2019 at 12:45 | comment | added | amator2357 | @JanGrabowski I see, thank you! | |
| Oct 4, 2019 at 12:42 | comment | added | Jan Grabowski | @amator2357 I was being a bit brief, as it was a comment. When I said "constructed inside", I meant using mutation to produce the cluster variables and hence the cluster algebra. The upper cluster algebra is given by intersecting the family of LP algebras associated to each cluster, and contains the cluster algebra (by the Laurent phenomenon). | |
| Oct 4, 2019 at 7:47 | comment | added | amator2357 | I would be interesting in both @JohnMachacek | |
| Oct 3, 2019 at 16:54 | comment | added | John Machacek | Are you interested in why there is a semifield P around? Or why given a the semifield we choose ZP as ground ring as opposed to something else involving P? Or both? | |
| Oct 3, 2019 at 12:12 | history | edited | YCor |
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| Oct 3, 2019 at 11:38 | comment | added | amator2357 | Thank you @JanGrabowski ! The construction you have outlined, isn't that how we would construct the upper cluster algebras? | |
| Oct 3, 2019 at 11:27 | history | edited | amator2357 | CC BY-SA 4.0 |
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| Oct 3, 2019 at 11:19 | comment | added | Jan Grabowski | Yes, I believe that's the main reason. I'd note that in practice few cluster algebras are left defined integrally in this way: most examples of cluster algebras, especially the geometric ones such as coordinate rings of Grassmannians, are constructed inside the field of fractions of a Laurent polynomial algebra whose indeterminates are the initial cluster variables (both mutable and frozen/coefficient ones) and where the LP coefficients are from a field. Tl;dr: most times people don't worry overly about this... (Maybe they should!) | |
| Oct 3, 2019 at 10:52 | history | asked | amator2357 | CC BY-SA 4.0 |