Timeline for Nice proofs of the Poincaré–Birkhoff–Witt theorem
Current License: CC BY-SA 4.0
8 events
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| Jan 29, 2021 at 13:21 | comment | added | Andy | Thanks for your answer @David, I love the fact you guys tried to force hopf algebra on everything (being new to them I think I'm in the same position). I shall look at group schemes to see the analogy. I wonder if there is a topological application; something like a map between $H$ spaces that is injective on $H^1$ is injective on the cohomology ring. | |
| Jan 18, 2021 at 22:24 | comment | added | David E Speyer | Moreover, PBW basically says that the associated graded ring of $U(g)$, with respect to a certain ascending filtration, is a polynomial ring. Being regular says that the associated graded with respect to a descending filtration (powers of the maximal ideal) is a polynomial ring. So maybe there is a framework that covers both of these? | |
| Jan 18, 2021 at 22:22 | comment | added | David E Speyer | For me now, I wonder if there is some similarity between this proof and the proof that a group scheme (in characteristic zero) must be regular. In both cases, we have a Hopf algebra (the universal enveloping algebra, and the coordinate ring of the group scheme), and we deduce a property of it as an algebra simply by knowing that it has a coproduct. | |
| Jan 18, 2021 at 22:20 | comment | added | David E Speyer | @Andy For Noah, I suspect it came from just thinking "I wonder if the coproduct would help." One thing that Noah, and a bunch of us, in grad school were very into was figuring situations where the language of Hopf algebras simplified things. | |
| Jan 17, 2021 at 3:47 | comment | added | Andy | Do you have intuition to where this proof comes from and how to invent it? | |
| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Mar 6, 2019 at 17:57 | history | edited | David E Speyer | CC BY-SA 4.0 |
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| Feb 5, 2016 at 14:02 | history | answered | David E Speyer | CC BY-SA 3.0 |