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14:22
Hi, rschwieb! How are you doing?
Did you see my suggestion about the Weyl algebra at DaRT?
Hello Jose! Yes!
i check it at least every few days
Hi there!
and I did notice your suggestion. Thank you for contributing!
Ok! I saw it was still pending, and wondered...
Usually I sit on it a bit to decide whether to 1) trust the user completely 2) work it out completely for myself or 3) decide I don't trust it
I'm leaning toward 1) for you
Yeah, unfortunately with my workload it could be a day or two before suggestion becomes reality :)
(counting from when you drop it in the suggestion box)
14:33
Haha, thank you! But in this case it is simple enough to check for yourself, I believe!
Have you found the site useful?
(I think I made the suggestion like one month ago or so)
really??! (checking)
Yes, of course! But ideally, I'd like many more rings to be there!
I must have been absent longer than I thought. I have a newborn, you see
14:34
Perhaps two weeks? I don't know, time goes weird around me :P
Oh, congratulations! Is all going well with him/her?
Yes, I'm always looking for rings, at least ones that are special and not "pretty much the same as something in there already"
yeah, he's completely healthy and not very fussy
can't ask for more really
I think you don't have Weyl algebras in $2n$ variables, but don't know if they count. Besides, in positive characteristic there are differing definitions of Weyl algebras
And there are tons of little gaps: so many it's hard to scan for them
I'm glad for you and your son! I thought you were "older", taking into account the level of your contributions
pfft, nah. I graduated 2011
and now I'm in industry
So doing math now for me is purely for fun
14:38
I'll be on guard and try to contribute to DaRT. I am myself bad for remembering properties...
@JoseBrox Well, I look forward to that... I only hope i can keep up
I see! I graduated in 2008 but did not doctorate until 2015 (previously I studied telecommunications). I'm a postdoc these days
I'm searching for a reference to Kaplansky's PI algebra theorem... do you have one handy? I see allusions to it, but not the specific paper so far
Yes, give me a second
Neat! Well it's nice to know you better. I'm happy to talk about rings and modules of course, but I'm also interested in geometry and mathematical physics, and also Python programming, which I do professionally now.
14:45
Hi again! Sorry, my wifi router has to be reset from time to time
Oh, pfft, it is just this books.google.com/…
i was fooled at first because I was looking for "central simple"
The original paper of Kaplansky is Rings with a polynomial identity (1948). The theorem is in all serious books about PI rings. For example, in Rowen's Polynomial identities in ring theory it is Theorem 1.5.16
but "a PI algebra has to be fd over its center" is pretty clear cut
Yes; in fact, that was the main idea behind the original development of PI theory by Kaplansky
He noticed that Hall's identity in division rings forced them to be fd algebras, and asked "what more can be said"?
I'm experiencing internet problems, do you copy me?
Yeop
got those messages anyway
Kaplansky was a very interesting person. One of those people who you discover thought about a lot of things that occurred to you (and a bunch more things you didn't)
14:52
Sure, he was strong and creative! Also Amitsur, that I feel is not well enough. He may be even better than Kaplansky!
:49500024
yes, him too
@rschwieb Nice to know you better too! If you are into Python, do you use Sagemath much?
By the way, do you think I should specify characteristic 0 for that algebra? We're talking about ringtheory.herokuapp.com/rings/ring/66 right
I'm thinking about it. The PI part would be no problem because Kaplansky's theorem goes well for all rings and for all PIs (not just multilinear)
The problem is with the alternative definitions (some are simple in char 0 and some are not, for example)
Let me check my notes
Sorry: all definitions agree in char 0, but do not in char p. In char p, some are simple and some are not
well!
I think the database needs a nonsimple positive characteristic weyl algebra
@JoseBrox I have only tinkered around a little with Sage. A while ago I was using it to search for small examples to a ring-theoretic question from my dissertation
@JoseBrox I'd probably be happy for excuses to use it more
15:08
You see, if char(K)=p, then [y,x^p]=px^(p-1)=0, so x^p commutes with y, but also with x, hence with the whole algebra A. Then the ideal generated by x^p is just x^pA, and x is not in there due to degree, that can be defined in this context also
So A is not simple
OK: I'll try to get it entered sometime soon, and I'll see what difficulties I encounter sorting out its properties.
I don't think it's implicitly there in another ring, but I'll keep my eyes open
It's always a domain?
I don't think you will have them disguised. Weyl algebras are differential polynomial rings, but I'm not sure if they are still in char p or not. There is yet another definition, but I don't recall it now
Domains yes, because you have the degree
"They are differential polynomial rings, but i'm not sure if they are in char p" . Hm, this treats "differential polynomial ring" as a property but I was thinking about it as a construction. What am I missing?
@Zacky Hi.
I mean that Weyl algebras as you have them defined (well, you get x1,...,xn and y1,...,yn where yi is the derivative of xi) are isomorphic to differential polynomial rings (with the usual derivative as the twisting derivation), but I'm not sure if this is true anymore in char p. In a quick look, they seem to be the same, but...
hi @JoseBrox
15:22
Hi, maths student!
need some help if you are free
Sorry, this room is for other subjects! I'm not really free now. Can you write me an email?
@rschwieb On the other hand, I don't understand why in the DaRT definition for the Weyl algebra you have K to be countably infinite
I think I would maintain the same page, and distinguish the properties by the characteristic, as in Simple: yes (char 0) / no (char p)
I'll leave for now. My connection problems are really killing me... but you can write anything in here and I will read it later, or you can write to me at [email protected] if you prefer
Kindest regards! :)
15:44
Hi @rschwieb, sorry I have no knowledge about Ring Theory. The name of the room looked cool though that is why I checked in.
@JoseBrox I can explain that. In order to be able to list its cardinality, I made the field countable. Typically I make them as small as possible, unless something larger is needed. I remember specifically one thing that required an uncountable field in its construction to differentiate it nontrivially from the same construction with a countable field.
@JoseBrox Structurally that would be difficult :) especially if it turns out they have more nontrivial differences
@Zacky Aha, thanks for indirectly complimenting my site's name. I wanted to name it "All Rings Considered" but it turns out that's already taken (apparently it's a pro-wrestling blogger)
@JoseBrox I think I see what you're getting at: I might have confused two things in my description. On one hand I called them differential operators, but in the description it's that quotient. Would it be most consistent to just call it the first Weyl Algebra and stick to the quotient?
 
1 hour later…
16:58
@JoseBrox can I create seperate chatroom if you have time

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