Timeline for A textbook on linear algebra where involutions on linear spaces are considered
Current License: CC BY-SA 3.0
23 events
| when toggle format | what | by | license | comment | |
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| Apr 15 at 16:30 | comment | added | Nemis L. | @JoeSchindler Thanks for the note! I found this textbook, which has something: Linear Algebra and Geometry by Kostrikin and Manin. | |
| Apr 12 at 12:12 | comment | added | Joe Schindler | @NemisL. Nothing too useful, unfortunately! ... Like Sergei, I eventually wrote up some very basic notes (unpublished) that you can see here jcsphysics.net/pub/files/pos-neg-freq.pdf , though I doubt it might be too useful. Good luck :) | |
| Apr 12 at 9:50 | comment | added | Nemis L. | @JoeSchindler Have you found a good reference? I am looking for the same, also in the context of spaces with indefinite sesquilinear forms. | |
| Jul 17, 2020 at 17:56 | comment | added | Joe Schindler | Sure, here's the update: The paper you sent has very nice clear statement of the most basic properties! I was hoping to somewhere find a deeper treatment that involves the additional structure of a sesquilinear inner product on such a space (specifically a semi-definite one where all real vectors have zero norm, closely related to a symplectic space). I'm starting to think this is too specific to hope for, although such spaces arise in bosonic field theories in physics. | |
| Jul 17, 2020 at 9:31 | comment | added | Sergei Akbarov | @JoeSchindler I hope, if you'll find something, you'll share this information. | |
| Jul 14, 2020 at 4:58 | comment | added | Joe Schindler | Thanks, I will take a look at the section! Very surprising that there is no simple canonical reference for this topic on vector spaces | |
| Jul 14, 2020 at 4:52 | comment | added | Sergei Akbarov | @JoeSchindler no, I didn't! I eventually wrote a subsection in my paper where these facts are listed: arxiv.org/abs/1303.2424 I needed this for a special category, the category of stereotype spaces (and the name of the subsection is "Involution on stereotype spaces"), but one can consider other categories. | |
| Jul 14, 2020 at 4:39 | comment | added | Joe Schindler | @SergeiAkbarov Did you ever find a nice reference? I am looking for the same.. | |
| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Jul 9, 2013 at 15:25 | comment | added | Sergei Akbarov | Mariano, maybe you are right. Perhaps, it will be reasonable to do what you suggest... | |
| Jul 9, 2013 at 10:01 | comment | added | Mariano Suárez-Álvarez | It should be apparent to you by now that there is no standard linear algebra textbook which does what you need, and if you find some more or less obscure counterexample it will be of little use; that someone somewhere wrote this down does not mean that giving that as a reference wil be useful for anyone! Maybe it is worth stating the properties you want, possibly —since the proofs should not be hard— omiting all details about the proofs? | |
| Jul 9, 2013 at 9:54 | answer | added | anon | timeline score: 1 | |
| Jul 7, 2013 at 19:54 | comment | added | Paul Reynolds | That's the book, yes. My copy of the book is in a box under some other boxes in a cupboard in a very inconvenient place right now so I can't dig it out. Having seen your comments below you may not be satisfied, but take a look. I once wrote my own set of notes on this topic, including quaternionic structures, because I didn't find a nice treatment in a book. It's certainly a common notion, though. | |
| Jul 7, 2013 at 11:14 | answer | added | Daniel Moskovich | timeline score: 2 | |
| Jul 7, 2013 at 11:14 | comment | added | Sergei Akbarov | @Paul Reynolds: Do you mean the book by J.F.Adams "Lectures on Lie groups"? Where does he write about this? | |
| Jul 7, 2013 at 10:02 | answer | added | Ben McKay | timeline score: 1 | |
| Jul 7, 2013 at 8:48 | comment | added | Paul Reynolds | This is indeed the notion of a real structure on a complex vector space, and does the opposite of complexification. If you replace your first equation with $x^{**} = -x$ then you get a quaternionic structure on a complex vector space. These are discussed in Adam's book 'Lie Groups' for representations, but everything he says is relevant for mere vector spaces as well. He calls both of the above objects 'structure maps'. I suppose I could put this as an answer but I've typed it here now. | |
| Jul 7, 2013 at 6:40 | comment | added | Yemon Choi | Not that one, but maybe it is in there also. I was thinking of a small book Roe write, titled something like "Elliptic operators and topology", but I do not have my copy here. | |
| Jul 7, 2013 at 6:37 | comment | added | Sergei Akbarov | It doesn't matter, of course, on which discipline the book is. John Roe's book, do you mean this one: folk.uio.no/rognes/higson/Book.pdf ? | |
| Jul 7, 2013 at 6:24 | comment | added | Yemon Choi | I think it was somewhere in the middle of John Roe's book on the Atiyah-Singer Index Theorem - I don't even remember why it was mentioned or what role it played, to be honest, this was at least five years ago. Sorry. | |
| Jul 7, 2013 at 6:20 | comment | added | Sergei Akbarov | You don't remember details? Title, author? | |
| Jul 7, 2013 at 6:16 | comment | added | Yemon Choi | I can't help with your actual question, since I never saw this in a linear algebra textbook, but I saw these mentioned in passing in some monograph on an advanced topic, as "real structures on a complex vector space". Perhaps that phrase will yield more search results | |
| Jul 7, 2013 at 5:25 | history | asked | Sergei Akbarov | CC BY-SA 3.0 |