Timeline for Generators for the Jordan algebra of symmetric 3-by-3 matrices
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
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| Nov 19, 2010 at 4:53 | comment | added | Steve Richards | continued: That is true for any field with significantly many elements. For small finite fields this does not work. | |
| Nov 19, 2010 at 4:30 | comment | added | Steve Richards | @skip: This esessentially is in Mark's answer below. He does not give a proof. But if you take two matrices, a matrix unit (1,1) and a generic matrix $A$, then they generate the whole Jordan algebra. Take $A$ with variable coordinates. And start producing matrices with more and more zeroes using the Jordan product and linear combinations. It is clearly possible to get all matrix units, but in the process you need to divide by polynomials of coeff. of A. Hence you need that those are not 0. This gives you the conditions that $A$ must satisfy ($A$ must belong to a complement of a variety). | |
| Nov 19, 2010 at 4:05 | answer | added | Aaron Meyerowitz | timeline score: 2 | |
| Nov 18, 2010 at 22:20 | comment | added | TOM | @José Figueroa-O'Farrill:Thank you for your edition.The field does not matter. | |
| Nov 18, 2010 at 21:49 | vote | accept | TOM | ||
| Nov 18, 2010 at 21:36 | answer | added | user6976 | timeline score: 7 | |
| Nov 18, 2010 at 20:46 | answer | added | Andrei Moroianu | timeline score: 9 | |
| Nov 18, 2010 at 20:25 | comment | added | Skip | I don't think the base field $k$ really matters here. The answer is that any two randomly-chosen symmetric matrices should generate "with probability 1". (And it may be clearer to work over $\mathbb{Q}$ or some other field with lots of cubic extensions.) Pick a matrix $A$ with minpoly of degree 3, so $A$ generates a degree 3 (associative!) subalgebra. Pick $B$ not in $k[A]$ also with minpoly of degree 3. (You can take $A$ diagonal, but then $B$ cannot be.) Then I think $A$ and $B$ will generate... | |
| Nov 18, 2010 at 19:43 | comment | added | José Figueroa-O'Farrill | I have edited the question to make it more readable. I have added the condition that the algebra be real. If this is not intended, then please delete that. | |
| Nov 18, 2010 at 19:42 | history | edited | José Figueroa-O'Farrill | CC BY-SA 2.5 |
I've made the question more readable.
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| Nov 18, 2010 at 16:21 | answer | added | Denis Serre | timeline score: 2 | |
| Nov 18, 2010 at 16:16 | history | asked | TOM | CC BY-SA 2.5 |