Timeline for generalizations of the interplay between Cauchy Riemann equations and complex-linearity
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Jul 30, 2012 at 3:26 | comment | added | James S. Cook | continuing, is it fair to state that the $\cal{A}$-linearity of the differential is the key feature to abstract? The "natural" theory in the case of Cayley-Dickenson seems to take the $\cal{A}$-linearity of the differential as the defn. of differentiability. | |
| Jul 30, 2012 at 3:17 | comment | added | James S. Cook | Great Kevin! The example of Cayley-Dickenson variables is much more in the direction I was looking. Fascinating paper, I think it will take me a while to appreciate the way he has dealt with the nonassociativity. Prop. 2.3 on page 12 is the basic type of result I wanted. I wonder if there are other associative algebras which provide similar results. As best as I can cipher with my first read, the definition of differentiability over the cayley-dickenson algebra $\cal{A}_r$ was given by the condition of $\cal{A}_r$-linearity of the differential (tweaked in the non-associative context) | |
| Jul 26, 2012 at 15:34 | comment | added | Kevin | You're right. I probably should have said "One generalization that might interest you is called...". I guess I haven't seen too much on this topic when $\mathcal{A}$ is not a Clifford algebra. One exception is arxiv.org/abs/math/0405471, where differentiability of functions of Cayley-Dickson variables is discussed. | |
| Jul 26, 2012 at 5:20 | comment | added | James S. Cook | Thanks Kevin. That is interesting. However, I wonder if the idea I sketched above is a bit more general than the clifford analysis. If $\cal{A}$ is not a clifford algebra then what is known? | |
| Jul 25, 2012 at 22:02 | comment | added | Kevin | I think the generalization you are looking for is called Clifford analysis -- en.wikipedia.org/wiki/Clifford_analysis. The Cauchy-Riemann operator of complex analysis is replaced with a -- more general-- Dirac operator. | |
| Jul 25, 2012 at 21:06 | history | asked | James S. Cook | CC BY-SA 3.0 |