Timeline for Group structures on the cartesian product of two groups
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Mar 20, 2017 at 15:19 | comment | added | Joshua Grochow | @VladimirDotsenko: The relationship with semi-direct product is closer than your last remark hints at: $\alpha$ must be a left action of $K$ on $H$, and $\beta$ must be a right action of $H$ on $K$. (And then those two actions must satisfy some compatibility assumption.) And indeed, if either of these actions is trivial, then one recovers the semi-direct product. | |
| Jun 14, 2010 at 1:34 | comment | added | Vladimir Dotsenko | @Victor: I was inclined to say the same after I first read this, but then I looked again into the part on external Zappa-Szep products, and I should admit that in a sense this is a construction in the same way the semi-direct product is a construction. The semi-direct product depends on some data (action of H on K); similarly, this product depends on the mappings $\alpha$ and $\beta$ satisfying compatibility condition... | |
| Jun 13, 2010 at 23:54 | comment | added | Victor Protsak | As José and Theo remarked, this is certainly an interesting notion, but I don't see any general (non-tautological) $\textit{construction}.$ | |
| Jun 13, 2010 at 22:01 | comment | added | Theo Johnson-Freyd | @Jose: Yes, and also there is an analogous version for Hopf algebras. Indeed, the "Drinfeld Double" of a Hopf algebra is a special example. | |
| Jun 13, 2010 at 20:30 | comment | added | José Figueroa-O'Farrill | Interstingly enough, this construction -- which I knew but not by name -- is intimately related to the subject of "dressing transformations" in integrable systems. The actions of H on K and of K on H are abstract analogues of the dressing actions in Poisson-Lie groups. | |
| Jun 13, 2010 at 19:37 | vote | accept | Vladimir Dotsenko | ||
| Jun 13, 2010 at 19:37 | comment | added | Vladimir Dotsenko | Dear Steven, thanks! That (more precisely, the part on external Zappa-Szep products) is exactly what I was looking for. | |
| Jun 13, 2010 at 19:19 | history | answered | Steven Gubkin | CC BY-SA 2.5 |