Timeline for Eigenvalues of the free sphere
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Jan 8, 2013 at 16:53 | comment | added | Uwe Franz | The generator of the Brownian motion is then our candidate for a Laplace operator. From the Laplace operator we get a Dirichlet form, under certain conditions and after fixing a reference state. If the reference state is tracial, then we can apply the construction by Cipriani and Sauvageot to get a derivation that implements the Dirichlet form via $\mathcal{E}[a]=||\partial a||^2$ and the Laplace operator as $\partial^*\partial$. The free sphere does have a tracial state that is invariant for the co-action of $O_n^+$, yes? | |
| Jan 8, 2013 at 16:42 | comment | added | Uwe Franz | We are asking the same question, but in different terminology. "what is the Laplace operator on ...?" becomes "what is Brownian motion on ...?" For quantum groups we have Schürmann's theory of Lévy processes, the question becomes "which of the many Lévy processes on given quantum group deserves to be called a Brownian motion?" On $O^+$ we discovered that invariance under the adjoint action is a nice condition that leads to a subclass that we where able to classify. This is not the only condition one could imagine, but it is one that also works for compact simple connected Lie groups. | |
| Jan 8, 2013 at 16:26 | history | edited | Uwe Franz | CC BY-SA 3.0 |
added 1 characters in body
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| Jan 8, 2013 at 13:26 | comment | added | Alexander Chervov | +1, but somehow it seems the definition of "d" in question was not actually given, seems OP wrote that no one known, or I am not correct ? So is it a "theorem" or "guess" ? | |
| Jan 8, 2013 at 12:59 | vote | accept | Richard | ||
| Jan 8, 2013 at 11:29 | history | answered | Uwe Franz | CC BY-SA 3.0 |