Timeline for Infinite-dimensional complex polynomial or rational Lie algebras and their pseudogroups
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Oct 31, 2019 at 10:06 | answer | added | Ben McKay | timeline score: 0 | |
| Jan 5, 2011 at 22:31 | comment | added | Pasha Zusmanovich | What you mean exactly under "Lie algebras of vector fields"? If any Lie algebra admitting realization as vector fields with polynomial/rational/etc. coefficients (i.e. subalgebras of appropriate $W_n$) - there are many. This topic goes back to Sophus Lie with a tremendous amount of more recent literature. But your examples probably suggest that you are interested in "full", in some sense, algebras -- of the form $f(z) d/dz$ ? what about many variables? -- but the exact meaning is not clear to me. | |
| Aug 26, 2010 at 16:13 | comment | added | Simon Wadsley | The answer to your first question is yes. If you differentiate the action of the group $PSL_2$ on the Riemann sphere at the identity to give a Lie homomorphism from the Lie algebra $\mathfrak{sl}_2$ to vector fields on the Riemann sphere. The image of this map on a coordinate chart obtained by removing a point at infinity is your Lie algebra of complex quadratic vector fields. | |
| Aug 26, 2010 at 1:38 | comment | added | Victor Protsak | Drinfeld-Krichever construction in the theory of integrable systems involves a generalization of the polynomial vector fields: for a complete algebraic curve $X,$ a point $x_\infty$ and a local coordinate $z$ at this point, consider the Lie algebra of vector fields on $X\setminus x_{\infty}$ expressed in the coordinate $z.$ Its analytic version is the Lie algebra of complex vector fields on a small circle $\partial D_\infty$ around $x_{\infty}$ that extend to holomorphic vector fields on its exterior $X\setminus D_\infty.$ | |
| Aug 26, 2010 at 1:25 | comment | added | Victor Protsak | The Lie algebra you've denoted $V_P$ is the complexification of the Lie algebra of the poly vector fields on the real line, $W_1,$ which is one the 4 families of Cartan's transitive pseudogroups. While AFAIK $V_R$ doesn't have another name, its subalgebra consisting of the Laurent polynomials in $z$ times $d/dz$ is the complexification of the Lie algebra of polynomial vector fields on the circle. In a certain sense, it corresponds to the diffeomorphism group of the circle, whereas $W_1$ corresponds to the diff group of the line or the group of holomorphic automorphisms of the closed unit disk. | |
| Aug 26, 2010 at 0:41 | answer | added | Tom Goodwillie | timeline score: 3 | |
| Aug 25, 2010 at 23:40 | history | edited | Daniel Asimov | CC BY-SA 2.5 |
changed title
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| Aug 25, 2010 at 22:57 | history | edited | Daniel Asimov | CC BY-SA 2.5 |
Added note about the ease of computing formulas for the flows
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| Aug 25, 2010 at 22:17 | history | edited | Daniel Asimov | CC BY-SA 2.5 |
probably -> surely
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| Aug 25, 2010 at 22:11 | history | asked | Daniel Asimov | CC BY-SA 2.5 |