Timeline for Are two Lie algebra deformations with cohomologous tangents isomorphic?
Current License: CC BY-SA 3.0
7 events
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| Aug 10, 2017 at 12:07 | comment | added | Jon Pridham | On the other hand, the enumerated procedure you describe with multiple variables looks like the universal deformation over the hull of the deformation functor. | |
| Aug 10, 2017 at 11:48 | comment | added | José Figueroa-O'Farrill | Hi @JonPridham. Indeed. My rephrasing needs work. In your example, if a nontrivial bracket is divisible by $t^2$ then the first nonzero term is still a cocycle, so then the question is whether if the $t^2$ term is a coboundary, whether the deformation is trivial. I need to think more about this, because I'm starting to think that the answer is negative. | |
| Aug 10, 2017 at 9:03 | comment | added | Jon Pridham | Can't you get a counterexample to the rephrased question by starting from an abelian Lie algebra, taking $\alpha_1=0$ and $\alpha_2$ to be any non-trivial bracket divisible by $t^2$? | |
| Aug 10, 2017 at 8:03 | comment | added | José Figueroa-O'Farrill | I'm interested in whole deformations: so solutions of the Maurer-Cartan equation modulo gauge transformations. However, for lack of a better way, I'm solving the equation perturbatively. I have added more information to the question to clarify. | |
| Aug 10, 2017 at 8:03 | history | edited | José Figueroa-O'Farrill | CC BY-SA 3.0 |
added a rephrasing of the original question to clarify after a comment.
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| Aug 10, 2017 at 2:16 | comment | added | user40276 | I don't know if I've understood the question, but it's a general fact that any deformation problem is given by the Maurer-Cartan elements of a DGLA (the infinitesimal automorphisms of the functor defining the problem) modulo gauge transformations. The Maurer-Cartan equation says that you can extend the problem to a higher order deformation, while the gauge transformation takes into account this irrelevance given by these cocycles representing the same cohomology class. Now I couldn't really understand whether you're only picking first-order deformations or if you want the whole deformation. | |
| Aug 9, 2017 at 17:48 | history | asked | José Figueroa-O'Farrill | CC BY-SA 3.0 |