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I am reading a paper that does the following:

  • Constructs a Lie algebra $\mathfrak{g}$ of derivations acting on a certain space (as an infinite direct sum of one-dimensional spaces of derivations)
  • Defines $G$ to be $\mathrm{exp}(\mathfrak{g})$. (In fact, the direct sum defining $\mathfrak{g}$ is graded, with each degree finite-dimensional, so $G$ is really defined as an inverse limit of exponentiations of finite-dimensional Lie algebras.)
  • Points out that $\mathfrak{g}$ can be decomposed as a direct sum $\mathfrak{g}_-\oplus\mathfrak{g}_0\oplus\mathfrak{g}_+$, with each of the three summands closed under the Lie bracket.
  • Asserts that each element $g\in G$ can therefore be factored as $g_-\cdot g_0\cdot g_+$ with $g_-\in\mathrm{exp}(\mathfrak{g}_-)$ and so forth.

So my questions are:

  1. Is there an easy way to see why this factorization exists and is unique?
  2. If I know a particular group element very explicitly, is there a good formula/procedure for finding the factorization?

(I don't think that the inverse limit in the definition is the hard part. I'm comfortable with how an answer for the finite-dimensional case would get me what I want.)

I guess a more basic question is, to what extent is this actually hard, and to what extent should I just learn more basic Lie theory.

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  • $\begingroup$ In case people are curious, the paper is Gross-Hacking-Keel-Kontsevich (arxiv.org/abs/1411.1394) and I'm reading their proof of Theorem 1.17, which is actually a result of Kontsevich-Soibelman. $\endgroup$ – Nathan Reading Mar 12 at 12:01

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