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Mathematics
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I'm trying to understand the idea of 'formally' integrating a Lie algebra.

The universal enveloping algebra of a Lie algebra has a comultiplication on it, turning it into a Hopf algebra; if you take the set of all group-like elements of this algebra (i.e,. $\Delta g= g\otimes g$), you get a group which people call the 'formal' Lie group integrating the Lie algebra.

How does this 'formal' Lie group relate to the 'actual' Lie group (which there will be in a finite-dimensional setting) integrating the Lie algebra, e.g. how do you equip it with a topology, etc.? Is there a nice reference for this?

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  • $\begingroup$ For example, this paper by Goncharov et al. has a nice overview on the formal integration of Lie algebras, in section $3$. For the Heisenberg Lie algebra, this gives the formal group law $x\ast y=x+y+\frac{1}{2}[x,y]$, see Example $3.10$. $\endgroup$
    Dietrich Burde
    –  Dietrich Burde
    2025-10-08 18:29:41 +00:00
    Commented Oct 8 at 18:29

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