# Groups associated with infinite dimensional Lie algebras

There is a classical correspondence between Lie algebras (over $$\mathbb{R}$$ or $$\mathbb{C}$$) and Lie groups in the finite dimensional case: to every Lie group $$G$$ there is an associated Lie algebra $$\mathfrak{g}$$, and conversely. Moreover, this correspondence is one-to-one if one requires $$G$$ to be simply connected. One also has maps $$\exp$$ and $$\log$$ that map between a Lie group and its associated Lie algebra, and the Baker–Campbell–Hausdorff (BCH) formula gives a Lie series $$z$$ for given $$x$$ and $$y$$ in $$\mathfrak{g}$$ such that $$\exp x\exp y=\exp z$$ (at least for sufficiently small $$x$$ and $$y$$).

Every Lie group gives rise to a Lie algebra, but in the infinite-dimensional case there are ‘non-enlargeable’ Lie algebras which don't correspond to a Lie group.

However, I came across a remark in Does a finite-dimensional Lie algebra always exponentiate into a universal covering group saying that every Lie algebra over $$\mathbb{R}$$ can be exponentiated to give an abstract group, but that it is a non-trivial theorem. Rather than poking this old comment to another question (where I may have taken something out of context), I thought this deserves to be explicitly asked.

Let $$\mathfrak{g}$$ be a Lie algebra over a field $$k$$. Is there an abstract group $$G$$ with which $$\mathfrak{g}$$ is naturally associated? What if we restrict to $$k$$ having characteristic zero — or even $$k=\mathbb{R}$$ or $$k=\mathbb{C}$$?

By ‘naturally associated’ I am thinking of a bijective correspondence reminiscent of the classical situation, where the algebra and the group are linked by the BCH formula, and Lie algebra homomorphisms $$\phi$$ lift to group homomorphisms $$\Phi=\exp \phi\log$$. However I am not expecting a differentiable structure on $$G$$. (That said, if $$G$$ comes with some extra structure for free, I would be interested to hear about it.)

I have looked at other MO questions such as Lie Groups and Lie Algebras but it's not clear to me that infinite dimensional Lie algebras are considered there.

A further observation is that the Lie series encountered in the classical case converge in the usual sense on sufficiently small neighbourhoods. However, in some cases of interest to me, such as free Lie algebras, the Lie algebra can be embedded in an (associative) algebra of formal power series with non-commuting indeterminates, in which case there are no restrictions on when the Lie series converge. So it seems to me that a subcase of the question above corresponding to the situation where all Lie series converge may well have been studied. There is a 1948 Doklady paper On normed Lie algebras over the field of rational numbers by Mal'cev which sounds relevant but has proven resistant to my attempts to track it down.

I would be grateful for any pointers to books or articles that discuss this.

• Jul 21, 2021 at 15:17
• I changed your link from a question with many answers and comments to a link to what I think was the appropriate comment. I also changed your link that pointed to an answer but seemed to mean to refer to a question so that it actually referred to the question. I hope this was all right. Jul 21, 2021 at 16:26
• What about a "pro-finite dimensional" construction? Probably it won't retain all the information but it is a first approximation which has probably the right functorial properties. It mimicks how to extens Galois theory to the infinite dimensional case. Namely, let $FQ(g)$ be the poset of finite dimensional quotients of $g$. Let $T$ be the classical functor from fdim lie algebras to lie groups. Then extend $T$ by $T(g) := \varprojlim_{g/h \in FQ(g) } T(g/h)$. This should be a corresp between pro-finite dimensional lie algebras and pro-Lie groups. Jul 21, 2021 at 17:17
• @AndreaMarino's link on the arXiv: Hofmann and Morris - Pro-Lie groups: A survey with open problems. Jul 22, 2021 at 12:55
• @LSpice Thanks for correcting the links. Jul 23, 2021 at 22:04

Here is an informative example that illustrates the difficulties: Consider the Lie algebra $${\frak{g}} = \mathrm{Vect}(\mathbb{S})$$ of smooth vector fields on the circle $$\mathbb{S}$$. The flow of any vector field $$X\in{\frak{g}}$$ is a $$1$$-parameter subgroup of $$\mathrm{Diff}_+(\mathbb{S})$$, the (connected) Lie group of orientation preserving diffeomorphisms of $$\mathbb{S}$$, so this defines an exponential map $$\exp:{\frak{g}}\to\mathrm{Diff}_+(\mathbb{S})$$. Moreover, every $$1$$-parameter subgroup of $$\mathrm{Diff}_+(\mathbb{S})$$ is the flow of some smooth vector field on $$\mathbb{S}$$.

Unfortunately, there is also a sequence $$\phi_k$$ of diffeomorphisms of $$\mathbb{S}$$ that converges to the identity in the $$C^\infty$$ topology such that none of the $$\phi_k$$ lie in $$\exp\bigl({\frak{g}}\bigr)$$. In particular, the exponential map does not cover a neighborhood of the identity in $$\mathrm{Diff}_+(\mathbb{S})$$. See, for example, R. Hamilton's Bulletin article The inverse function theorem of Nash and Moser.

Meanwhile there is no proper Lie subgroup (in Lie's original sense) of $$\mathrm{Diff}_+(\mathbb{S})$$ that contains $$\exp\bigl({\frak{g}}\bigr)$$.

There has been a lot of work on so-called 'infinite dimensional Lie groups'. It might be good to start with J. Milnor's classic paper, Remarks on infinite-dimensional Lie groups, in “Relativité, Groupes et Topologie II,” B. DeWitt and R. Stora (Eds), North-Holland, Amsterdam, 1983, 1007–1057, but you will also want to read some of the more modern literature as well, such as Fundamental Problems in the Theory of Infinite-Dimensional Lie Groups, by H. Glöckner (https://arxiv.org/abs/math/0602078) and works on this subject by P. Michor and collaborators.

• Concerning the modern literature, Neebs survey arxiv.org/abs/1501.06269 is still quite relevant albeit being already 15 years old. Jul 21, 2021 at 17:41
• Title of Neeb's survey referenced by @AlexanderSchmeding: Neeb - Towards a Lie theory for locally convex groups. Is it really 15 years old? It was posted on the arXiv only in 2015. Jul 22, 2021 at 12:57
• @LSpice: Yes, it was published in the journal of the Japanese mathematical society in 2006. However, a lot of what is recorded there is still up to date (though some later developments exist its always a solid base for further inquiry) Jul 23, 2021 at 9:57
• Many - belated! - thanks for your answer, which I will have to think about further. Regarding Milnor’s paper, I had come across it but the main thrust of it seems to be to broaden the notion of Lie group while maintaining some differentiable concepts, rather than ‘enlarging’ a given Lie algebra in a more general sense. I will have a look at the other papers you mention. Jul 23, 2021 at 22:00

Another source discussing the (non-)integrability of infinite dimensional Lie algebras is T. Robart: Around the exponential mapping, in: Banyaga et al (Eds) Infinite dimensional Lie groups in geometry and representation theory, 2002

There a detailed account of "the exponential catastrophe" can be found together with an overview of older research on integrability criteria (e.g. for Banach Lie algebras.

• This sounds interesting - many thanks. Jul 26, 2021 at 8:59