Does the Lie algebra structure on rational homotopy groups reflect similar information to the formal group structure in characteristic p?

Question

It's well known (c.f. Quillen and Sullivan) that the rational homotopy theory of spaces is equivalent to the homotopy theory of rational DG-algebras; in particular, rational spaces and rational spectra have a homotopy Lie algebra, not just homotopy groups. On the other hand, more recent work in the p-local setting has demonstrated the tremendous importance of the chromatic structure on cohomology, which is parameterized by the theory of (one-dimensional commutative) formal groups. When I was learning about this, I noticed an interesting connection between two facts:

• Formal groups are only interesting in positive characteristic, because over $\mathbb{Q}$ they're equivalent to Lie algebras (all their higher order structure vanishes).
• Chromatic homotopy theory isn't typically done over $\mathbb{Q}$, and Lie algebras replace formal groups as the central algebraic objects.

In both algebra(ic geometry) and topology, formal groups are replaced by Lie algebras in the rational case. Is this just a coincidence, or can the Lie algebra structures appearing in the study of $\operatorname{Sp}_{\mathbb{Q}}$ be viewed as the characteristic-zero case of some chromatic phenomena?

Answer 1

The Lie bracket on rational homotopy has a relatively simple conceptual explanation: if $X$ is simply connected, then the rational homology $H_{\bullet}(\Omega X, \mathbb{Q})$ of its loop space is a connected graded Hopf algebra, and (under some mild hypotheses, maybe?) must therefore be the universal enveloping algebra of its Lie algebra of primitive elements, which turns out to be the rational homotopy $\pi_{\bullet}(\Omega X, \mathbb{Q})$ of $\Omega X$. This is of course the rational homotopy of $X$ but shifted by a degree, and that shift is responsible for the degree shift of the bracket. (These brackets vanish for rational spectra.)

Roughly speaking this rational homology is the "group algebra" of $\Omega X$ and so what this suggests is that the homotopy Lie algebra is the "Lie algebra of $\Omega X$" in a suitable sense. Like all good Lie algebra structures it is fundamentally attached to a group, which in this case is the loop space.

Now, I understand even less about chromatic homotopy than I do about rational homotopy, but my understanding is that the formal group structures appearing there are most simply understood as coming from Chern class computations on complex oriented cohomology theories, so fundamentally the group structure here is tensor product of line bundles. There is a graded Hopf algebra that can be written down here but it is (I think) $E^{\bullet}(BU(1))$ where $E$ is complex oriented, whose comultiplication (which encodes the formal group law) comes from the group structure on $BU(1)$ representing tensor product of line bundles. This has quite a different flavor from the loop space group structure above so, at least based on my limited background, the two don't seem all that related to me. To make a basic observation, the formal group laws appearing in the chromatic picture are all (commutative and) $1$-dimensional, whereas the Lie algebras appearing in rational homotopy are decidedly not. Snaith's theorem is perhaps relevant here.