The origin of emergent algebras

In the last section “Why is the tangent space a group?” (section 8) of the great article by A. Bellaiche, The tangent space in sub-riemannian geometry*, the author explains a very interesting story, where names of Gromov and Connes appear, which is the first place, to my knowledge, where the idea of emergent algebras appear.

In a future post I shall comment more consistently on the math, but this time let me give you the relevant passages.

[p. 73] “Why is the tangent space a group at regular points? […] I have been puzzled by this question. Drawing a Lie algebra from the bracket structure of some X_{i}‘s did not seem to me the appropriate answer. I remember having, at last, asked M. Gromov about it (1982). The answer came under the form of a little apologue:

Take a map f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}. Define its differential as

(79)              D_{x} f(u) = \lim_{\varepsilon \rightarrow 0} \varepsilon^{-1} \left[ f(x+\varepsilon u) - f(x) \right],

provided convergence holds. Then D_{x}f is certainly homogeneous:

D_{x}f(\lambda u) = \lambda D_{x}f(u),

but it need not satisfy the additivity condition

D_{x}f(u+v) = D_{x}f(u) + D_{x}f(v).

[…] However, if the convergence in (79)  is uniform on some neghbourhood of (x,0)  […]  then D_{x}f is additive, hence linear. So, uniformity was the key. The tangent space at p is a limit, in the [Gromov-]Hausdorff sense, of pointed spaces […] It certainly is a homogeneous space — in the sense of a metric space having a 1-parameter group of dilations. But when the convergence is uniform with respect to p, which is the case near regular points, in addition, it is a group.

Before giving the proof, I want to tell of another, later, hint, coming from the work of A. Connes. He has made significant use of the following observation: The tangent bundle TM to a differentiable manifold M is, like M \times M, a groupoid. […] In fact TM is simply a union of groups. In [8], II.5, it is stated that its structure may be derived from that of M \times M by blowing up the diagonal in M \times M. This suggests that, putting metrics back into the picture, one should have

(83)          TM = \lim_{\varepsilon \rightarrow 0} \varepsilon^{-1} (M \times M)

[…] in some sense to be made precise.

There is still one question. Since the differentiable structure of our manifold is the same as in Connes’ picture, why do we not get the same abelian group structure? One can answer: The differentiable structure is strongly connected to (the equivalence class of) Riemannian metrics; differentiable maps are locally Lipschitz, and Lipschitz maps are almost everywhere differentiable. There is no such connection between differentiable maps and the metric when it is sub-riemannian. Put in another way, differentiable maps have good local commutation properties with ordinary dilations, but not with sub-riemannian dilations \delta_{\lambda}.

So, one should not be abused by (83) and think that the algebraic structure of T_{p}M stems from the absolutely trivial structure of M \times M! It is concealed in dilations, as we shall now prove.

*) in the book Sub-riemannian geometry, eds. A. Bellaiche, J.-J. Risler, Progress in Mathematics 144, Birkhauser 1996

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