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Let $V$ be a finite dimensional vector space over $\mathbb{R}$. Let $S$ be the vector space of multilinear maps from $V\times V$ to $V$. Let $L:\mathbb{I}\rightarrow S$ be a continuous map such that for every $t$ we have that $L(t)$ is a Lie bracket on $V$. For every $t$, let $G_t$ be the simply connected Lie group whose Lie algebra is $(V,L(t))$. It is given that $G_t$ is compact for every $t$.

Question: Must $G_0$ and $G_1$ be diffeomorphic ?

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Yes, in fact in this situation you have more, all the Lie algebras $(V, L(t))$ (which I denote by $\mathfrak{g}_t$ to abbreviate) are isomorphic:

Let $G$ be the simply connected Lie group integrating a finite dimensional real Lie algebra $\mathfrak{g}$. Let $W$ be a finite dimensional representation of $G$, and write $W$ also for the induced Lie algebra representation of $\mathfrak{g}$.

Then the differentiable Lie group cohomology of $G$ with values in $W$ vanishes in positive degrees (this can be proven by averaging cocycles with respect to a bi-invariant Haar measure to obtain primitives for them).

The group $G$ is 2-connected (connected and simply connected by assumption, and $\pi_2(G)=0$ holds for any Lie group by a result of Hopf - see edit comment at the bottom), and so there is an isomorphism induced by the so-called Van est map in degrees up to 2, between differentiable Lie group and Lie algebra cohomology with values in $W$:

$$VE: H^k_\mathrm{diff}(G,W)\stackrel{\cong}{\to} H^k(\mathfrak{g},W),\ \ \ k=0,1,2$$

In particular, considering $W$ to be the adjoint representation of a Lie algebra on itself, one conclusion is that, in the setting of the question, $H^2(\mathfrak{g}_t, \mathfrak{g}_t)=0$.

A result of Nijenhuis-Richardson says that any Lie algebra $\mathfrak{g}$ such that $H^2(\mathfrak{g}, \mathfrak{g})=0$ is rigid, i.e., nearby Lie algebras to $\mathfrak{g}$ are isomorphic to it, where nearby is with respect to the topology you have on $S$. One possible place to read about this is the following survey: M. Crainic, F. Schätz and I. Struchiner, A survey on stability and rigidity results for Lie algebras, Indagationes Mathematicae, 25(2014), 957-976. Theorem 5.3 there is the rigidity result.

Since every $\mathfrak{g_t}$ is rigid and the path is continuous, they will end up being isomorphic, and so $G_t$ are also isomorphic Lie groups.

Edit: the result of Hopf I had in mind only guarantees finite $\pi_2(G)$. But the result that $\pi_2(G)=0$ is true, actual proofs and references for the result are in this Mathoverflow question: Homotopy groups of Lie groups

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  • $\begingroup$ Thanks for your answer. Do we also have the same result for symmetric spaces (instead of lie groups) if we consider instead a deformation of Lie Tripple systems (instead of a deformation of lie algebras) ? $\endgroup$
    – Amr
    Commented Apr 24, 2023 at 20:12
  • $\begingroup$ "and $\pi_2(G)=0$ holds for any Lie group by a result of Hopf": What is a reference for this result of Hopf? $\endgroup$
    – Mikhail Borovoi
    Commented Apr 27, 2023 at 5:52
  • $\begingroup$ @MikhailBorovoi let me give a Mathoverflow reference for this, because it has both references for the result, some proofs right there, and a lot of interesting discussion (in my view): mathoverflow.net/q/8957/104042 . I think I was wrong about attributing it to Hopf, what I had in mind his result only implies finite $\pi_2$, as explained in this answer: mathoverflow.net/a/8996/104042 . I don't know who first proved it. Some of the references there are "Representations of compact Lie groups" by Bröcker and tom Dieck and Milnor's book on Morse theory. $\endgroup$
    – João Nuno Mestre
    Commented Apr 27, 2023 at 19:08
  • $\begingroup$ @Amr That's an interesting question (to me at least)! I'm not very familiar with the setting but I looked a bit into what I could find in the literature. There is a cohomology (Yamaguti cohomology) for Lie triple systems (Lts) that when using the Lts itself as the module (analogous to the adjoint for Lie algebras) controls the deformations of the Lts. Kubo, Taniguchi, A controlling cohomology of the deformation theory of Lie triple systems If $H^3$ vanishes, then the Lts is rigid in some sense, see theorem 2 in that paper. $\endgroup$
    – João Nuno Mestre
    Commented Apr 27, 2023 at 19:42
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    $\begingroup$ Borel, Groupes d’homotopie des groups de Lie. I, attributes the theorem: $\pi_2(G)=0$ to Élie Cartan (Theorem 1(2) on page 12-05). He refers to Cartan’s book: La topologie des groupes de Lie. Exposés de géométrie Nr. 8. Actualités Scient. Industr. 358 Paris: Hermann & Cie. 28 p. (1936). $\endgroup$
    – Mikhail Borovoi
    Commented Apr 29, 2023 at 6:29

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