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I am starting to study Lie algebras and when I reached the notion of solvable Lie algebra, I tryed to find concrete applications ( in physics for exemple) and I couldn't find one. For exemple, Solvable group are very important for the unsolvability of quintic equation ( and by the way, it's the only application I know of them).

In the same manner, can we find application for solvable lie algebras ?

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    $\begingroup$ See this MO post. $\endgroup$
    – abx
    Commented Jan 11, 2019 at 13:47
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    $\begingroup$ Borel subalgebras of reductive algebras are both solvable and quite important. $\endgroup$
    – user44191
    Commented Jan 11, 2019 at 15:52
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    $\begingroup$ @user44191, as BConrad would remind us if he were here, lest we wander inadvertently into the territory of non-perfect fields, one should speak of Lie algebras of Borel subgroups of linear algebraic groups (reductivity isn't important; Borels just get fattened up by their radicals). As the example of, say, $G = \operatorname{SL}_2$ in characteristic 2 shows, it is possible for a reductive group to have a soluble Lie algebra. $\endgroup$
    – LSpice
    Commented Jan 11, 2019 at 16:37

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There are actually lots of applications of solvable Lie algebras, especially in the field of integrable systems where the solvabiltiy of Hamilton's equations of motion is frequently related to the solvability of the Lie algebra of the integrals of motion of the system.

For example:

for any Hamiltonian system on $\mathbb{R}^{2n}$ with its standard symplectic structure, if there are $n$ integrals of motion $F_i$, which are functionally independent and they form a solvable Lie algebra (under the Poisson bracket) -plus some technicalities on the structure constants- then the system can be integrated, on the level-set of the integrals, by quadratures.

This is a classic result: see for example Perelomov's book, p. 34, 35 Theorem 2, the subsequent example and the references therein. Some authors call this result Lie's theorem (although i am not sure if this is a standard terminology).

On the other hand, you can find lots of concrete examples in Integrable Hamiltonian systems on low-dimensional Lie algebras (here is the -free- Russian version).

There are lots of other references as well. A descent coverage might require a fairly sized review article.

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    $\begingroup$ Thanks @Konstantinos Kanakoglou , that's the kind of theorem I was looking for. $\endgroup$
    – poloC
    Commented Jan 14, 2019 at 13:12
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Besides Lie's theory of integration by quadrature for flows of vector fields, there is a completely different theory of solving rational coefficient linear differential equations by repeatedly integrating, taking logarithms, taking exponentials, and forming rational functions thereof: differential Galois theory. A nice introduction is: Camillo de Lellis, Il teorema di Liouville ovvero perche non esiste la primitiva di $e^{x^2}$. The upshot is that there is an invariant of any rational coefficient linear differential equation, its differential Galois group, and if the equation has general solution obtained by such repeated operations, then its differential Galois group is solvable.

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    $\begingroup$ Repeated from my other comment: Nonetheless this is solubility of groups, not really of Lie algebras, no? (Of course one may take the Lie algebra, but the post seems implicitly to be asking for applications that 'naturally' live on the Lie-algebra side (despite the tag).) $\endgroup$
    – LSpice
    Commented Jan 11, 2019 at 16:38
  • $\begingroup$ Well, this is a valuable piece of info but i do not think it is really relevant to the OP. $\endgroup$
    – Konstantinos Kanakoglou
    Commented Jan 12, 2019 at 17:32
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I have read that one motivation of Lie when he worked on transformation groups was to build a similar theory for solvability of ODEs by quadrature to Galois theory for solvability of polynomials.

You can find some very concrete examples in this spirit in the paper: "Bryant, R, An introduction to Lie groups and symplectic geometry". I think Robert Bryant might have many interesting things to tell about this. (I want to post this as a comment only, but I can't post comment yet)

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    $\begingroup$ Nonetheless this is solubility of groups, not really of Lie algebras, no? (Of course one may take the Lie algebra, but the post seems implicitly to be asking for applications that 'naturally' live on the Lie-algebra side (despite the tag).) $\endgroup$
    – LSpice
    Commented Jan 11, 2019 at 16:36

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