# Historically, which came first: the Lie algebras or their classification?

The classification of the complex simple Lie algebras by their Dynkin diagrams gives rise to five exceptional complex simple Lie algebras: $F_4, G_2, E_6, E_7$ and $E_8$.

I am trying to find out whether the classification was discovered first (attributed to Wilhelm Killing [1888-1890]), or whether some/all of the exceptional complex simple Lie algebras were discovered before the classification.

It's said here that Killing discovered $G_2$ in 1887, which would mean that $G_2$ at least came before the classification. I suspect that, since this discovery came only very shortly before his discovery of the classification, this means that the others were discovered as a consequence of the classification, but I'm hoping for clarification.

• Do you think the Lie groups came before the Lie algebras? – Gerald Edgar Apr 15 '18 at 12:51

The classification came first. As Killing says in his introduction (translation by Coleman (1989)):

For each $l$ there are four structures supplemented for $l = 2, 4, 6, 7, 8$ by exceptional simple groups. For these exceptional groups I have various results that are not in fully developed form; I hope later to be able to exhibit these groups in simple form and therefore am not communicating the representations for them that have been found so far.

More details in Hawkins (cited in your linked reference) (1982, p. 156):

The discovery that type $G_2$ actually exists seems to have transformed Killing’s attitude towards the possible existence of further new simple groups. (...) after this discovery, when he obtained the $\smash{a_{ij}}$ and associated root systems for the exceptional types $E$ and $F$ and the “new” general type $\smash{C_l}$, Killing maintained that simple groups for these types exist even though he never managed to carry his calculations far enough to be able to write down multiplication tables for them.

and in Cartan (1894, pp. 94-95):

As to the determination of simple group structures, Mr. Killing is content to show that to types A), B), D) there correspond long known simple groups, namely the general projective group of $l$-dimensional space and the projective groups of a nondegenerate quadric in spaces of $2l$ and $2l-1$ dimensions. But he doesn’t give the explicit calculations allowing to show that they are the only ones, and as to the other integer systems, he only gives vague indications on the corresponding structures, except however for type C).

• I think it's interesting to note that Killing counted $F_4$ twice, without realizing it. – Allen Knutson May 6 '18 at 13:40