The Wedderburn–Artin Theorem is one of the cornerstones of the structure theory of (associative) rings.
Wedderburn–Artin Theorem : Let $R$ be a left Artinian ring with zero Jacobson radical. Then $R$ is the direct product of a finite number of simple left Artinian rings, let's say, $R=\prod_i R_i$, and each $R_i$ is a matrix ring over a division ring, $M_{n_i}(D_i)$.
One can naturally ask if there is a similar result for non-associative algebras that are "nearly associative", such as Lie, Jordan or alternative algebras.
If one is looking for a similar result, then one of course must develop an adequate notion of radical ideal.
In case of alternative algebras, there were introduced two notions of radical ideal, one by Smiley and one by Kleinfeld. A fundamental result in the structure theory of alternative algebras, due to Zhelakov, is that both radicals coincide — and hence nowadays this radical is usually called the Zhelakov radical.
With this notion of radical at hand, one has an analogue of the Wedderburn–Artin theorem for alternative algebras:
(Zhelakov; Schafer-Zorn for the finite dimensional case): Let $A$ be a left Artinian alternative algebra with zero Zhelakov radical. Then $A$ is a finite direct product of simple left Artinian alternative algebras, each of which is either a matrix algebra over division ring, or a Cayley–Dickson algebra over its center.
For Jordan algebras, there is also an analogue of the Wedderburn–Artin Theorem. Here the issues were finding an adequate notion of radical and realizing that the correct descending chain condition should be for quadratic ideals. This theory is exposed in detail in the book by Zhelakov, Slin'ko, Shestakov and Shirshov, ‘Rings that are nearly associative’ (MR).
Now let's consider Lie algebras. If they are finite dimensional, then we have the classical Cartan's theory using the solvable radical.
But, if we consider infinite dimensional Lie algebras, is there an adequate notion of radical ideal and a suitable descending chain condition for ideals such that we have an analogue of the Wedderburn–Artin Theorem?
Best Answer
I had the opportunity to chat with an expert in nonassociative algebras, and I report here what he told me.
The short answer is: as yet there is none Wedderburn-Artin theory for Lie algebras. The details are as follows.
Georgia Benkart, in her paper 'On inner ideals and ad-nilpotent elements of Lie algebras' introduced a notion of inner ideal for Lie algebras (the Lie analogue of quadratic ideals for Jordan algebras), and studied Lie algebras with descending chain condition for inner ideals; however, she was unable to develop an Wedderburn-Artin theory.
As I mentioned in my question, for a successful Wedderburn-Artin theory for a class of nonassociative algebras, one needs to find an adquate notion of radical ideal, in order to define semisimple algebras as the ones with $0$ radical ideal.
Kurosh and Amistuser developed an axiomatic theory of radicals, in order to organize what properties should hold for an adequate strucutre theory involving them. Kurosh-Amitsur theory became very influential, and with their abstract point of view they were able to discover new facts about concrete radicals.
Radical ideals that satisfy the Kurosh-Amitsur axioms include the Jacobson radical (associative algebras), the Zhelakov radical (alternative algebras) and the Maccrimmon radical (Jordan algebras), among others.
A big problem about the radicals introduced in the theory of Lie algebras is that they are not radical ideals in the sense of Kurosh-Amitsur. The most useful notion of radical ideal for Lie algebras is the Kostrikin radical (the Lie analogue of Maccrimmon's radical in Jordan algebras). It seems to be a radical in the sense of Kurosh-Amitsur in the class of Engel Lie algebras and algebraic Lie algebras, but not in the class of all Lie algebras.
For a reference on this ideal, one has for instance the paper by Golubkov 'The Kostrikin Radical and Similar Radicals of Lie Algebras'.