Over the past couple of months I had the chance to study the classification of compact simple Lie algebras. During this time I've always been wondering if these results can be extended to more general Lie algebras (removing the simple-requirement but maybe looking at the compact finite dimensional case, dropping the compact-requirement, etc.).

I know that one can classify more general families of algebras (for example twisted and untwisted affine Kac-Moody algebras), but I'm more interested in the case of just not-simple or non-compact Lie algebras. Can one still use the theory of root systems, Dynkin diagrams, etc. ? What is some standard literature in this direction and what are some main results?

Unfortunately I wasn't really able to find a clear answer, but this may be related to me not knowing how to formulate my question well.

  • 2
    $\begingroup$ Lie algebras of compact Lie groups are precisely the direct products of simple compact ones, and abelian ones. Beyond this you'd need to focus the question since you seem to both address the finite-dimensional and infinite-dimensional case, make no assumption on the field, etc. Also the direction depends on the motivation (e.g., representation-theoretic? geometric group theory? etc). $\endgroup$
    – YCor
    May 9, 2020 at 12:44

1 Answer 1


In general, non-simple or non-semisimple Lie algebras cannot be classified. For example, the case of solvable or nilpotent Lie algebras is "hopeless". There is a large literature on the classification of solvable and nilpotent Lie algebras in low dimensions, with applications to physics. See also several posts at MSE:

Classification results for solvable lie algebras.

Is there a classification of non-compact Lie Groups? I am interested specifically in subgroups of GL(n,R).

See also this MO-question, this one, and others.


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