Talk:Simple Lie algebra

Redirected

I reverted this page to a redirection page because recent editing had resulted in a page not that informative and because all essential information is found in the article on simple Lie groups. Also, it is somewhat misleading to mention only the year 1914 when Élie Cartan catalogued the real simple Lie groups and not the year 1894 when he laid the foundation of this result by cataloguing the complex simple Lie groups nor the huge previous work of Wilhelm Killing whereupon this result was based.

Kai Neergård (talk) 14:43, 26 June 2013 (UTC)

Ok, reverting does make sense but, eventually, we do need an article on simple Lie algebras as the notion (and the classification) make sense over an arbitrary field. I have therefore moved the content to Draft:Simple Lie algebra, which can replace the redirect when it matured. I will also fix the history comment. —- Taku (talk) 22:56, 27 December 2018 (UTC)

Semisimplicity

This article states that a Lie algebra that is the direct sum of simple Lie algebras is called semisimple, but this is only necessarily true over fields of characteristic 0 (see: finite fields). I've changed the definition to include the characteristic 0 assumption. Callie Liddle (talk) 05:14, 3 April 2026 (UTC)

I think it’s a matter of the definition of a semisimple Lie algebra. Often, by definition, a semisimple object is a direct sum of simple objects. See the semisimple Lie algebra article. Taku (talk) 10:02, 3 April 2026 (UTC)

Often, however J.E. Humphreys, for example, defines a semisimple Lie algebra as having trivial radical (i.e. no nontrivial solvable ideals). This is only necessarily equivalent to a direct sum of simple Lie algebras in characteristic 0. The article does seem to focus entirely on Lie algebras over $\mathbb {C}$ and $\mathbb {R}$, in which case I understand this being the definition. But I feel a clarification should be made. Perhaps I'm overstepping here. Callie Liddle (talk) 21:52, 3 April 2026 (UTC)

So, a ring with that kind of semisimplicity (i.e., having zero radical) is often called a Jacobson semisimple ring, precisely because that might differ from the usual semisimple ring. Some authors thus define a semisimple Lie algebra to be semisimple in the Jacobson sense, which of course is not wrong and may be more natural depending on perspectives. Anyway, you’re correct that readers may expect a different definition of semisimple Lie algebras (and the article thus should take that into account). —- Taku (talk) 00:20, 4 April 2026 (UTC)