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Among the finite dimensional Lie algebras over a field of characteristic 0, there is a sensible definition of "reductive Lie algebra" going back at least to the 1960 first chapter of N. Bourbaki's tre …

While trying to get some perspective on the extensive literature about highest weight modules for affine Lie algebras relative to "level" (work by Feigin, E. Frenkel, Gaitsgory, Kac, ....), I run into …

These terms have become common in Lie theory and related algebraic geometry and combinatorics, as seen in many questions posted on MO, but it's unclear to me where they first came into use. Probably …

Cell decompositions have been used in topology for a long time as a tool in computing cohomology, but the notion in algebraic geometry and arithmetic geometry of paving by affine spaces (or "affine pa …

Terminology in mathematics develops a bit haphazardly, and sometimes things get misleading names. … Anyway, one might make a case for the terminology "James module" here, but it's too late for that. Specht himself had no special influence on the modular theory. …

(This at any rate is the kind of answer given to the terminology question raised with Mostow at a conference I attended decades ago.) …

It's unclear to me why this terminology is so popular, since over a field of characteristic 0 it's equivalent to "reductive algebraic group". …

As usual with terminology such as "weight", the history reaches back into nineteenth century's invariant theory (Cayley, G. … The history is not at all easy to untangle, but I think Hawkins was thorough in his study of the development of ideas along with terminology. …

Your basic question about a reference does go back to Weyl's complete reducibility theorem (I'd have to check his book on classical groups, but it isn't just a result about classical Lie groups). F …

In the construction of Soergel's bimodules in representtion theory , it's essential for him to work with split Grothendieck groups. Here he starts with a certain small additive category $\mathcal{A} …

Mostow's theorem actually has a relatively modern textbook treatment, by his early collaborator Gerhard Hochschild: see Theorem 4.3 in VIII.4 of his book Basic Theory of Algebraic Groups and Lie Algeb …

The correct answer to the narrow question asked about terminology has been given by quid, but I'm tempted to add some broader semi-historical remarks as well. … As usual, the development of mathematical notation and terminology is somewhat arbitrary but complicated. …

Marty is definitely correct about the origin of the terminology in the study of homogeneous spaces $G/H$ of reductive Lie groups. …

I think the answer to the original question is that there is no special name for the involution (otherwise it would have turned up by now). My first encounter with it was in the 1979 Kazhdan-Lusztig …

Many of the examples mentioned go back to earlier centuries, when insulated national traditions and slow communications promoted mistaken labelling of results and concepts. A much more recent exampl …