What is the motivation behind naming the category O appearing in the theory of Lie algebras? Does O stand for something? Here is a question Why the BGG category O? that further confuses me. It seems like there is a notion of when a category is O, is it?
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3$\begingroup$ @VítTuček, isn't that because of something like "O for olomorfe"? $\endgroup$– LSpiceCommented Jun 11, 2020 at 5:43
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2$\begingroup$ I think that the $\mathcal O$ notation comes from the notation for rings of integers in number fields, probably standing for "order" ("Ordnung" in German). $\endgroup$– AngeloCommented Jun 11, 2020 at 7:25
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6$\begingroup$ Four comments, four different answers... I guess that proves this question is worthy of an authoritative answer! $\endgroup$– WojowuCommented Jun 11, 2020 at 9:31
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2$\begingroup$ the paper by Bernstein, Gelfand & Gelfand that introduces category O just says: "We shall call this category of $g$ modules the category O." $\endgroup$– Carlo BeenakkerCommented Jun 11, 2020 at 18:31
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5$\begingroup$ Someone (Mirkovic? I'm not sure) told me that when they discovered it, they said "Oh, that's the right category!" This was probably a joke, but is my preferred explanation. $\endgroup$– Ben Webster ♦Commented Feb 4, 2021 at 2:35
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1 Answer
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From [Humphreys: Representations of semisimple Lie algebras in the BGG category O], notes for Chapter 1:
The letter chosen to label the category is the first letter of a Russian word meaning “basic”
which is основной.
