Timeline for Why do Lie algebras pop up, from a categorical point of view?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| S Jul 29, 2014 at 20:40 | history | suggested | Marcus Johnson | CC BY-SA 3.0 |
made link clickable and corrected noun
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| Jul 29, 2014 at 20:38 | review | Suggested edits | |||
| S Jul 29, 2014 at 20:40 | |||||
| Jul 28, 2014 at 15:40 | comment | added | Hiro Lee Tanaka | @Tom: Hmm. I'm not sure how I should answer your question. For starters, I bet Quillen uses somewhere that finite products and coproducts are the same in chain complexes, and that tensor product distributes over these. These are the typical kinds of "linear structure" I had in mind. And isn't his passage between dgLie and dgCoComm the Koszul duality between Chevalley-Eilenberg chains and primitives? (I might be misremembering.) As for the "co," if you think of Koszul duality as like bar-cobar duality, Bar turns algebras to coalgebras. Likewise, Koszul turns Lie to CoComm. | |
| Jul 25, 2014 at 17:38 | comment | added | Matthias Künzer | @HiroLeeTanaka: You're talking about operads now, I suppose. Yes, that duality (due to Loday?) looks natural. | |
| Jul 25, 2014 at 17:36 | comment | added | Matthias Künzer | @Allen Knutson: Yes. Does this extend to a categorical point of view (in a natural way)? | |
| Jul 25, 2014 at 16:33 | comment | added | Tom Goodwillie | @Hiro: I'm mixed up about the "co-". Where does Quillen's equivalence between differential graded Lie algebras and differential graded (co)commutative co-algebras (over $\mathbb Q$) fit into what you are saying? | |
| Jul 25, 2014 at 16:06 | answer | added | Urs Schreiber | timeline score: 12 | |
| Jul 25, 2014 at 15:09 | comment | added | Allen Knutson | The derivations of an algebra form a Lie algebra. | |
| Jul 25, 2014 at 12:11 | comment | added | Hiro Lee Tanaka | @Matthias, In my experience, Lie algebras pop up usefully when your category has some sort of linear structure. (Additive is a good place to start; better if you have something k-linear, or something stable.) In this setting, (coLie)Lie is Koszul dual to (commutative)cocommutative, and Koszul duality for augmented objects gives the interpretation of Lie objects as infinitesimal objects studied "at a point." I don't think the Koszul dual to cocommutative algebras in, say, spaces or sets (with whatever monoidal structure) has such a rich interpretation, though I'd be happy to hear about one! | |
| Jul 25, 2014 at 12:02 | comment | added | Hiro Lee Tanaka | @QiaochuYuan That's true, but in an additive category, the Ab-enriched structure is determined by the categorical property. | |
| Jul 25, 2014 at 10:01 | comment | added | Qiaochu Yuan | Additive is a stronger condition than you need; you only need $\text{Ab}$-enriched. | |
| Jul 25, 2014 at 9:56 | answer | added | Qiaochu Yuan | timeline score: 37 | |
| Jul 25, 2014 at 8:34 | history | asked | Matthias Künzer | CC BY-SA 3.0 |