Timeline for Simple groups analogous to fields
Current License: CC BY-SA 3.0
8 events
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| Mar 13, 2013 at 17:56 | comment | added | Todd Trimble | @Aeryk: (1) the paradigmatic such case would be the classification of simple complex (ha ha) Lie groups as a motivation to study the classification of simple complex Lie algebras. (2) There are clear cases (as in Morita equivalence) that set up correspondences between simple objects. But insofar as quotient objects (for varieties of algebras) correspond to internal congruence relations, it may be more or less hard to understand simple objects obviously depends on how hard or easy it is to detect congruence relations. At this level of generality, I don't think I can say much more. | |
| Mar 13, 2013 at 16:23 | comment | added | Dima Pasechnik | The main motivation is to decompose complicated objects into simpler ones, using natural maps. In different categories the complexity of objects might be totally different. E.g. finite groups vs finite abelian groups. More interesting is perhaps the relationship between commutative algebra and algebraic geometry. There the correspondence between well-chosen "simple objects" is often 1-to-1. | |
| Mar 13, 2013 at 16:08 | comment | added | Aeryk | @Todd: So are there cases where statements about simple objects in one category were motivation to pursue those statements for other simple objects? What does it mean if the property of a simple object depends on the category? Like what Tom points out, why would the simple objects in some categories be so different than those in another? | |
| Mar 13, 2013 at 16:02 | comment | added | Aeryk | @quid: Yes, the quadratic extension. I was thinking (in my two examples) about non-finite fields (hence the rationals) and extension fields (hence the quadratic extension). | |
| Mar 13, 2013 at 15:06 | comment | added | Tom De Medts | I agree with Todd. The main observation here is that simple commutative groups and simple commutative rings become rather boring objects (from the point of view of group theory and ring theory, respectively), but simple groups and simple rings (non-commutative ones) are very interesting. | |
| Mar 13, 2013 at 14:40 | comment | added | Todd Trimble | There are also simple rings, simple modules, simple Lie algebras, and general simple objects in categories (especially in varieties of algebras) which are objects with no nontrivial quotients. A priori, you might as well open the field (no pun intended) to all such things. | |
| Mar 13, 2013 at 14:13 | comment | added | user9072 | What do you mean by the second example in 2. A (or rather the) quadratic extension of field with p elements or something else? | |
| Mar 13, 2013 at 13:56 | history | asked | Aeryk | CC BY-SA 3.0 |