Talk:Langlands decomposition
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Definition
[edit]The decomposition as given in the first sentence doesn't make much sense to me: all abelian groups are nilpotent as well, so one can just drop the "abelian group A" from the decomposition?! --Roentgenium111 (talk) 21:40, 13 February 2012 (UTC)
- I don't know about the parabolic case, but certainly decomposing a subgroup as AN is different from decomposing it as N. For instance, the non-abelian group of order 6 is the product of an abelian group of order 2 and a nilpotent group of order 3, but the product itself is not a nilpotent group. A might also be nilpotent, but AN need not be nilpotent. I assume M is a Levi subgroup, A is a semi-simple abelian subgroup, and N is a unipotent subgroup, but maybe that is called the Levi decomposition. JackSchmidt (talk) 22:22, 13 February 2012 (UTC)
- The non-abelian group of order 6 is not the product of groups of order 2 and 3 (otherwise it would be abelian), it's only a semidirect product of these. (And these subgroups are both abelian and nilpotent, so the decomposition would not be unique either.) I do think that the (direct) product of two nilpotent (one of them abelian, say) groups is always nilpotent: The commutator of a direct product of groups is the product of their respective commutators, so the product group is nilpotent of the same class as that of the non-abelian factor. --Roentgenium111 (talk) 20:28, 14 February 2012 (UTC)
- The product in the article need not be direct or even semi-direct. In group theory "product" just means every element can be written as a product in the specified order (but possibly in many ways, and none of the factors need be normal). For instance, every group is the product of its cyclic subgroups, and the simple group of order 60 is a product of subgroups of order 12 and 5, but neither subgroup is normal. The (MA):N part is probably always a semi-direct product. In fact the non-abelian group of order 6 is a parabolic subgroup in GL(2,3), and I would assume its decomposition is 1*2*3 where 1 is a reductive group of order 1, 2 is a semi-simple abelian group of order 2, and 3 is a unipotent subgroup of order 3 normalized by 2). JackSchmidt (talk) 22:10, 14 February 2012 (UTC)
- I see; I usually understand a product of algebraic objects always as a direct product unless explicitly stated otherwise, and thus misunderstood you and the article, sorry for that. Still, I would hesitate to call such a product MAN a "decomposition" unless at least the intersection of each pair of the factors is trivial; is this meant to be the case here? --Roentgenium111 (talk) 22:33, 14 February 2012 (UTC)
- It looks like M, A, and N have pairwise trivial intersections. MA and N definitely do (MA is the Levi factor, and N is the unipotent radical), but I haven't found a clear definition of M or A. Certainly something more specific should be added to the article. JackSchmidt (talk) 22:57, 14 February 2012 (UTC)
- Is there actually a definition of the "Levi factor" somewhere on WP, or could you give one? Levi decomposition only states the existence of a "Levi factor" (and this only for a Lie algebra), but gives no concrete definition. I suppose the semidirect product mentioned there for the Lie algebra translates to one for the Lie group as well. --Roentgenium111 (talk) 16:18, 19 February 2012 (UTC)
- I just found that Knapp's book on "Repres. theory of semisimple groups" gives (p. 132) a "Langlands decomposition" of a parabolic group S as S=MAN with multiplication giving a diffeomorphism M×A×N→S, so apparently the three groups have pairwise trivial intersections (that book states that A is abelian and N nilpotent, but doesn't seem to state that M be reductive).--Roentgenium111 (talk) 21:21, 28 February 2012 (UTC)
- It looks like M, A, and N have pairwise trivial intersections. MA and N definitely do (MA is the Levi factor, and N is the unipotent radical), but I haven't found a clear definition of M or A. Certainly something more specific should be added to the article. JackSchmidt (talk) 22:57, 14 February 2012 (UTC)
- I see; I usually understand a product of algebraic objects always as a direct product unless explicitly stated otherwise, and thus misunderstood you and the article, sorry for that. Still, I would hesitate to call such a product MAN a "decomposition" unless at least the intersection of each pair of the factors is trivial; is this meant to be the case here? --Roentgenium111 (talk) 22:33, 14 February 2012 (UTC)
- The product in the article need not be direct or even semi-direct. In group theory "product" just means every element can be written as a product in the specified order (but possibly in many ways, and none of the factors need be normal). For instance, every group is the product of its cyclic subgroups, and the simple group of order 60 is a product of subgroups of order 12 and 5, but neither subgroup is normal. The (MA):N part is probably always a semi-direct product. In fact the non-abelian group of order 6 is a parabolic subgroup in GL(2,3), and I would assume its decomposition is 1*2*3 where 1 is a reductive group of order 1, 2 is a semi-simple abelian group of order 2, and 3 is a unipotent subgroup of order 3 normalized by 2). JackSchmidt (talk) 22:10, 14 February 2012 (UTC)
- The non-abelian group of order 6 is not the product of groups of order 2 and 3 (otherwise it would be abelian), it's only a semidirect product of these. (And these subgroups are both abelian and nilpotent, so the decomposition would not be unique either.) I do think that the (direct) product of two nilpotent (one of them abelian, say) groups is always nilpotent: The commutator of a direct product of groups is the product of their respective commutators, so the product group is nilpotent of the same class as that of the non-abelian factor. --Roentgenium111 (talk) 20:28, 14 February 2012 (UTC)