Talk:Root system

Counterexample
The article claims:

"The integrality condition also means that the ratio of the lengths (magnitudes) of any two roots cannot be 2 or greater, since otherwise either the projection of the shorter root onto the longer root will be less than half as long as the longer root, or the shorter root will be exactly half the longer root or its negative."

This can generally not be true, since it is easy to construct a simple counterexample: choose two roots $$\alpha$$ and $$\beta$$ to be perpendicular to each other. These will trivially satisfy the conditions 1 to 4 for arbitrary lengths of $$\alpha$$ and $$\beta$$ ! But on the other hand, this seems to be the only counterexample so I will just insert this restriction into the article. —Preceding unsigned comment added by 130.75.25.73 (talk) 15:49, 14 April 2010 (UTC)

Giving this another thought, I come to the conclusion, that the entire statement is not true in general. As it is misleading, I will delete it. 130.75.25.73 (talk) 14:56, 15 April 2010 (UTC)

Remark: I advise anyone able to understand German to refer to: http://de.wikibooks.org/wiki/Beweisarchiv:_Lie-Algebren:_Wurzelsysteme:_Klassifikation_von_Wurzelsystemen. 130.75.25.73 (talk) 08:05, 16 April 2010 (UTC)

Simple Lie groups
Any particular reason for removing


 * Simply connected compact Lie groups which are simple modulo their centers?

Michael Larsen 11:39, 24 Oct 2003 (UTC)

I've merged in now what was on Dynkin diagram. I think that - and then moving out the Dynkin diagrams from the simple Lie group page - is much preferable to merging the latter here.

Charles Matthews 17:43, 3 Dec 2003 (UTC)

Root systems and Lie theory
I think, it would be good to have more specific information on the classification of simple complex Lie groups and / or "Simply connected complex Lie groups which are simple modulo centers". Perhaps an extra page for these classifications issues?

Perhaps it would be an idea to delete the link to "Simple Lie-Algebra" in this section, as it redirects to "Simple Lie groups"?


 * I suggest it is best to keep the focus on root systems and keep Lie theory out of the root systems article, since root systems are not merely a corner of Lie theory but have other reasons for interest. It makes sense to explain connections with Lie theory but not to do Lie theory in this article. Zaslav (talk) 02:10, 10 March 2009 (UTC)

One comment
I believe that there should be a mention to the other way of drawing Dynkin diagrams - that is, with single edges everywhere and numbers indicating the angle. Regards --132.205.159.206 19:20, 5 May 2006 (UTC)

You can find out about those if you follow the link to Coxeter-Dynkin diagram. They aren't quite the same thing though - you're allowed to label an edge with any integer from 4 upwards, and also with an "infinity" symbol. Any Coxeter-Dynkin diagram that corresponds with a Dynkin diagram, however, will have only 4 or 6 as a label on any labelled edge. Also, Coxeter-Dynkin diagrams don't include any information about relative lengths of roots - those arrows on the Dynkin diagrams are crucial. The Coxeter-Dynkin diagrams of types B and C are identical. You can find some Coxeter-Dynkin diagrams in the article on Coxeter groups. There you'll find that the classification of finite irreducible Coxeter groups is confusingly similar to that of irreducible root systems, but with a few differences here and there. 137.205.31.198 (talk) 21:26, 10 January 2009 (UTC)

In the news
Does the following article relate to this page?


 * NPR: "Team Solves Mammoth, Century-Old Math Problem": "Scientists have solved one of the toughest problems in mathematics, performing a calculation to figure out the symmetry of a 248-dimensional object known as the Lie group E8."

&mdash; Chris53516 (Talk) 15:06, 23 March 2007 (UTC)


 * No. The E_8 project (as described by the American Institute of Mathematics) concerns the Lie group, not the root system. Zaslav (talk) 02:14, 10 March 2009 (UTC)

Positive definite
This article references the term positive definite, which is a disambiguation page. Please review this usage and determine which of the articles at the disambiguation is intended and adjust as appropriate. Chromaticity 02:25, 7 May 2007 (UTC)
 * Done. It was referring to an inner product, which is a bilinear form.  Though, I believe that since this article is only talking about Euclidean spaces, any of the three definitions of "positive definite" would apply. 66.117.137.139 (talk) 23:01, 17 November 2007 (UTC)

Question
The article says: "The set of simple roots is a subset &Delta; of &Phi; which is a basis of V with the special property that every vector in &Phi; when written in the basis &Delta; has either all coefficients &ge;0 or else all &le;0.". However, I think there is a mistake, because that is not possible. For example, for a root $$\alpha$$ in &Delta;, the root $$-\alpha$$ may never has a coefficient &ge;0 in a basis of V. —Preceding unsigned comment added by 88.24.73.176 (talk) 09:17, 22 September 2007 (UTC)


 * If &Delta; = (&alpha;1,...&alpha;i,...&alpha;n), then the root -&alpha;i has coordinates (0,...0,-1,0...0) in the basis &Delta;, whose coefficients are all &le;0. --JWB 12:11, 22 September 2007 (UTC)

You are totally right. I misread that either all roots had coefficients &ge;0 or all roots had coefficients &le;0. Sorry for my english. —Preceding unsigned comment added by 88.23.98.119 (talk) 14:10, 23 September 2007 (UTC)

Lie groups in "most parts of mathematics"?
What's the basis for the claim that "Lie groups... have come to be used in most parts of mathematics"? What does that even mean? Is it saying that more than 50% of the world's current professional mathematicians use Lie groups in their daily work? More than 50% of all articles published in peer-reviewed mathematical journals mention Lie groups? I doubt that either of these is actually true. 66.117.137.139 (talk) 22:38, 17 November 2007 (UTC)


 * Just to clear it up (retrospectively): The claim is hyperbolic. Zaslav (talk) 02:16, 10 March 2009 (UTC)

Reduced root systems
Correction: The definition of root system given in the article is at odds with standard definitions. The inclusion of property 2 in the definition makes it a "reduced" root system. Standard theory counts the non-reduced root systems of BC-type.

Also, a (related) comment: In view of the fact that the systems of the BC family are the only non-reduced, irreducible root systems, perhaps they should be mentioned. —Preceding unsigned comment added by 192.167.204.11 (talk) 14:33, 23 June 2008 (UTC)


 * Non-reduced root systems are also important for twisted groups of Lie type, though they are sometimes avoided. I believe Bourbaki has a somewhat standard treatment of "property 2 is almost superfluous except for the following short list of exceptions" (which I think is slightly longer than just BC, but only in very low ranks, or possibly not assuming the crystallographic restriction or something). JackSchmidt (talk) 03:01, 25 July 2008 (UTC)


 * It is not "at odds with standard definitions" to assume property 2. Some authors assume it and some do not.  It depends on the purpose.  I've added the term "reduced" and an explanation. Zaslav (talk) 02:29, 10 March 2009 (UTC)

Wrong definition of the simple roots ?
I'm not used to the abstract root system, but in the case of a Lie algebra, the definition given here does not fit what I know. It is said, on the one hand, that $$\Phi$$ is finite. And, on the other hand, it is said that $$\Phi^+$$ is a subset of $$\Phi$$ such that
 * For any $$\alpha, \beta\in \Phi^+$$ such that $$\alpha+\beta$$ is a root, $$\alpha+\beta\in\Phi^+$$.

But since $$\phi$$ is finite, $$\alpha+\beta+\beta+\beta+\beta+\ldots$$ will finish to get out of $$\Phi$$.

In the Lie algebra setting, if $$\mathfrak{H}$$ is the Cartan algebra, $$\Phi$$ is a finite subset of the dual of $$\mathfrak{H}$$. Then one take a positivity notion on $$\mathfrak{H}$$, and one put $$\Phi^+=\mathfrak{H}^*\cap\Phi$$.

I do not know how that works in the abstract root space case, but something has to be changed in the article because, as it, it is wrong. —Preceding unsigned comment added by 146.186.131.84 (talk) 02:27, 25 July 2008 (UTC)


 * The way it is written is very standard, but easy to misread. Assume a,b are positive roots. If a+b is a root, then it is a positive root, but there is no reason to think it is a root (usually it is not, and certainly a+b+b+b+b+b is not). JackSchmidt (talk) 02:58, 25 July 2008 (UTC)

Redirection Fault
I typed in 'Dynkin Diagram' and I was redirected to this page. There is a page called 'Coxeter-Dynkin diagrams'. Surely that's where I should have been redirected to. How does one change redirects? Dharma6662000 (talk) 02:58, 20 August 2008 (UTC)

Wrong definition of positive roots for E6, E7, E8
The root 1/2(e_1+...+e_8) is negative. It should be -1/2(e_1+...+e_8) instead. That means we also have to fix the pictures.. —Preceding unsigned comment added by 18.194.1.116 (talk) 07:32, 5 December 2008 (UTC)

Just noticed that myself. I've managed to figure out how to fix it. 137.205.31.198 (talk) 20:49, 10 January 2009 (UTC)

F4
I'm reluctant to edit the article directly but I think there is a typo in the section on F4. I think the reference to B3 should be replaced by a reference to B4. Typometer (talk) 15:55, 4 April 2009 (UTC)


 * I think it should be B3. I think F4 contains B4 as a subsystem, but the sentence in question is talking about the *simple* roots, not the roots as a whole.  F4 should be given 4 simple roots, B3 has 3, so B3 and one more is 4. B4 and one more is 5, so cannot be correct. JackSchmidt (talk) 19:10, 4 April 2009 (UTC)


 * Ah, right you are. Sorry about that! Typometer (talk) 20:23, 5 April 2009 (UTC)

Inconsistency
Weyl group claims that the Weyl group to A2 is the dihedral group of order 12 while the table included in this article claims it to have (2+1)!=6 elements. Rgds --Boobarkee (talk) 13:45, 7 July 2009 (UTC)
 * The error is on the Weyl group side. Sorry. --Boobarkee (talk) 13:48, 7 July 2009 (UTC)

Classification
Where is the proof that A-G are the only possible connected Dynkin diagrams? --JWB (talk) 20:51, 19 December 2009 (UTC)

Is this the correct definition of equivalent root systems?
In the Definitions section, this sentence occurs:

"Two irreducible root systems (E1, Φ1) and (E2, Φ2) are considered to be the same if there is an invertible linear transformation E1 → E2 which sends Φ1 to Φ2."

Is this correct?

That's it? Just any old invertible linear transformation? Considering the definition of a root system in terms of inner products, I would expect that such a linear transformation be required to preserve ratios of inner products.Daqu (talk) 17:46, 29 September 2010 (UTC)


 * You're right; I've fixed this, and added a reference to Humphreys. Cheers, Mark M (talk) 12:08, 27 November 2012 (UTC)

Merger with Root system of a semi-simple Lie algebra
As far as I understand, the most important instance of a root system is one that arises from a semisimple Lie algebra. While I understand that the notion of "root system" is independent of Lie theory, it makes much more sense and is also useful for the readers to have a single article denoted to "root system" and "root system" in Lie theory. This leads me to propose that we merge Root system of a semi-simple Lie algebra into this one. The latter article is also not long and I don't think it gets much longer in the future too. -- Taku (talk) 06:49, 18 January 2020 (UTC)


 * Actually, for some weird reason, the article "Root system of a semi-simple Lie algebra" actually mostly discusses the generators and relations of a semisimple Lie algebra and relations (which are important). Thus, I have merged it into semisimple Lie algebra, which previously had no discussion of generators and relations. -- Taku (talk) 02:33, 21 January 2020 (UTC)

G2 labeling
Comparing

Rank two examples

with

Positive roots and simple roots

it appears that G2 should be defined with n=4, not 3 as stated.

Darcourse (talk) 08:49, 13 February 2023 (UTC)


 * I do not know what comparison you think you are making, but certainly you are mistaken if you believe $$\sigma_\alpha(\beta) = \beta + 4\alpha$$ in $$G_2$$. --JBL (talk) 20:08, 13 February 2023 (UTC)