The Unapologetic Mathematician

Mathematics for the interested outsider

Root Systems

Okay, now to lay out the actual objects of our current interest. These are basically collections \Phi of vectors in some inner product space V, but with each vector comes a reflection and we want these reflections to play nicely with the vectors themselves. In a way, each point acts as both “program” — an operation to be performed — and “data” — an object to which operations can be applied — and the interplay between these two roles leads to some very interesting structure.

First off, only nonzero vectors give rise to reflections, so we don’t really want zero to be in our collection \Phi. We also may as well assume that \Phi spans V, because it certainly spans some subspace of V and anything that happens off of this subspace is pretty uninteresting as far as \Phi goes. These are the things that would just be silly not to ask for.

Now, the core requirement is that if \alpha\in\Phi, then the reflection \sigma_\alpha should leave \Phi invariant. That is, if \beta is any vector in \Phi, then

\sigma_\alpha(\beta)=\beta-(\beta\rtimes\alpha)\alpha=\beta-\frac{2\langle\beta,\alpha\rangle}{\langle\alpha,\alpha\rangle}\alpha

is also a vector in \Phi. In particular, this means that we have to have \sigma_\alpha(\alpha)=-\alpha\in\Phi. But we don’t want any other scalar multiples of \alpha to be in \Phi, because they’d just give the same reflection again and that would be redundant.

Of course, we could just throw in more and more vectors as we need to make \Phi invariant under all of its reflections, and each new vector introduces not only new images under the existing reflections, but whole new reflections we have to handle. We want this process to stop after a while, so we’ll insist that \Phi is a finite collection of vectors. This is probably the biggest constraint on our collections.

We have one last condition to add: we want to ask that for every pair of vectors \alpha and \beta in \Phi, we have \beta\rtimes\alpha\in\mathbb{Z}. In other words, the length of the projection of \beta onto \alpha must be a half-integral multiple of the length of \alpha. This makes it so that the displacement from \beta to \sigma_\alpha(\beta) is some integral multiple of \alpha. This provides a certain rigidity to our discussion.

So, let’s recap:

  • \Phi is a finite, spanning set of vectors in V which does not contain 0\in V.
  • If \alpha\in\Phi then the only scalar multiples of \alpha in \Phi are \pm\alpha.
  • If \alpha\in\Phi then the reflection \sigma_\alpha leaves \Phi invariant.
  • If \alpha and \beta are in \Phi, then \displaystyle\beta\rtimes\alpha=\frac{2\langle\beta,\alpha\rangle}{\langle\alpha,\alpha\rangle} is an integer.

A collection \Phi of vectors satisfying all of these conditions is called a “root system”, and the vectors in \Phi are called “roots” for ABSOLUTELY ARBITRARY REASONS THAT HAVE ABSOLUTELY NOTHING TO DO WITH ANYTHING. As far as we’re concerned for now.

So yeah: “root system”. Just ’cause…

Our lofty goal, for the immediate future, is to classify all the possible root systems.

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January 20, 2010 - Posted by | Geometry, Root Systems

26 Comments »

  1. Oh boy. This should be interesting.

    Are you going to do the motivation of, say, classifying complex simple Lie algebras?

    Comment by Akhil Mathew | January 20, 2010 | Reply

  2. What are these “Lie algebras”, of which you speak? I just see a bunch of vectors being reflected around in regular old n-dimensional spaces. These “Lie algebras” sound big and scary.
    ;)

    Comment by John Armstrong | January 20, 2010 | Reply

  3. [...] Weyl Group of a Root System Let’s take a root system in the inner product space . Each vector in gives rise to a reflection in , the group of [...]

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  4. [...] Category of Root Systems As with so many of the objects we study, root systems form a category. If is a root system in the inner product space , and is a root system in the [...]

    Pingback by The Category of Root Systems « The Unapologetic Mathematician | January 22, 2010 | Reply

  5. [...] Root Systems We should also note that the category of root systems has binary (and thus finite) coproducts. They both start the same way: given root systems and in [...]

    Pingback by Coproduct Root Systems « The Unapologetic Mathematician | January 25, 2010 | Reply

  6. [...] Root Systems Given a root system , there’s a very interesting related root system , called the “dual” or [...]

    Pingback by Dual Root Systems « The Unapologetic Mathematician | January 26, 2010 | Reply

  7. [...] Root Systems Given a root system in an inner product space we may be able to partition it into two collections so that each root [...]

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  8. [...] of Roots When we look at a root system, the integrality condition puts strong restrictions on the relationship between any two vectors in [...]

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  9. [...] First, a lemma: given a root system , let and be nonproportional roots. That is, . Then if — if the angle between the vectors [...]

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  10. [...] for Root Systems We don’t always want to deal with a whole root system . Indeed, that’s sort of like using a whole group when all the information is contained in [...]

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  11. [...] Existence of Bases for Root Systems We’ve defined what a base for a root system is, but we haven’t provided any evidence yet that they even exist. Today we’ll not only [...]

    Pingback by The Existence of Bases for Root Systems « The Unapologetic Mathematician | February 2, 2010 | Reply

  12. [...] A very useful concept in our study of root systems will be that of a Weyl chamber. As we showed at the beginning of last time, the hyperplanes for [...]

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  13. [...] Lemmas on Simple Roots If is some fixed base of a root system , we call the roots “simple”. Simple roots have a number of nice properties, some of [...]

    Pingback by Some Lemmas on Simple Roots « The Unapologetic Mathematician | February 4, 2010 | Reply

  14. [...] of Irreducible Root Systems I Now we can turn towards the project of classifying irreducible root systems up to isomorphism. And we start with some properties of irreducible root [...]

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  16. [...] Graphs and Dynkin Diagrams We’ve taken our root system and turned it into a Cartan matrix. Now we’re going to take our Cartan matrix and turn it [...]

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  17. [...] of (Possible) Root Systems At long last, we can state the classification of irreducible root systems up to isomorphism. We’ve shown that for each such root system we can construct a connected [...]

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  18. [...] the Classification Theorem I This week, we will prove the classification theorem for root systems. The proof consist of a long series of steps, and we’ll split it up over a number of [...]

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  19. [...] so we must construct some actual root systems. For this task, we let stand for a finite-dimensional real vector space for various , equipped [...]

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  21. [...] of D-Series Root Systems Starting from our setup, we construct root systems corresponding to the Dynkin diagrams (for [...]

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  22. [...] of B- and C-Series Root Systems Starting from our setup, we construct root systems corresponding to the (for ) and (for ) Dynkin diagrams. First will be the [...]

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  24. [...] of E-Series Root Systems Today we construct the last of our root systems, following our setup. These correspond to the Dynkin diagrams , , and . But there are [...]

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  25. [...] laying down some definitions on reflections, we defined a root system as a collection of vectors with certain properties. Specifically, each vector is a point in a [...]

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  26. [...] Lie algebras, and/or algebraic groups; the former are very important in understanding the latter. John Armstrong over at the Unapologetic Mathematician has a series on root systems. In addition, for a [...]

    Pingback by Coxeter groups « Annoying Precision | June 27, 2010 | Reply


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