From: Gerard Westendorp Subject: Dynkin diagrams Date: Thu, 27 Jul 2000 03:01:22 GMT Newsgroups: sci.physics.research Summary: [missing] I am trying to develop an intuitive understanding of Dynkin diagrams. These diagrams look pretty simple, they are just circles connected by 1, 2 or 3 lines, for example: o | o---o---o---o---o This diagram is supposed to depict the Lie group E6. All Lie groups can be classified by Dynkin diagrams. To figure out the idea behind them, I could read a book, or ask the newsgroup. I'm doing both. Anyway, so far I've understood it has something to do with ladder operators. The good thing about ladder operators (or the Cartan canonical form as the incrowd says) is that they offer a simple picture of the structure of a group. For example, consider the group SU(2), which describes spin (and a lot of other things, but lets just consider spin). SU(2) is a 3 dimensional group, in that the elements of the group can be expressed as a linear combination of 3 generators. You can take one of the generators, call it the operator for spin in the z-direction, and rewrite the others so they are "ladder operators". A ladder operator flips a spin (1/2) in the z-direction from up to down or vice versa. So if you have a state with spin j, the ladder operators "click" the state from j to -j in steps of 1 spin flip. This means you are describing a 3D continuous group with a 1D discrete group, which seems a simplification. The simplification gets better as you move on to higher groups. For higher groups, the ladder operators are a bit more complex. Instead of just a 1D "ladder", you get an n dimensional "lattice". the weird thing about these lattices is that the angles and lengths of the lattice are fixed. The angels can only be 90, 120, 135 or 150 degrees. This is where the Dynkin diagrams come in. They somehow describe the structure of the lattice. I hope to figure this out soon. But maybe a someone will intervene and explain it better, and point out the mistakes in the above... Gerard ============================================================================== From: Chris Hillman Subject: Re: Dynkin diagrams Date: 27 Jul 2000 19:21:57 GMT Newsgroups: sci.physics.research On Thu, 27 Jul 2000, Gerard Westendorp wrote: > I am trying to develop an intuitive understanding of Dynkin > diagrams. > > These diagrams look pretty simple, they are just circles I'll call them "nodes" rather than "vertices" to head off a possible levels of structure confusion. > connected by 1, 2 or 3 lines, You mean single, double, or triple "bonds" between vertices :-/ > for example: > > o > | > o---o---o---o---o Which has a bunch of single "bonds" connecting some of the nodes. > This diagram is supposed to depict the Lie group E6. The (complex) Lie algebra, actually, or better yet, the associated root lattice. The nodes represent "root vectors", and the bonds encode the angles between these vectors. (If there is no bond between two nodes, the corresponding vectors are orthogonal. The other possible angles are coded by single, double, or triple bonds.) The basic idea is that these diagrams are convenient way to encode the geometry of the immediate neighborhood of any vertex in the root lattice, and therefore of the lattice as a whole. > All Lie groups can be classified by Dynkin diagrams. More properly called Coxeter-Dynkin diagrams, since Coxeter used them to classify "reflection groups" and regular polyhedra -before- Dynkin reinvented them to rederive Cartan's classification of complex Lie algebras. (Cartan's original proof was apparently very complicated; Dynkin's derivation is much simpler.) > To figure out the idea behind them, I could read a book, Try the first third of the book Lectures of Lie Algebras and Lie Groups, LMS Student Series, University of Cambridge Press. Then see Arnold, Catastrophe Theory, Springer, for an interesting discussion of the somewhat mysterious "universality" of the ADE pattern found by Cartan. (Before you ask, the A, D, E do not stand for "Arnold", "Dynkin" or "Evgeny" or even (?) "Elie", but refer to Cartan's original letter classification, in which E was used for the "exceptional" Lie algebras which don't fit into the countably infinite families which take care of the generic case.) Chris Hillman (which took complex analysis from Dynkin, but which still only knows [a tiny bit] about the "A" in "ADE") Home Page: http://www.math.washington.edu/~hillman/personal.html ============================================================================== From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Dynkin diagrams Date: 27 Jul 2000 23:29:13 GMT Newsgroups: sci.physics.research In article <397E2365.849A33CE@xs4all.nl>, Gerard Westendorp wrote: >I am trying to develop an intuitive understanding of Dynkin >diagrams. Good!!! They are lots of fun and very useful in particle physics and the like. I immodestly recommend the following issues of This Week's Finds as a place to get started: http://math.ucr.edu/home/baez/week62.html http://math.ucr.edu/home/baez/week63.html http://math.ucr.edu/home/baez/week64.html http://math.ucr.edu/home/baez/week65.html >These diagrams look pretty simple, they are just circles >connected by 1, 2 or 3 lines, for example: > > o > | >o---o---o---o---o > >This diagram is supposed to depict the Lie group E6. All >Lie groups can be classified by Dynkin diagrams. Not all Lie groups - just the semisimple ones, which are all direct sums of these guys: SO(n)'s, SU(n)'s, SP(n)'s, and 5 exceptional guys called G2,F4,E6,E7, and E8. The SO(n)'s describe rotations in real vector spaces; the SU(n)'s and SP(n)'s do a similar job for complex and quaternionic vector spaces, and the 5 exceptional groups are related to the octonions. >Anyway, so far I've understood it has something to do >with ladder operators. The good thing about ladder >operators (or the Cartan canonical form as the in crowd >says) is that they offer a simple picture of the structure >of a group. >For example, consider the group SU(2), which describes spin >(and a lot of other things, but lets just consider spin). Right, I bet you understand the SU(2) case. That corresponds to the Dynkin diagram with a single dot, so we get a single raising operator J+, a single lowering operator J-, and a single operator J_z corresponding to the z component of angular momentum - all satisfying the usual commutation relations. Standard quantum mechanics stuff! If we have lots of dots, it's a wee bit more complicated. First, we get one J+, one J-, and one J_z for each dot. This just comes from the fact that we can pack a bunch of copies of SU(2) into a bigger semisimple Lie group. For example, it should be sort of obvious that we can stuff a bunch of SU(2)'s into SU(n). The rules of this game say we must fit these SU(2)'s into our bigger group in such a way that the J_z's for all our different SU(2)'s all COMMUTE with each other. I bet you can see how to pack n-1 SU(2)'s like this inside SU(n). That's why the Dynkin diagram for SU(n) has n-1 dots! But there's a catch: the J+'s and J-'s corresponding to different dots don't commute! The lines in our Dynkin diagram tell us their commutation relations. If you want to grok this, I strongly urge that you work out the SU(n) case in some detail: see how the raising and lowering operators for the different copies of SU(2) fail to commute. And look up the "Serre relations" in a textbook - those are the precise formulas for the commutation relations of the raising and lowering operators, in terms of the Dynkin diagram. >For higher groups, the ladder operators are a bit more complex. >Instead of just a 1D "ladder", you get an n dimensional "lattice". >the weird thing about these lattices is that the angles and lengths >of the lattice are fixed. The angles can only be 90, 120, 135 or >150 degrees. > >This is where the Dynkin diagrams come in. They somehow describe >the structure of the lattice. Right. To really understand this lattice stuff, especially the magic angles you mention, it's helps to start by thinking about Platonic solids, crystals, and "finite reflection groups". Then the trick is to relate this stuff about discrete symmetry groups to Lie groups! This may seem mysterious - what do discrete symmetry groups have to do with *continuous* symmetries? - but it's actually not very hard. Deep in the heart of every semisimple Lie group (or more precisely its Lie algebra), there is a lattice of crystalline beauty. I explained all this stuff in This Week's Finds. ============================================================================== From: abergman@princeton.edu (Aaron Bergman) Subject: Re: Dynkin diagrams Date: Fri, 28 Jul 2000 02:15:22 GMT Newsgroups: sci.physics.research In article <397E2365.849A33CE@xs4all.nl>, Gerard Westendorp wrote: >I am trying to develop an intuitive understanding of Dynkin >diagrams. > >These diagrams look pretty simple, they are just circles >connected by 1, 2 or 3 lines, for example: > > o > | >o---o---o---o---o > >This diagram is supposed to depict the Lie group E6. All >Lie groups can be classified by Dynkin diagrams. All complex semisimple Lie algebras, actually. When you look at the groups, things can get a bit more complicated. Anyways, here's my short short version of Lie algebras. First, we look for the maximal commuting subalgebra. Call this 'h'. Let's also choose a basis for the Lie algebra 'g', G_i. Then, for any H in 'h', we can look at the matrix a_ij: [H, G_i] = \sum_j a_ij G_j It's easy to see that these matrices all commute with eachother, so we can simultaneously diagonalize them. Then, we can choose a basis such hat [H_i, G_j] = A_j[H_i] G_j So, to each G_j, we can associate this A_j object that eats an element of 'h' and gives out a number, so it is a vector in the dual space of 'h' denoted 'h^*'. Now, you show that these objects, called roots, form a lattice in 'h^*' and you can define an angle between them. You draw a line through the origin that divides the set of roots into two, the positive and negative roots. Then, you want to choose a set of roots such that none of them is the sum of any two other roots. These are the simple roots. For each one, you draw a dot. The number of lines connecting the dots has to do with the angles between the pair of roots. This is useful because you can prove that there are some properties that classify all the possible diagrams you can draw, and so you've classified all possible semisimple Lie algebras. Aaron -- Aaron Bergman