Dynamics on Graphs 18

With the introduction of the Cartan matrix of a graph, and the associated Tits quadratic form, we have a new rich geometry automatically associated to a general graph. To begin to appreciate the beauty of this situation, we look at a very simple example which plays a crucial role in modern physics. This is the graph A2 which corresponds to the simple Lie algebra sl(3) and to the compact Lie group SU(3), which generates both flavour symmetry and colour symmetry in the Standard Model of particle physics.

Although this certainly seems to be a super simple example (the graph has only 2 vertices!) we will see that the associated geometry involving roots and weights and also Weyl chambers is quite lovely.

And we adopt a more general view than usual to this story, so that even if you have studied classical root systems in a Lie theory or representation theory course, you will be seeing things in a different, and more natural, fashion in this lecture! This superior point of view will enable the introduction of Rational Trigonometry more smoothly, and also simplifies a lot of calculations of interest to physicists as we will see in subsequent lectures.

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Hello everyone I'm Norman Wildberger thanks for joining me on this adventure we're looking at

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Dynamics on graphs trying to understand Lee Theory and exceptional structures from a very simple

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Elementary point of view by starting with graphs as a fundamental objects and playing the mutation

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in numbers games so last time we introduced a new element into this story that of geometry

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via the tits quadratic form today we're going to exhibit all of that in a very explicit but simple

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case the case of the diagram of the graph A2 now A2 is a very modest looking graph and you might

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think well it's almost devoid of interest but actually it's hugely interesting because it turns

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out that the associated lead group or symmetry group is su3 or some sl3 and this group plays a

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crucial role in modern physics it's basically the Symmetry group of the strong nuclear force and so

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responsible for interactions of quarks and and more generally somehow intimately connected with

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3 three-fold symmetry in a in a very profound way so today we're going to exhibit routes and

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weights for A2 but also the geometrical aspect will come before with the introduction of wild

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Chambers okay so on the left hand side of this story here we have a population a1a2 we're going

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to play the mutation we see before that we can sort of play this in a in a kind of a

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two-sided way we can do first S1 and then S2 and then S1 or we could do S2 and S1 and S2 and you

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could just check that that these are actually the populations that we get so we're going from

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population A1 A2 to population minus A2 minus A1 now I don't think I've shown you the uh the

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Dual side um if here's uh weights W1 W2 and we perform the Dual mutations or firings then an

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S1 will send this thing to minus W1 and W1 plus W2 another further S2 to this one and

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S1 to minus W2 minus W1 and if we go the other direction S2 S1 S2 we arrive at the same thing

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okay so um you can check that these are sort of complementary this is the mutation side of

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things this is the Dual mutation side of things here we're dealing with populations here we're

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dealing with weights and when we're working with matrices or vectors we generally represent these

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populations with row vectors and the weights with column vectors so we notice that we're exhibiting

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algebraically the braid relation that the effect of S1 followed by S2 followed by S1 is equal to

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S2 followed by S1 followed by S2 in fact we can sort of read this either from left to to right or

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from right to left sorry depending on uh whether we're talking about mutations or dual mutations

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now to connect with some of the things that we've been talking about in previous videos here is the

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graph again with just the vertices labeled one and two so that's the ordering that we're going

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to use so using that ordering the adjacency Matrix of the graph is just a simple two by

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two Matrix zero one one zero the carton Matrix which is two times the identity minus this a has

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this value here two one bar one bar two where I remind you one bar is the same as minus one

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just written a little bit differently maybe a little bit more logically and then we have

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the mutation matrices okay they are obtained from from from these things um so here's S1

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with one bar zero one one and S2 which is one one zero one bar and we can check that you know

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the actions of these matrices on vectors uh really agrees with what we've been just talking about

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formally and I want to emphasize this sort of somewhat unique point of view that we're adopting

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here is that S1 and S2 are to be thought of here as matrices principally not transformations

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okay so it's it's these objects as matrices that we're really manipulating so for example

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um if we have look at this S1 this S1 can act on the left on a population if we do that we just get

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um what I've already shown you up here on this uh previous diagram

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from going from here to here okay that was the action of S1 on the

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the row Vector acting on the right on the other hand if we take this same Matrix S1

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and act on the left on a column Vector which is our representation algebraically of the

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weights W and W2 then we get well what we ended up getting on the right hand side of the diagram

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so this illustrates that these matrices have intrinsically a two-sided nature if you have a two

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by two Matrix it acts either on row vectors on the left or on column vectors on the right and often

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modern mathematics breaks that Symmetry and looks predominantly at one of those and then ignores

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the other here we want to keep both of them in mind because the actions are somewhat different

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but they complement each other and they're intimately connected okay and similarly you could

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verify the same thing for S2 okay so last time we introduced the tits quadratic form on populations

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allowing us to introduce geometry into the situation so if we have a population P then the

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quadrants of p is by definition this combination Vector P times the carton Matrix times P transpose

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and we saw that this had the very important property in general that the quadrants of

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what you get when you apply a mutation to the population is the same as the quadrants of the

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population itself so the mutations don't change the quadratic form and explicitly in this case

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in the A2 case Okay this uh here's the carton Matrix and so the quadratic form looks like

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if we apply to a vector a1a2 we get 2a1 squared plus 2a2 squared minus 2 A1 A2

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now in Geometry quadratic forms are associated to symmetric bilinear forms there's a sort of

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a correspondence so a symmetric bilingual form more informally called a DOT product we might

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use that term dot product and overall I think it's safe to say that really the dot product is

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somehow more fundamental than the quadratic form the quadratic form is very useful and important

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but maybe underlying it you should think of the symmetric bilinear format being somewhat more

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basic somewhat more fundamental so let's introduce that so there's a corresponding dot product and

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has the property that if we take a vector dotted with itself then we get back this quadratic form

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the quadrants of P . so in terms of matrices this is very simple to Define we just sort of enlarge

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the scope so instead of defining just P dot P now we have P dot Q so p and Q are both population

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vectors they're both say row vectors then we can Define their dot product to be P times c times Q

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transpose so obviously if Q equals P then we're recovering the tits quadratic form and we can

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call this the tits dot product and it's acting on populations and what does it look like explicitly

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for A2 well uh here it is the same thing here except now P1 P2 on the left and q1 Q2 on the

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right let me expand this all out we get two P1 q1 plus 2 p 2 Q2 correspond to the diagonal elements

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minus P1 Q2 okay of corresponding to that minus one minus P2 q1 corresponding to that minus one

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okay so each of the entries of the matrices sort of contributes a term here

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now um this this expression okay it's bilinear in the following sense that if you take just one of

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these vectors and look at the dependence on P1 and P2 it's actually linear in the coordinates

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of that one vector so as an expression in P1 and P2 this is linear and simultaneously if you look

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at it as an expression just in q1 and Q2 then it's also linear so it's a bilinear expression

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okay the the dot product has many uses but principally so initially what you do is you

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use it to Define orthogonality this is sort of where the geometry comes in okay so we Define

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an orthogonality of vectors if we have a general dot product P dot Q by saying that the vectors p

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and Q are orthogonal or perpendicular if you like precisely when their dot product is zero

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so then once we have this orthogonality then given a vector P we can look at the

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property of another Vector to be orthogonal to it that defines a hyperplane in general

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so this is very general thing if you have a vector p and you have

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a DOT product in some linear space then you can Define the associated hyperplane

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which is essentially the equation P dot x equals zero where X is an arbitrary vector

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and typically if you're in some higher dimensional space you have a vector okay then the hyperplane

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is you know is a is actually a a relatively big space which is perpendicular to the vector but

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perpendicular in the sense of the particular dot product that you've defined it doesn't always have

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to look like this okay it can look uh in different forms depending on the nature of the dot product

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now in our situation we have these distinguished vectors called roots or root populations so then

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we have distinguished hyperplanes called H sub P are root hyperplane if p is a root population

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and pretty clearly if we replace the vector p with its negative then this condition hasn't

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substantially changed so we're getting the same hyperplane so we get a sort of

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a hyperplane for every pair of vectors a vector and it's negative so that means in A2 there are

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going to be three hyperplates and here they are algebraically okay so if we look at this

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by linear form this is the bilinear form we're looking at it right and we say replace P1 P2

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with the vector 1 0 and then see you know what what do we get as a function of q1 Q2

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or perhaps X1 X2 so we take one zero dot X1 X2 and you have a look at that

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by linear form then you see you're getting the linear equation 2x1 minus X2 equals zero which

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is indeed the equation of a plane in a hyperplane in in the plane which is just a line if you

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look at the Hybrid plane Associated to the vector 0 1 and 0 1 dot X1 X2 is 2x2 minus X1 so that's

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another line that we get that's associated to this population and the other positive root population

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is one one if we take the hyperplate associate to that we get 2x1 plus 2x2 minus X1 minus X2 equals

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zero or simply X1 plus X2 equals zero so these are the equations of the three lines which are

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perpendicular to our three positive roots with respect to the tits bilinear form or dot product

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what does it look like well now we're getting some geometry okay so have a look at this

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so here is our familiar plane with an X1 axis this way and X2 axis this way and here in blue

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we see the root populations okay so this one here corresponds to the vector 1 0 this is

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the vector zero one this is the vector one one and over here down here are the negatives of it

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I've also shown you the quadrants Q of P equals two Circle

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okay if you set the quadrants equal to a particular value you're getting a circle in that

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Associated geometry now you may think well this is not looking like a circle it looks like an ellipse

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yes but it's a circle in the geometry determined by the tits quadratic form or tits dot product

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now I have to say that you know in this is a very important example it's sort of like the

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simplest example in Lee Theory and some sense of the simplest example of a root system and

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Associated lots of things so this diagram or variant of it is found in many textbooks and

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many papers but you almost never find it this way you will typically find it

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um with vectors which are all at 60 degrees to each other so forming sort of a regular hexagon in

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the plane okay um we don't want to do that okay because that makes it look like we're talking

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about the ordinary Euclidean geometry and we're not talking about the ordinary Euclidean geometry

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because this tits quadratic form is clearly not the same one as the Euclidean one right

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I mean that that's not a Euclidean quadratic form now it could potentially be sort of transformed

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into a Euclidean quadratic formulas we could perform some kind of linear transformation that

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try to get it into a different form but as it is if we just look at it as it is without any

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other transformation this is what it looks like so this is actually in my view a more

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um real picture of the A2 root system and I want to Advocate this picture Okay so from

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now on when you're making calculations with A2 physicists you can you know use this point of

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view so very important for the the Su su3 and uh strong interaction story okay and now I've also

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shown you these root hyperplanes that we've just identified so the the perpendicular the hyperplane

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associated to the vector one zero is this line here it has equation 2x1 minus X2 equals zero so

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in particular it goes through the point one two this point up here so this slanted green line

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is the hyperplane associated to this Vector here and there's another way of thinking about this um

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from the from the point of view of the circle so if you look at the Circle passing through

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this Vector here and you look at its tangent line just at that point okay that tangent line

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is going to be parallel to this that's another way of thinking about where this line comes from

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geometrically it's just the Direction formed by the tangent to the circle at the point in question

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so similarly if we look at this point here okay the tangent line Looks like this that's

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that line there which has the second equation that we have here 2x2 minus X1

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equals zero and the third one this point here its tangent line is like this okay which is

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parallel to this hyperplane with equation X1 plus X2 equals zero

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so note that these three hyperplanes that we have here one for every positive root divide the plane

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actually into six regions okay so there's a region here one and then a region here two and then three

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four five six regions and these regions are called vile Chambers after Hermann vile who was

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arguably the uh the most important 20th century mathematician

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in my view he was the most important 20th century mathematician let me put it that way

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okay so note that the vial group The the group generated by the mutation matrices

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so S1 S2 the identity Matrix let's call it D and then these various combinations S1 S2 s2s1

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and this uh triple s1s to S1 which happens to be equal to S2 S1 S2 so the vile group in this case

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also has exactly six elements and we'll see that that's not a surprise and that's a very sort

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of uh remarkable fact that the algebra or the group Theory contained in this story is being

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reflected by the geometry and the geometry of these these bile Chambers turns out to have a

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important connections with the algebra of the group Theory and the structure of the root systems

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so this is a very rich story and this is you know in some sense almost the the simplest non-trivial

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example it's already reasonably Rich here but as we go up as the graphs become more complicated

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this story becomes very very um complicated and and but still Rich okay but um don't be fooled

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by the relatively simple nature of what's going on here although what's going on here is still

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very important because uh this is the mathematics behind the uh the physics of quarks in some sense

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so besides having connections with with physics of course this also has major connections with

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areas of mathematics including coccidated groups because we're talking about the file groups and

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also a branch of combatorics called hyperplane arrangements okay so combinatorialists like to

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study Arrangements of hyperplanes and and how they subdivide space in the various relations

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and that's exactly what's happening here and of course with the representation theory

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of Lee algebra's and Lead groups which we're going to have to talk quite a lot more about.