 Is this clear? Is the notation clear? This is what is the notation for the SU 2 sub-algebra. There are 3 independent positive roots, so you will have 3 SU 2 sub-algebra ok. So, what I am trying to say is that if I do J 3 alpha 1 or let us say lambda and mu vector, what will that be? It will be why is that the J 3 alpha 1 is this operator. So, it will be a dot product of that weight vector with the mu vector. So, that is where this will be useful ok. So, I am giving back to the screen again to just to make you get a feel of that you will have a calculational tool if you define SU 2 sub-algebras and this is one definition of the SU 2 sub-algebras. Only thing you have to remember here is formally I should have written this with this, but you know in SU 2 sub-algebras what should be that coefficient? This is the J 3 Eigen value in SU 2. J 3 Eigen values are always integers or half odd integers. It is a number, it is no longer a vector now, it is a number and this number has to be integers or half odd integers. I am putting h cross to be 1. This whole thing has to be always a either integer or half odd integers. Can be half odd also because spin half state will have half integer. If it is spin 1 it will be an integer. So, these are things which I can put in as conditions here which is what I am showing it on the slide now that the Eigen values of J 3 has to be half odd integers or integers or twice the Eigen value should be an integer ok. So, you can check whether you all your twice the Eigen values are also integers. You know what is the mu 1 vector, mu 2 vector ok. This can be exploited raising operator, lowering operator. How many steps you can go from one states above p units, below you can go q units and so on. And you can get a condition on the angle between any two root vectors. I am not going to derive this. Anybody is interested can go and look it up. Georgie talks about it, but what we need is that the angle between any two root vectors in any Lie algebra is constrained by these integers. q is an integer, p is an integer, q prime is an integer, p prime is an integer and there is a condition which you can derive. This puts a lot of constraint. You cannot have any arbitrary Lie algebra. You have to have a Lie algebra with the root vectors satisfying this condition. That is why you can determine what are the allowed angles with this. Please go and check what are the allowed angles using this. Theta equal to 0 is trivial, 180 is fine, but you can get 120. Can you get some other angle like arbitrary angle like 10 degree, 15 degree you can check it out. You will find only the allowed angles for it are a list of them, not everything. Angle means the angle between two root vectors. So, one root vector I wrote here, if you take the other root vector you can find what is the angle between those two. So, if you take this root vector and if you take alpha 2, alpha 2 was what? Half and minus root 3 by 2. So, if you take the dot product of these two, what are you getting? What does that tell you about the angle between these two vectors? What is the angle between these two vectors? Mod alpha 1, mod alpha 2, cos of angle between these two, I am going to your 120 degrees. So, this implies cos of angle between alpha 1 and alpha 2 has to be minus half the angle between these two, theta. Theta is 120 degrees, everybody agrees. So, theta will be angle between alpha 1 and alpha 2 is 120 degrees in the SU 3 degrees. So, there are only specific angles and you can check that integers q, p are integers you can still get 1 by 2, not arbitrary angles. You cannot get arbitrary angles because q should be all the 4 has to be integers. So, this is what you get and in the case of SU 3 you found that the angle between the two simple roots is 120 degrees, cos of 120 is minus half, clear? So, now, what we are going to do is any abstract Lie algebra, first of all the rank of the Lie algebra will tell you the number of simple roots and then you know the angle between the simple roots has to satisfy the condition like this. So, using this there is a huge classification done using a diagram called Dinkin diagram. So, I will just expose you to that what the Dinkin diagram is. Number of simple roots is the rank of the Lie algebra, the rank as L the each blob is a simple root and a single line is to denote that the angle is 120 degrees and so on. So, there is a double line then the angle is different. So, there is a notation for orthogonal group, symplectic group even in orthogonal group you have 12 plus 1 is different from 12 and this has been classified and these are the only possibilities which you can play around. This has been a it is called the complete classification of simple Lie algebras. So, it is actually an ocean if you want to get into it you can sit and try and look at the proofs, but for me in this course I want you to get a feel of applications of some simple physics problems which you are doing. So, my focus will be only on SU 2 and SU 3 of course, orthogonal groups I have discussed and I have given assignments on Lorentz groups and rotation groups. This beyond this I am not getting on to in a in one semester I cannot do everything, but if anybody is interested can get on to and it is a compact way of seeing at the site what is the rank by looking at the diagram, dink in diagram you can say what is an abstract Lie algebra it has rank L and so on and then you have these notations ok. So, I am not going to go further on this, but at least for SU 3 what will be the dink in diagram? SU 3 the dink in diagram will be two simple roots which I am calling it as alpha 1 and alpha 2 and they have an angle, angle is 120 degrees and 120 degree angle is denoted by a single line and this is the dink in diagram for SU 3. SU 2 what is the dink in diagram? Just a single one with alpha vector to be just one that is it. So, this will be the what does it mean by talking about angle between two root vectors there is only one simple root. If you go to SU 3 SU 4 you will have three simple roots and it will turn out that if you try and do it for SU 3 sorry SU 4 you will have three simple roots, but only the consecutive ones are connected angle between the third one and the first one is 0. This will be for SU 4. So, just looking at the diagram you can say that there are three simple roots and the angle between the consecutive roots alpha i and alpha i plus 1 the angle is 120 degrees and so on. This is where the dink in diagram is really just at a site you can look at an abstract algebra in this language. Fine ok. So, I am not going to get on to this is what I said the angle is given by different simple roots you can write the matrix form of it and so on. So, I am not going to get into the, but you can construct this Carton matrix for the SU 3. For completeness you can sit down and write down the Carton matrix between simple roots. Simple roots are alpha 1 and alpha 2 we have already seen alpha 1 dot alpha 2 is minus half. So, 2 into minus half is minus 1 the off diagonal elements are minus 1 diagonal elements are numerator and denominator will cancel, but the two factor will remain that is why you have a 2 here ok. So, this is what is the Carton matrix sometimes they write just the Carton matrix sometimes they show it by a dink in diagram. So, these are various ways of trying to convey what are you looking at an abstract algebra ok. This is something which I will also need to explain for SU 2 and SU 3 just like you had simple roots we also will have fundamental weight vectors ok. The number of fundamental weight vectors will always be equal to the rank of the learge. So, let me get to the fundamental weights now SU 2 1 fundamental weight SU 3 there are two fundamental weights I will also denote it by young diagram now this one fundamental weight I will denote it by a single box the two fundamental weights you will have a single box that is one the second one will be two vertical box SU n what is the rank n minus 1 is the rank. So, that will be the n minus 1 fundamental weights I am going to denote it by what will this be you remember these diagrams I kind of when I was looking at the discrete group I was giving some definitions to these diagrams. These diagrams when I put below what is that called anti-symmetrizer right anti-symmetrizer if I put two vertical it is anti-symmetrizer. This single box denotes my fundamental representation in the single box I can put an up spin in the box or down spin in the box. So, this is what means it is two-dimensionally two-dimensional irrep which is also a fundamental representation or defining representation shown in the young diagram black. There are only two fundamental states with which I can play around here this means I can allow an up quark I can allow a down quark I can allow a strange quark. So, which means this is three-dimensional or three states three independent states I showed you by weight diagram also for the defining representation or the fundamental this diagram you can have three possible states ok. Now when I take this now when I take this I have to make sure that each box can be UDS and whatever is the entry on this box the entry on the other box should not be same you agree. I cannot put is this allowed not allowed why because it is anti-symmetrizer what happens in the case of SU 2 suppose I say this is allowed what is the state I can put here if I put up here this has to be necessarily down what is this this is a one-dimensional irrep which is analog of your unit representation which is an analog of your unit representation singlet is what we call anti-symmetric between the two boxes ok. So, this is one-dimensional it is not really relevant you can add this like any number of such singlets to a collection of non-trivial spin what will happen if you add a singlet to another spin half anything happens it will still be a spin half system ok. So, this state is not important in the context of SU 2 that is why when I write the fundamental weights what about SU 3 is that non-trivial or it is also singlet in SU 3 what are the possibilities here you can have up down what are the possibilities somebody this is one possible then down strange use strange, but this ordering is unimportant why is it unimportant which is already anti-symmetrized you do not need to worry about the order this turns out to be a three-dimensional vector space three states are possible for it, but that is non-trivial it is not like a unit represented. If I want to look at this what happens if I put up here I have to put d here I have to put s here this is again a unit representation or a singlet that is why it is trivial. So, in this case what will be trivial that will be. So, I am just giving you some kind of a diagrammatic representation for the irreps of SU 2 SU 3 and SU n fundamental ones fundamental means what I will say is an any arbitrary state. So, suppose I want to write. So, half half half m this I am going to associate with single box in SU 2 you understand what I mean it is a two state system I will represent it by this. Suppose I take 1 m this is three-dimensional you all know that right this will be having 1 1 1 0 1 minus 2 clear. This I need to represent using the fundamental weight. What I will do is I will say that it is two times the fundamental weight which I will diagrammatically denoted by two single box. This means it is two times the fundamental weight by fundamental weight it is a single box. So, two times the fundamental weight and this diagram means what? Symmetric this diagram means it is symmetric which means what are the allowed here I can make it u u I can make it I am working with SU 2 sorry I have to put up up up up up what else down down is allowed and what about this? These are the three states which gives you this diagram ok. So, that is why this diagram is the young diagram way of looking at the irreducible representation which is three-dimensional of SU 2. Now you tell me if I have to put an arbitrary spin what should be the diagram? Spin half it is one box, spin one it is two box. So, tell me how is it defined? For spin n by 2 it will be n boxes and everything has to be horizontal because you have only this fundamental weight to play around and n by 2 times that single box which has to be concatenated. Now the same thing you play on the SU 3 what are the fundamental weights you have? You have this and this. Now you can play around.