 Now, let me come to some familiar examples of Pley algebra. Are you all fine? I think many things are known to you in some pattern I am just only fixing the notations. That way you will start not getting scared with the notations. When you see an S o n or an S o n comma m in any book you will not get scared. Oh, I do not know this you will not see. Yeah. Take a two dimensional complex vector space. Till now I was trying to look at only a real vector space. Let us go to a two dimensional complex vector space ok. Then you can have matrices which are linear operators operating on this two dimensional complex vector space. What should be the dimensions of those matrices? 2 cross 2 matrices, but the entries can be complex now ok. How many independent real entries are possible in a 2 cross 2 matrices? 8 elements right 2 n cross 2 n. So, you will have 8 2 n 2 n squared elements. So, there will be 8 independent real elements possible ok. So, now you can try and look at what are the generators for such set of 2 cross 2 matrices with complex entries. That le algebra is written as g l small g small l 2 for 2 cross 2 dimensional matrices acting on a two dimensional complex vector space. So, that complex vector space is put in here on the side as g l 2 c. So, I told you that whenever I am looking at le algebra I will use small letters. So, g l and 2 comma c. Just like you write basis states you can write 4 such basis states, but you have to remember that you have to multiply with coefficients which are complex coefficients. Instead you can also write 8 basis states with similar thing multiplied with an i fact. And there will be 8 basis states for the algebra which corresponds to 2 cross 2 matrices with complex entries. So, what do I mean by basis states? Any arbitrary transformation you can write it as a linear combination of these basis states. This is what you have done at some point in quantum mechanics, but as of now I am trying to tell you that it is these 4 and again you can multiply with i factor and say that it is an 8 dimensional real vector space a set of linear operators or 4 dimensional complex vector space. Take it as 4 dimensional complex vector space you denote it by g l 2 comma c. Subalgebra of g l 2 c you can look at a subset over on top of this general 2 cross 2 matrices with complex entries you add some constraints traceless and it should be hermitian. Somewhere it rings the bell you use the traceless and hermitian poly matrices, but I am not doing that here I am just trying to put in why is traceless gets the letter s do you know? Remember your rotations. If I say SO 3 rotations and if you want to write this SO 3 rotation as exponential of i times something you have to see what condition determinant equal to plus 1 imply on the trace of the matrix. So, if I say e to the power of i g if I say determinant of this this is the group element g is the element of the Lie algebra. If I want this to be plus 1 this can be rewritten as these are acting on arbitrary vector space right. You can always find that this is equivalent to writing it as. So, the generators which constitute the algebra if it is traceless the exponential of those generators which gives you the group element will be having determinant plus 1 correct. Trace g equal to 0 implies determinant of e to the i g equal to plus 1 and this is the group element ok. Group element I will denote it by here I will denote this group element by SL and if it is acting on NC and these g's are elements of small SL NC ok. That is the notation we follow Lie algebra elements are given by small letters and group elements are got by exponentiating the Lie algebra elements and they have to satisfy if it belongs to special this L for linear S for special g is for general if it does not satisfy this then you will call it GLN's ok. So, all your SO 3 is a subgroup of O 3. Similarly, GLNC to start with Lie algebra will have a subalgebra which is SL NC because GLNC can have determinant plus 1 or minus 1 SL NC has to have determinant plus 1 the S denotes determinant has to be plus 1 and arbitrary elements which I write here in the Lie algebra is traceless and hermitian. So, you will see how many independent elements you can write you can show that because you are going to be on a complex space the coefficients will be complex there will be three independent elements A is 1 real Z does 2 real, but then you multiply with coefficients which are again complex which is real and imaginary. In principle it is a three dimensional complex vector space or it will be six dimensional real space just like what I said for GLNC. GLNC was a four dimensional complex vector space or eight dimensional real vector space same thing for SL NC. If you constraint that the coefficients have to be real which is what you will do when you do exponential of i theta orthogonal groups have real entries right. Similarly, if you constraint that you want the coefficients to be real then it will again be again a subalgebra of SL 2C you agree. So, SL 2C itself is a subalgebra SL 2C is a subalgebra of GL 2C and then SU 2 is your familiar SU 2 is the real algebra which involves real entries. So, this is going to be a subalgebra of SL 2C. So, set of real Hermitian traceless matrices forms a subalgebra and that subalgebra is sitting on a this is a tower each one is a subalgebra of ok. This is the broader one you have to have all the matrices with determinant nonzero why nonzero should be invertible, but you will have various subalgebras and exponentiating the generators of these Lie algebras will give you the corresponding groups. This SU 2 which you have been extensively using you call it as an angular momentum algebra is not very different from your orbital angular momentum algebra, but this was introduced because the Stern-Gerlach experiment showed that you either have a up state or a down state which is a two dimensional vector space correct and you needed the generators acting on a two dimensional vector space. So, this is why you have the SU 2 naturally occurring to understand the two states two spin states seen in the Stern-Gerlach experiment. But as an algebra you will see that there is no difference between SU 2 spin angular momentum and SO 3 orbital angular momentum algebra you know this all of you know this ok which one SL 2 C, SL 2 C involves entries which are coefficients have to be C as well ok. So, what you can say is that you will have your sigma x sigma y sigma z and you will also have an i sigma x i sigma y i sigma z. So, this will be your six-dimensional, six-dimensional real vector space. So, but if you go here it will only be one subs not this clear. The basis states could have any coefficient, but then you have to also allow that the coefficients could be complex if you are looking at the group with complex entries and that will force you to basis state entries could have been complex to start with, but that is what happens in your poly matter. On top of it you also want your coefficients to be complex real and imaginary that will multiplied by a two factor if you are in SL NC SL 2 C, but if you go to SU 2 it only allows three octaves not the other factor here also I wrote the basis here basically I am saying 1 0 0 which I write I will also have a i 0 0 0. So, there will be eight of them coming I have not written the eight of them, but you can do it. So, that is why this will be a four-dimensional complex vector space or eight-dimensional real vector space. Now to your familiar world of angular momentum which is your belongs to a unitary group in quantum mechanics all your inner products should remain preserved that is why your time evolution as unitary you know all your transformations are all unitary is what you learn and you also know that such a unitary operator for the SU 2 algebra by SU 2 now you know that it is going to be the algebra involves Hermitian traceless matrices with determinant plus 1 and it should belong to real you know the entries have to be real the coefficient entries have to be real. You understand what I am saying if I write an arbitrary element x which belongs to the Lie algebra SU 2 small s small u 2 it means that it should be summation over of independent elements have to be this is real and then you have these poly matrices this is three-dimensional real vector space. So, three generators are required independently you can do it for just like I did for the orthogonal group you can define SU 2 as a group what is the meaning of SU 2 is 2 cross 2 matrices with u dagger u to be identity to denote determinant of u has to be plus 1 you could do it from the group point of u and find out how many independent elements are there which will define for you number of parameters number of parameters will define for you the number of generators and those number of generators should be there when I write any element of your Lie algebra which is also a vector space. So, what is this condition going to give you unitary means you can you can try and figure it out from this condition that to start with 2 cross 2 entries and determinant will make that to be traceless. So, just work this out and see how many independent elements are there ok. So, number of independent parameters I leave it you to check this you will see that it is only 3 you can work this out for SU 3 also ok. So, do that also now I am writing group. So, if I want to see the element how will I write here once I see three parameters and three generators I will write element u this u exponential of i theta dot these are the bases which are the three generators. So, theta dot sigma. So, this will be your group element of SU 2 which will when it operates on a two dimensional vector space A and B means linear combination what is this mean 1 0 is up spin 0 1 is down spin A B means A times up spin plus B times down spin what is supposed to do it should give you some new A prime and a B prime. So, this is what is the transformation on a two dimensional vector space and this one will be the I will know how to write this exponential. I am sure you would have done it as an exercise in your assignment in quantum mechanics right. Poly matrices have nice properties. So, coming back to the slide exponential of SU 2 Lie algebra generators for example, the three poly matrices are good generators for traceless Hermitian matrices. You do find that the group elements which is exponential of i theta dot sigma can be rewriteable as cos theta by 2 identity plus i n sigma sin theta by 2. How many of you have not done this? Anybody who has not done it except for one person whom can try it out later, but others you all done this right. So, this is what is the group elements which can be rewriteable in terms of sin theta and cos theta. Why am I doing this is mainly because I want to show the group elements, group parameter space. If you put theta to be 2 pi what happens? If you put theta to be 2 pi you get g of 2 pi in rotation when we did rotations. If you do a rotation by 2 pi it is equivalent to not doing any rotations, but if you do a rotation by 2 pi in this 2 dimensional spin vector space it gives you a negative sign, a negative sign. So, what does that mean? It means if you do a 4 pi rotation you will get back to identity. This is not new fermions are spin half particles, bosons are integer particles. When you exchange to fermions the wave function can pick up a sign. When you exchange to bosons the wave function does not pick up a sign. You all know this. This is what is nature's way of doing things and this is what happens. So, explicitly when you write and now tell me what will be the group manifold for SU 2. It is no longer a solid sphere of radius pi it better be a solid sphere of radius 2 pi going from minus 2 pi to plus 2 pi. E of 4 pi is always identity which means all the points on the boundary are all identified not diametrically opposite. So, that is the parameter space for your SU 2 group. So, let me stop here and