 Okay. So let me start. I am the last speaker today. So this is first. And I think among speakers I am the only mathematician and Maximus here and he is the only mathematician in the audience. No, no, no. Are you saying you talk to each other and we can go? No, no, no. No, no. No, it is possible. Yes, and one more. Yes, yes, yes. Okay. Anyway, so we are here because of Dima and maybe just two words. I knew him and not so good as you, but we communicated. And the problem was that I was a mathematician. That time I am a mathematician now, so it was a big communication problem. So now I know that I worked a lot later about things like Hamiltonian, quantum Hamiltonian reduction and things like that. And now I know that in some, in his own language, he knew such things in that time. And many things which I worked later probably, or definitely he knew like other people like Sachs as a mathematician, for example. And so anyway, so this is the problem of mathematics and physics and relationship. And probably they have to be different. Otherwise it's, they will not interesting to each other. So it's normal, like couple or lovers. Michael Agia said that, Michael Agia declared once that in its best version, theoretical cases must match that same thing. This is the opposite. No, in some sense opposite, opposite. I said like lovers, they have to be different. Otherwise they are not interesting to each other. Okay. So anyway, so what I will talk about. So I will talk about the W algebras. I will restrict myself only to JLN case and usual and deformed. And I'll explain something about usual and deformed. And maybe the main part of my talk, I'll explain some general things about what is the deformation of W algebra and why it is useful. And useful in different kind of question. One question is the old question in conformal field theory. And I will try to formulate them. And also in new business like Samsung and Nikita in this AGT and so on. But let me first explain about something about W algebra in JLN case. Well, so the standard definition of W algebra, what is it? So we take maybe JLN and consider the JLN hat and the level will be S. So this algebra is related with the corresponding Xumina model and the definition of W algebra that we can make a sort of reduction. Just killing some part of this JLN hat and get the W algebra. And this reduction is calling usually the quantum Drinfeld-Sokolov reduction. And what it is doing? That in JLM we have a maximal nilpotent subalgebra and we have n hat. So this is central extension of current to JLM and this is central extension of current to the maximal nilpotent. And well, the Hamiltonian reduction we take a character of this nilpotent to C and then make a reduction to this. It means that we kill the part of JLM hat which is corresponding to n hat and kill the dual variables in some sense. And what remaining is some deformation to carton and this is the W algebra. And such kind of construction and this is related to what Dimak Nizhnik did when we go from Wersumina to Virasora or from Wersumina to Liouville we make the procedure like this. Just making the reduction well. And I want to say that result of this reduction, so this is some mathematical construction, how to do it, but maybe it is not reasonable to explain it right now. At least it gives us some algebra and just to explain what is this, let me recall that if we have JLM hat and make this reduction, then as a result it gives me a Heisenberg, it means one free field. And also Virasora, so the result is like this. If I apply it to JL3 then it will be some algebra, actually the Molochic W algebra, in this case it contains free field, it means Heisenberg, then Virasora, then some additional generator. And if it is 3 and if it is bigger then we have bigger set of generators. Next point of a story, so first of all let me fix some notation that corresponding algebra, notation is falling. First I have M here and I also have some dependence in S. S is a level and if to use standard notation it is reasonable to put here some combination like this. So this is more or less the notation for W algebra and I will say a little bit more precisely later, but if M is big what is this? It contains generators maybe S0 Z, S1 Z, S2 Z and so on and this is corresponding to Phi, it is Heisenberg, this is corresponding to Virasora, this is additional generators and so on. And they satisfy some relation and if you write this relation you see that this relation then depend in variable M algebraically. It means that we can define some algebra in principle with infinite set of generators which depend in this M algebraically and so we can take M as complex number if we like and we cannot construct this from JLM for complex M or we can but in this case we have to determine what is this. But such kind of algebra exists and one more thing that because we get this algebra as relation as reduction we can use the following things. We know that it is JLM hat plus JLM prime hat. This is embedding to JLM plus M prime, this is usual way we can and if we carefully look what it gives us after reduction we get the map to the opposite order. No, here we have some level and S, no it is the same level, yes. And then anyway what I want to say that I do not want to be specific about this second parameter but from this we can derive the multiplication which is looking like this. And no, so here it is something, yes it is commutablication. This is, it is some cheating but not very much. The question is what we understand here by this sign. But at least it is almost honest thing that if we have this W algebra and M are just parameters which it depends or we can cook up from this the quantum group or at least by algebra because we have multiplication, we have commutablication and this is what we have here. And commutablication and quantum group appear because of this and maybe later I try to explain that this is natural but no, people using it by the following way. No, if it is a sort of quantum group we can look what kind of object. Quantum group usually is the quantization of some le algebra. And so if you have quantum group like this, at least the object, you can try to get it from some le algebra. And no, in this case the procedure is following. We can take these parameters M and if M go to infinity and if we make up this process of the limit to infinity by a proper way and such kind of limit surely settles things. But nevertheless we can do it. Then we can see the following which is that we can take the le algebra of differential operators. So it will be C, C, Z, D over D, Z and we can consider this, so this is algebra and we can make from this le algebra with the operation AB minus BA. Yes, and then in a limit it is going to universal enveloping of this and we can make it a little bit more precise. And long ago Zakharevich did it by the following way. It is possible to consider not this but the observed differential operators where we have D and D inverse. Yes, and then in this we can also consider this as the algebra. Okay, so what Zakharevich did? He took this algebra of observed differential operator and on the algebra of observed differential operator we have trace, first of all. Trace that is the coefficient in front of one here. This is trace because if you have the bracket the trace of a bracket will be zero. Then we can cook up the invariant form with this trace by taking trace A times B and we can take the le algebra from this. We can make up the triangle decomposition of this. It is possible to do by many ways. Let me say this in a different example. Yes, but now the idea is that making such procedure just by quantization, some le algebra, we can get this family of W algebra and up to some technical problems this is correct, this one. And the simpler way and it appears when we make deformation to do the similar business is the following. Observed differential operator is a complicated object but maybe difference operator is a little bit simpler at least from this point. How we proceed? We can take the algebra of non-commutative Laurent polynomials in Z and D. Non-commutative it means that we have some Q, I will have a lot of Q, let us put Q tilde here such that Z D equal to D Z times this Q. So this is what is called in quantum torus or difference operators because they are difference operators. They are acting in a space of Laurent polynomials where Z is acting by product to Z and D is acting by shift. Yes, and this is also le algebra. No, this is associative algebra and we can make up le algebra by the same way A A B minus B A. Yes, and then we have the same things here. Namely trace is the coefficient which sent F which is equal to the sum A ij Z i dj to A 0 0. And this is trace and this trace of A B minus B A is definitely 0, it is easy. So we have a killing like form here and we have decomposition. And so the composition is the following. So we can take S carton sub-algebra C Z Z minus 1 and positive part this is C Z. Such kind which i is bigger than 0 and negative part we put a negative part of D. And if you are talking about decomposition we can take this one half of this as usual. This is one part of decomposition and this is, so it will be C Z times Z for example plus this. And the same with Z minus 1 to this and about one we can forget for now. And by this way this algebra we can up to, I did not explain what to do with one. We can write this A plus plus A minus. We have invariant form there. Oh, sorry, I just got confused right on the positive part. Positive part of D as well, yeah? So it will be one quarter. It's like one quarter, right? Z in Z inverse problem. Yes, yes. But then this shouldn't have to play by Z. Yeah. No, no. I wanted to product by D, so all this. Yes, and fine, at least better. Anyway, we have the algebra with quadratic form invariant and decomposition or almost decomposition to isotropic subalgebra. So by this way we get the algebra or corresponding Poisson structure of the group or at least any algebra have formal group. So we have such kind of structures there. And at least formally or we can apply some generic theorem which in infinite dimensional case they have some problems. But they work at least in this case. And after deformation we get some algebra. And maybe to be precise one more command because this algebra, because this is algebra of different separators, so it is acting here. So this is some version of algebra of Jill infinity. We can write it's just rather special matrices in infinite dimensional space. And Jill infinity we know it has central extension which appear always in this business. And quantization is better to do as usual for this central extension. And I also skipped such details. Yes, and what we get as a result? We get some quantum group or quantum group like object which is some version of this. And it depends as the object in two parameters. One parameter is q which is here. And second parameter is a parameter of quantization. It is some other q. Usually it is denoted by q1 and q2. This is as a result this is two parameters, quantum group depending in two parameters. Before continue let me say just very, very generic words about, no, what kind of story we expect if we have such kind of quantum group. To explain this let me recall the standard situation. Sorry, I don't understand one thing which you said. W algebra is not the algebra in principle. No, W algebra is not the algebra but... Well, but it is. No, no, no, no, no. No, no, W algebra is not. I said that if some parameters go somewhere then W algebra goes to universal of the algebra. So it means that as it is deformation of some W algebra and now I am talking about the deformed counterpart of this. So I took the algebra and it has the... by the algebra structure and I can use the standard dream field technique to deform it and it is possible and gives us something. And I want to say that what we get is coincide with the quantum deformation of W algebra. For example it up to some changing of variables we can derive from this. Quantum Verasora or quantum W3 and anything what you like. Now I want to make some comment maybe a little bit philosophical but it is following that... No, no, no, no. No, because what we have? We have to eat this algebra depending on two parameters q1 and q2. But as I said we work actually with the form the central extension. Therefore after deformation we also have a central extension and this central extension as usual can be understood as a parameter. So like central charge of Verasora also can be understood as some parameter in commutation relations. So this k and so if this k is acting by q1 power m then we get W algebra corresponding to JLM. More precise statement that if we have q1, q2 and if this k equal to this then the corresponding object can be factorized. Because this is universal enveloping of such kind of algebra and this is surely bigger than W algebra. Verasora have two fields one and here have many. But if q1, if you have some resonance condition that here it is like this then this algebra has some ideal and we can factorize and get W algebra. And don't you have an additional extension by some like log d? We can, we can. If we need we can do it, yes. The question is what is the universal information, how many parameters... But even for Katsmudi it is the second parameter is not so important. No, no, and here, who knows, but no, no, but right now not. Let me say the following. For conformal field theory we have the following things. We have conformal field theory and we know that conformal field theory it is some correspondences. We can, not always, but in many cases it is a quantum group which knows a lot about what is happening here. It knows about the monodermy of correlation function and things like that. And the standard problem and Sasha Zomolochka, for example, explained that something, that even conformal field theory is richer object than quantum group. It has many, many more things, but some feature of it, some things can be calculated in terms of quantum groups. There's some three-point function and things like that. So, yes, so it means that in quantum group side of a story we can make up some calculation and they coincide with some calculation in conformal field theory. Yes, and the idea is that here is some, have to be some counterpart of this. On this side we have a quantum group which is this. Some quantum group may be denoted by some way Q1, Q2 and maybe additionally this parameter K. And here, probably this is the way to understand all such calculation which you made that here we have four-dimensional, at least five-dimensional or something like that. And you calculate some quantities here. And the theorem, so here it is a sort of theorem. Here it is a sort of expectation or hope or desire, I do not know. But we calculate something here and some quantity here can be expressed in terms of this quantum group. Quantum group is probably ambiguity in the quantum sense, like the monogamy of the temporal block. Usually, yes, here in four, at least I do not know what it is. Yeah, in H2 you have a choice of boundary conditions. So, you can change boundary conditions and that's a simpler problem than the full information. There is some kind of quantum group which controls that. So anyway in this four-dimensional field theory something like this phenomena is working. And so the logic is the following. We have some calculation here. We can guess or so understand now what kind of story has to be here. It means that here we have, for example, representation. We take a vector here and some things like that. And because of this AAGT relation, it is something like this. But additional story that if this parameter is Q1, Q2 go to 1, that it goes to the honest conformal field theory. And this is the way how this machine is working. So we have at least how I understand it. We have the correspondences of such sort and then we can interpret this as something in conformal field theory. So maybe this AAGT business has more deep meaning but I understand this only on such level. Yes, now let me continue. First of all, maybe let me just write here. So the quantization, how it is going on. In this Lie algebra, c, z, z minus 1, d, d minus 1. So we used the triangle decomposition where Cartan is this. The generating of positive part is this and the generators of negative part is this. After quantization, this part remaining commutative or if we have central extension, it becomes Heisenberg. From this part, we get some generator, e, z. And from this part, we get some generators, f, z. And about commutation relations, so here before quantization we have commutation relation, e, z operator product is something over z minus qw and here some field plus so on. This is what is happening in this Lie algebra. Yes, and similar story is happening after deformation but first of all because this is Lie algebra, it means that e, z and ew, they commute for different z and w and they have pool in some place. And I cannot explain all details but at least after deformation, the fact that e, z times ew equal ew times ez up to some polar term, it replaced by a statement that ez times ew times some permutation function is equal to ew, ez. This is a standard story if we quantize the algebra of some such simple structure. And about this function lambda, z over w, it is equal to z minus wq1, z minus wq2, z minus wq3. And here let us, so this is the generators and f is satisfied, some similar condition. And this is also some bracket ez and fw, it is equal something from Heisenberg. Product q1, q2, q3 is equal to 1? Yes, yes, yes. In this case q1, q2, q3 equal to 1 and if not 1, it appear in more complicated cases. But here it is like this and so the statement about this is following. It looks as a rather simple or relatively simple algebra. So it has Heisenberg, it has e and f and Heisenberg with e and f have simple permutation relation. Usually with vertex operators also let me not explain it. And then what I want to say that so really the story is following. If you have such kind of algebra and if you want to get the elliptic w algebra, how it's appear in the standard places, you have to go to parafermion like algebra from this. So I want to say that here you have Heisenberg and you have e. This is obviously some solution to the back-circuit. Is it something we already know or it's a new solution which we never saw before? Because I can't recognize it right away. Such kind of permutation function? No, no, no, this particular solution with these three q's and so forth. This I just do not know, but it's appear here very naturally. And anyway, I want to say that parafermion appears a following way. We have SL2 hat and we have Heisenberg psi prime and we have generator ez from SL2 hat. After product ez to a vertex operator of phi you get something which commutes psi. And parafermion are usually constructing by this way. If you make that similar procedure for this algebra, it gives you elliptic permutation relation. And quantum deformed w algebra in precisely how it's appear. And quantum virocero we also can get by this way. And surely it is going, time is going very fast. So, let me just explain one thing about it. That about w algebra it is many interesting things. One of the... What is the goal? The goal is to write the sum of w algebra. If we have such goal we achieve it. So, this is quantum w algebra. But it is also interesting, but surely I wanted to explain more than that. About w algebra it is following phenomena. W algebra we can construct by reduction, but it is not the only way. For JLN it is at least two more. One of them we can take JLN hat and consider here the JLN minus one hat. And trying to make a set to find here the algebra which commute with this. And it is known that corresponding algebra is also w algebra with some parameters. And these parameters such cassette is w algebra. So, we have K here. It is level. And K will come here. And here it is some number which can be calculated in N. And K it has to be here, but I cannot find. Anyway it is something. Yes, and so we can get w algebra by this way. And here we get just the same object, but K is now not necessarily integer. It is level here. And we get something. So, by this way this is a sort of rank duality thing. And we can define w algebra by this and then make analytic continuation by this N. And it gives me the same object. Yes. And one more thing that if we look to this cassette little bit differently. We can find that the corresponding cassette is also w algebra. But a different kind. Because we can take the super l algebra GL N and minus one. And then make up the Hamiltonian reduction of a similar kind for this. As a result it gives us the precisely the same object as the result of this cassette. It has some explanation, but maybe not for now. For me the point is the following. That it is possible to prove. But proof is rather tedious and it is one thing. And the fact is that after quantum deformation it becomes much more simple and clear. And one more thing that if we know this I can consider the chain of w algebra. So, let me recall that if I have algebra GL N when I construct the Gilfan-Satelian basis what I do. I consider the sequence of sub-algebras like this. I take representation I restrict it to sub-algebra, sub-algebra, sub-algebra. And by doing this I find some specific vectors there. And this is working so good because in this case the sub-algebra here which commutes with this is commutative. Therefore we are doing this. If we apply this story to this it is sure cannot be so simple. Because if you believe me that here is GL N then I have GL N-1, GL N-2 and so on. And because this pretend to be commutant of this I can take representation M, representation of this. And restrict it to the product this times this. And it gives me something times something. This is representation of w algebra and this is representation of GL N-1. Then I can continue this business, make several steps till the end. And then it gives me some counterpart of Gilfant-Satelian decomposition. M is the rather big sum but rather similar. It is the same data as appear in Gilfant-Satelian decomposition of representation. And here we have a product of representation of w algebra. I thought the claim of Shatashvili and others was that a final look of this Gilfant-Satelian basis for this free-field representation. Different people make different statements about maybe the same things. No, no, no, it is in some sense more complicated story. But maybe I have only time. There are really interesting things that because for w algebra of such kind I have a description. So I have E, F, H generators. Therefore, in principle I know what is this. And if I have GLN head I know the generators here. And it is the question how to find inside GLN head the generators from this or how to make the opposite business. So if we have this sum to find GLN. And my title was the extension of conformal field series and what I had in mind several things actually. But this is one example of this. Here we have GLN head and here I have a product of w algebra. This product of w algebra is a part of GLN head and this is the extension. And the problem is how to get GLN starting from this w algebra and more or less the standard procedure how it is doing. So the standard procedure is we have a product of two conformal field series. For example, for C for 26 minus C. And I can extend it by adding some combination of vertex operators. There are a lot of examples where such kind of machinery is working. And this is a rather special and rather complicated case of this. And because my time is almost over, this I cannot explain. But I can explain the following. That if you are talking about universal envelope of GLN head we can take Verma representation of this, GLN head. And then Verma representation which depending in highest weight. Or we can take for simplicity the vacuum representation. Vacuum representation is what? It has vacuum vector. Then I act by GLN times t minus one, then so on. Yes. And if you look to, and this is the problem of the composition. I have this representation and I can decompose. No, this is important from the point of view of representation theory. It may be also important from geometric business because in this representation, for example, we have we take a vector which related to some instanton type problem which we calculate the scalar product of this we take and it is something. And usually we have here basis and we can find out the decomposition of this vector. And then such standard machinery can work here. Or we can decompose this representation with respect to this W sub-algebra. And then we can take this, we take a vector or in the form things we also can do it. We can take some vector which we are interested in and decompose it here. For example, if we take we take a vector here in decompose, then we get we take a vectors in each representation of W algebra which appear here. And as a result it gives us some identity where scalar product of we take it with itself. It has such kind of decomposition. And in five minutes I can explain only the following things which is some combinatorics here. Let me to finish with some precise statement. So the statement is the following. I take Wacom representation. So it means with the highest weight 0, 0, 0 key. And for simplicity I can't I want this key be generic. So this is Wacom representation for Jl and hat. And here Cartan sub-algebra is acting and I can take zero weight space of here. And in this case this zero weight space decompose to a sum of Wacom representation of W algebra which appear in when I go to n, n minus one. Then W algebra appear which n minus one, n minus two and so on. Yes. And then this representation Wacom representation of W algebra, they have a basis labeled by plane partitions. Actually the statement is the following. The Wacom representation of W algebra which appear here on a place sum s, s minus one basis is labeled by the following things. We have to consider plane partitions here but not all but with some condition. Condition depending in s. We can take some point here in the bottom in a position s, s minus one I think or I think s, s minus one. And we are looking for plane partition which are living in this zone here and not here. Yes. And so as a result we have the following. So fine. In this case finite this is because I can restrict myself only this sector. So this here we have finite plane partitions which are living here but with restriction that they are concentrated in such part of, in such part of a room where they are laying. Yes. And so the basis here is labeled by the following data. I have to take one plane partition with such restriction here then one plane partition with such restriction here and so on. And then it is it and it gives in principle some combinatorial identity which how much it is complicated I do not know because the proof which I know it representation is theoretical. So I have such kind of basis and I have the action of some operator there and I can identify it with the operator. Yes. And if we consider here the situation when K is not arbitrary but for example some rational number full answer is not known but in some case it is known and it gives us surely some restrictions some additional restriction to plane partitions which appear here. So well two minutes. No. So it looks nice but what just I wanted to say that in our recent paper with Miwa Jimber and so on there are some explicit formulas. Actually they are some which are also looks nice but not for now. They are formulas. They are formulas for what? That we have JLM hat but JLM hat is the representation of toroidal algebra also. And this statement using to constructing some different basis here and the formulas but if you have toroidal algebra that inside this toroidal algebra you can by a simple way and explicitly find this double algebra which I promise you just write down to generators. And to find out the product of commuting sub algebras. And no I only want to say that the formula are nice and they are solving from this point of view representation theoretic question which I could not solve for myself how to embed. We know and we have some rather complicated proof that the cassette JLM over JLM minus 1 hat is W algebra. It is some indirect proof. But just in conformal business to make up this embedding precise is very complicated. Maybe computer can do it but no no stop stop stop. No just. Next time we have to invite computer. Do you have a library of writers? In the presentations of W algebra we try just Q out Q2, Q2 Q3 and Q out Q3. Thank you.