 I will speak about application reflection equation algebra, what is to my opinion, two-metrex models. The first question, what is quantum an elechrom, the will open algebra of the algebra G? Everybody answer this question, of course, quantum group of the infinity jimba UQG. However, for the particular case of JLN and SLN, there exists another quantum analog from the NLN algebra. It is the so-called modified reflection equation algebra, or its quotient. So, now I will explain what do I mean by reflection equation algebra. Let R be a heky-symmetry coming from quantum recall, that by heky-symmetry we mean a bradyn. It means an operator R acting on the space V squared in the same space subject to the bradyn relation. It is well known. And heky-symmetry is particular case of the bradyn subject to the relation presented here. It is just as follows. Q is generic, and Y is identity operator or a matrix. The heky-symmetry coming from the quantum group equal if I choose a group in the space V. In the space V squared, the basis is as follows. And so I present my operator by a matrix, and with matrix is as follows. You'll see that as Q goes to 1, with heky-symmetry tends to the usual flip, matrix of the usual flip. However, if dimension is greater than 2, there are other heky-symmetry tending to the usual flip. For instance, Kramer-Gervais heky-symmetry. It's difficult to present explicitly with symmetry, but there are. Now, the mentioned above, the modified reflection-question algebra generated by the unit is the entrance of the matrix. Subject to the system, with system. So here R is, for example, the matrix presented above after M1 and so on and so on. What is M1? As usual, M1 is M-tensorated with identity. We're not with algebra L part. And if we consider a similar system, it is system because we have here matrix relations, so many relations. If we consider with system without right-hand side, we have zero here. It is called non-modified reflection-question algebra. I want to say it's very important property that if I replace here and this relation on this one R by the usual flip P, we get respectively the envelope in algebra. It is matrix presentation of the envelope. Or commutative algebra. If you put here B, you have just the algebra with commutative elements. If in the above relation, we replace the heky-symmetry R by superflip PMN, it is more or less usual notation. We get the envelope in algebra here with algebra of superalgebra and super commutative algebra, symmetric algebra of that. Super commutative. Okay, now I want to consider an example related to the algebra as follows. So the corresponding matrix R is heky-symmetry presented above. And now M1 is 4 by 4 matrix as follows. So if you put all the matrices in the relations defining the modified reflection equation with algebra and with algebra, we have the system. It is just an example corresponding to this algebra. If Q goes to 1, we have such relation in the envelope. So we have A B minus B A is equal to B and so on and so forth. It is just both relations. Okay, I want to remind you that there exists another quantum matrix algebra. Quantum matrix means that we present the relation with algebra by matrix. There exists another algebra called RTC algebra, Hullian-Grat algebra. So I don't want to speak about this algebra. I want only to say that it is not deformation of the envelope in algebra. It is only deformation of symmetric algebra, GL2. But if we impose with relation more determinant, Q determinant, in some sense until we want to speak about that is equal to 1. We have deformation of functional algebra, polynomial algebra of SI2. So for if we want to deform with algebra on the group, it's possible to have with algebra RTC and reflection equation algebra as well. But if we want to deform envelope in algebra, it is possible to use reflection equation algebra in modified form, modified reflection equation algebra. Because we are interested in an element of the algebra, we deal with modified reflection equation algebra and not with RTC algebra. Okay, I would like to present now modified reflection equation algebra in a form similar to envelope in algebra. So it's possible to present modified reflection equation algebra as follows. It is one generator, 10th rule. Another one, minus sum operator. Apply it to these two elements and a linear combination here. Some linear combination. But if I do not with linear combination as usual as a bracket of two elements. So we have operator R action in the space M squared in the same space M, it is the space linear space where space generated by M Y. So we have with operator and bracket. And so we come with object, what object object. The space with operator and the bracket. It is analog on the usual re bracket. Now, what is a version of, of Jacobi identity. Here it is more interesting. For this version, that if we define action of an element, another one as follows by means of this bracket on the previous page. So we define a representation, namely we have, we have, we apply Y to that after that X to that after that we transpose and set Y by using R. It is equal to that. So it looks like the usual Jacobi identity, but instead of super transposition of flip usual flip, we have here. Okay, now with data is called generalized algebra and it is denoted GLV, the space V, equipped with R, with a Hake symmetry R. This algebra have been introduced by myself for our involutive and myself with Yatof Asapanov for Hake R. For Hake R, it is more interesting, of course. And if we take, as a Berman value, Murakami-Wenzel symmetry, coming from quantum group of other classical theory, it's possible to introduce the subject, no problem. But it is not a good deformation on classical option. The Hake symmetry, it is, with this reason why I am interested in Hakey situation and not a Berman Murakami-Wenzel. Okay, now I want to introduce some operator introduced by Lubashenko. So operators B and C, some operator. They can be constructed by means of R, but it's difficult to explain. But finally, we have to operate B and C. And what is, why I am interested in this operator? Because if I define action of the generator MYG on an element of the space V as always by means of the rate B. So we have a representation. And why operator C as usual, useful? Because if we consider the following trace, trace of product, M, M, here is any matrix. And C is just with operator. We have a middle of the trace. And it is trace, I call it trace, coordinated with our initial Hakey symmetry R. Hakey or involutive symmetry R. With operator turns into the classical one. If R goes to its usual flip, that's super one. If R goes to super flip. So in this situation, he becomes identity and trace becomes the usual trace. And here, here's similar. Okay, now it's possible to introduce a, so. Finally, here I tried to explain that it is a structure and endomorphism of the space V. So what is endomorphism of the space V in the category? All spaces are finite dimension. So this is the product as follow. And finally, it's possible to, by using the operator B and B to introduce coupling or pairing between element of the space V and element of the dual space. If I consider in the dual space, the radial basis, it means that it is defined as usual with just data. But if we want to pair, to apply pairing between element in another sense, in another order, it's necessary to use B. So what is difference with classical situation? In classical situation, if we have here delta, we have delta here as well. In general situation, it is not so. Okay. Now, I would like to mention property is very useful for matrix model. Modified reflection equation algebra is in principle is a morphic to non-modified reflection equation algebra, if we have the following change of the matrix. It's very useful property, but it is not valid. If you is goes to one, you know that symmetric algebra of GLN, it is not as a morphic to envelope an algebra of GLN. So if R is involutive, it is not so. But if R is taken, it is so. Now, I want to introduce it in my algebra generated by the matrix M and a lot of elementary symmetric polynomials and full symmetric polynomials. It is just a definition. What is definition. So it's necessary to put here A on or S. It is analog of skew symmetrisers. This here we have symmetrisers and here some product of the matrix M one over line. So here I explain what is notation of indices of alignment, but finally doesn't matter it's possible. The reason is possible to introduce an elec of elementary and symmetric polynomial possible to introduce an elec of short polynomials as well. And for them, an elec of literature son rule is valid as well. The quantum power sums are the are defined as a whole difference with classical situation in the following one. The usual trace is replaced by the only different. Okay, now, we have an elec of Ronski relations. The difference is that Q comes here with the relations for a key situation for involutive situation. Q disappears, becomes one. Okay, now. It's interesting, of course, it's really him a ton of that to present it. I want to introduce. Consider special partition lambda it is partition partition was discussed in the previous talk partition corresponding to the following. You see, we have here rectangle here one column and to one line row. So I do not with partition or with diagram is a following way. And zero if we have zero here. I own it. This index is similar. Okay. Now, we say that involutive for here a key symmetry R is of type MN. If rectangle here, we add one here as well is the smallest rectangle such that they sure polynomial. We have the following, the following partition here is equal to zero. It is trivial. It is not evident that they exist the smallest rectangle but it's possible to. Identity in a factorized form. If R is of MN type as a following form. So, I present here just a fairly, clearly committed entity. I would like to say it's very important property for us that coefficient here and here are central. They belongs to the center of reflection. I want to say that this property with a relation is very thin. Non modified reflection equation algebra, but by using as a market mentioned above, I can obtain a similar relation in modified reflection equation. Okay, now. Introduce now new why and you knew why they are again various for the first factor. And they are engaged various of the second factor, which means that Kelly Hamilton polynomial can be presented under the following form. So, this is the first coefficient squared after the main coefficient after that with factorization of the first product and here the first, the second. Okay, now I would like to mention here. The results to buy who they were done a war enough. They, they have shown that for super matrices, the Russian ratio of two sure polynomials as follows and the following one equals to be resilient. So, for us, it is more reasonable definition of Brazilian. Because we have here one central element for another central element. So, the Brazilian is a ratio of two central. In our setting, much more general. We have a similar definition by definition, just with Russia is called the Brazilian. And it is presented as follows it is product of new product of new some coefficient here. It's possible also to define an analog of determinant. You see, there is a name and the terminal is two different elements in a way. This is a ratio of sure polynomial corresponding to this partition and here with work. We have here product of new and product of new here, Russia product of new. Now, it is interesting to express your normal here. For instance, if lambda is non trivial rectangle. Then we have the following expression. So, some non commutative combinatoric with polynomial is a super symmetric. By definition, if we have to set X, Y, and Y, J. Independent commutative variables. Super symmetric if is symmetric with respect to X, symmetric with respect to Y. X1 is equal to Y1, Y1 is equal to T. The result of polynomial does not depend on T. Parameterization. I present some parameterization some symmetric polynomials, but now I put the same question for power. What is parameterization? Parameterization is as follows. So it is usual formula classical formula is valid. We put here one and here one and here and here and here minus one. We have just a classical formula. These coefficients are called quantum dimension. But in general, for any mu and to you, the formula are as follows. I would like to say that with polynomials are close to whole little wood ones. A similar looks like. So now I repeat, if we put you equal one, we retrieve here the classical formula. But the classical was super matrices. So now, if we pass by using that modified form of equation isomorphic. We want to get similar formula for general situation for modified reflection equation algebra. We get the following formula. So this formula is similar. But formula for quantum dimension. A little bit different. You see here q minus one here q. That's so much fun. So only the quantum dimension are changed. As we put q is equal to one. We get a parameterization for generalization of matrix on the will open algebra. I want to say once more that the relation in the envelope in algebra is as follows. Okay. Now, I want to say that finally, if we consider modified reflection equation algebra corresponding to the hachysymmetry coming from the quantum group, we have in principle two parameter deformation of this community. One parameter. It is just a passage to envelope in algebra. It is as you wish corresponds to the bracket to linear bracket and another parameters comes from passing to reflection equation algebra. So we have two parameter deformation. So we have the Poisson counter part, which is generated by two bracket linear bracket and quadratic bracket. And so finally, Poisson curtain path is pencil. And an analog of the second bracket sometimes is called Semyonov-Tenshensky bracket, but Semyonov-Tenshensky bracket for me. It is rather bracket defined it on the group and we are dealing with the algebra. Another interesting remark. There is a very interesting result by Shoichet. Shoichet shows that if we have a unimodeler Poisson bracket, unimodeler means that they measure compatible events with a bracket. And if we have such Poisson bracket, it can be quantized by using for example, conservation method, but it's possible to quantize so that finally, we have an algebra with the usual trace, with trace on the quantum algebra. While the quantization with the trace is possible, if we have a unimodeler bracket, but the bracket related to reflection equation algebra is not. It is not possible to construct the corresponding measure. Such a measure doesn't exist. You see, I presented a trace in quantum algebra, but with trace is default. It is not classical. It is not usual. So I want to compare my construction with a Shoichet result. If we have unimodeler situation with usual traces possible, if we have my situation, it is not possible, but the trace is possible in principle after quantization, but it is default. I pass to matrix model. We consider the simplest one matrix molded, defined by the following usual situation. We say polynomial. If lambda are eigenvalues of the matrix H, H it is of course a matrix. So we consider the integration of the set of matrix. We can present with a partition function over via eigenvalues as follows. So what is here result result here is that in this situation. We have the measure D H is just Vendermont determinant square. It is just specific for Hermitian situation. For other classical situation we have here one or four. I consider Hermitian model. Okay. So what is my idea in so long. I would like to quantize. That I want to define our analog of the production possible to do. It is very tempting to replace the usual trace ball, but by trace. Now, instead of. Down L. As human L to be generated matrix of modified reflection equation algebra corresponding to a. Involity for Hickey symmetry. Conceived. All terms coming as classical situation should be replaced by the above also. Try that wire conversion you and you presented the one. But what is analog of the measure. It is not evident at all. So consider the following matrix. Yeah, you'll see all races. Group position. If I consider the classical case, classical means. M equal N and N is zero. So determinant. Usual determinant of this matrix is just. Or that among the determinant square. The dependent on. Finally, I omit the potential V dependent on time. Nature and we. They fall in form. We take. Determinant usual determinant of these metrics. As an electric. An electric of winder Monday. As a falling formula. And now. Expressed our generalized matrix model or braided matrix model. Why I can read. You and you. You see, now finally, instead of the classical. When they want to determine. Now. I want to discuss one question. More. The following question. We consider it on the final dimensional situation. And the consequences of this consideration for matrix model. But how. Infinite dimension. So I will present ways. How we can construct some NL. In infinite dimension situation. We consider analog. Of general. Young matrices. I call with young. Braiding. Depending on parameter current braiding current means that we consider here. Our braiding now depends on. And here. We have just the usual younger. Braiding current break. If we apply. Procedure procedure. We have the following proposition. If I is involved to see me. And the following braiding. Here we. Is a current brand. It means that it means meets the current braided relation. Braiding relation as well as above, but now we have. With relation with. Is a key symmetry dependent on key. So then. We have a similar statement, but now either given by the following. Formula. Okay. We introduce with my. The generalize generalized. Of. You see. Is similar to relation. Now only the only difference. Now the matrix T dependent parameter. And the braiding are dependent to parameters as well. And generalize generalized. Of reflection. You see. The difference. Is are in. Is inside with relation looks like a relation in reflection equation. It is also possible to construct double. In can like algebra. Inspire it. And. Approach. But I want to discuss a little bit more. To construct a infinite dimensional structure rising from. So we have now. A generalized the algebra denoted as follows. I define this algebra. It is a. Modified reflection. So. Notation. It is not good, but it is enveloping algebra or generalized. Okay. And what is affinitation. We take the generalized. Matrix initial matrix. And we consider a series of the corresponding matrix dependent on you. Number. Okay. As usual. Is the finish station. Apply it in the usual way. So now it's necessary to introduce to define. The envelope in algebra. Generated by with matrices. Okay. The relation is classical. More or less classical classical. We have here. Our. Our header. We are a header. It is a buffer. Of course, it's necessary to interchange the number K and number. So. And here we have an. So it's very natural. So it is not possible. To define shrinkage them in another way. It is very natural. We have product after that we apply. That's all that's all. So as you wish. We obtained a. A fine algebra. But with algebra is a brain. Okay. But the problem. I. Discover it with algebra with my colleague some years ago. But finally recently we understood. That we say algebra. I will explain why. So, finally, we consider it with algebra. Is a. You and a lot of the envelope in algebra. Of the. But question. Very interesting for me. Whether it's possible to construct quantum. An elec of tau function in this way. I would like to mention paper by Marozo. Control on QT deformation of Goshen model. And in this paper, they said. The tau function was not constructed. It is what was 2018. But with attempt to construct such a quantum tau function. Has been undertaken by cut chef and Taza. In, in this year. So before that. And finally. The attempt felt. It is not so easy to do. And I thought maybe our algebra. Was good for construction quantum and elec of tau function. Why, because if we consider. QT models. In sense of more often told. We have the relation with this algebra. More or less as follows. Plus me, minus means that we have fermions or bosons. It depends. But the relation are more classical here. We have something. So Q and T come in only coefficient here. And every, we want to construct a really. Quantum and a lot of tau function. It's necessary to have the following relation. So Q here. And Q here is absent in the project. So now I consider our. I repeat. When it's possible to construct co model by using our algebra. They find a final algebra. Unfortunately, if the initial is taken. How I find is not the formation of the classical. With this problem. Consequently, our. I find algebra is not good. However, if I is involutive. How I find algebra is good. It is not. Strongly proven. But it seems hopefully it is so. So if we want to construct a reasonable. If you were to sell to provide a finization. It's necessary to deal with an involutive symmetry. With this problem. And such a. Involutive symmetry. What is page now. Involutive symmetry. As for. Symmetry is involutive. You can check it. Symmetry breading. So the breading relation is fulfilled. And it's possible to consider the corresponding algebra. Modified reflection. And construct a fine. A fine. So in this situation with algebra is a good deformation. And I don't want you to go into detail. I don't want you to concern and construction of the corresponding quantum vertex algebra. I want only to say that the. Ordered product. Of two hills. And to be. Must be defined as follows. You see. It is more less classical way to define. Ordered product. But here. Instead of the. Necessary. To put. Outflip. But of course. It's necessary. Initial. Initial symmetry. To be involved if it's very important. And if we do so. So it's possible to do it. To develop. All theory. So. Everything goes smoothly. And finally. It's possible to construct the corresponding. Field theory. As you wish. But it is only the beginning of the story. It is just. Progress. It is possible. In this connection. I finished in. Two minutes. I want to mention. Quantum analog of. Vertical algebra. Introduced by. It is. Object algebra. Constructed. Via. Double Q and guns. Finally. It is in the spirit. In this. Spirited. So we have. Q and gun. As we discussed above. Dual Q and gun. Which is constructed in a similar manner. And finally. Some. Relation between two in guns. And a relation between two in guns. Or permutation relation. Are introduced. In the. On the paper. If I don't. I have not. Not mistaken. In. Ninety two. And. But finally. Axiomatic description. Of quantum. It's very. Very hard. If you. Want to understand. What. You have. Really in this situation. You have something like looking like double Q and gun. And our. Q and L of vertex algebra. It's more easily. To understand. And there in some. Some sense. More. So I stop here. I only want only to resume. In two words. Finally you see. The different. Our approach and concerning. Matrix model. Based on reflection equation algebra. In just in progress. The only beginning of the story. But the other part. Constructing on. In infinite dimensional. Algebra braided tension in dimensional algebra. So it's necessary. Necessary. Apply. Either. The first method described to buffer. So. The. Second method described to buffer. Affinization. And I think that. Affinization is much more interesting. But it is. The beginning. Okay. Thank you very much. Now. It's time for questions. Or remarks. Or. Even even comment. So we thank the speaker virtually. And it is the end of the day.