Lie Groupoids and Poisson geometry (contd.)

0:00 What is B0? B0 is a multiplication. So B0 is a usual multiplication of functions. So B0 is a fixed. And I'm going to tell you basically what B1 is. okay and the B1 so if you look at write down the B1 I think we'll get a K with H1 you will add up this equation here FB1 G H minus B1 plus B1 minus B1 and this guy is equal to 0 and this isn't just a hawkshaw, long cold cycle, this is what this is I think it's very easy to check if you want to do some manuals and then I'll do two phase. If the functions and if you look at this co-cycles, you take anything you want, part is symmetric part. We write on this symmetric part and we must be doing one schysymmetric schysymmetric part and and then the symmetric part must be the co-boundary of Hobscher co-boundary And then the second part, the skee-symmetric part, this one here, it turns out, it's just the same thing, well it's now, it's isomorphic to the chromology, so this one is basically defined by the bi-active field.

2:30 And it turns out, then the property which I, this equation here, and if you look at this equation, it becomes just the D, D, D, the Poisson bracket. So, the claim is that, the claim is that in fact, this is B1, it's Poisson structures. So, normally, what the stock product is, the stop product is just the deformation and normally in the sense which I just stated, just so you have the associative product adding the parity conditions, namely plus the B, I normally adding this condition B, I, J, equal to minus minus I, B, J, it's called parity conditions and also And you want the one still be unit. You still want the one... So you mean, this one could be like the i of the i? Excuse me? I'm sorry, this is the i. This is all term to be symmetric, the even part is symmetric. And this one, well, it's not really, well, people would take different definitions, but it's the most common definition, you look at a star photo that you have in this condition. So, that's what a star photo that is, so we have a one's identity unit here. So, what I'm, what I'm, this lampard just says is the following thing, which you want to write down in the numbers, is this, is basically you have a star g h minus g, because that's the case, you take a symmetric part, and divide it normally, divided by r h, so this, this one, it must be a, I forgot a little

5:00 there must be some physical reason you're putting R instead of N. From the G-cord. Or it may be a shooting equation. Right. So, and then limit. H equals zeros. That's just B112. This is exactly this term over here. Up to minus, up to constant, up to factor of 2. And this guy is Poisson bracket. So, this is another way of thinking what Poisson bracket is. just the, I think it's two times, so the better is that it's this way, but in force this condition here would be, this is the way it now becomes calculus derivative, see how you define the first year calculus, the derivative is that you look at FH minus F0 divided by H, up to this I. So in some sense, what is the Poisson geometry, Poisson manifold, is just the first That's another way of thinking, so this is the Poisson structure. That's another point of view, thinking of a Poisson structure. It's just the first order derivative of the Poisson Algebra. So this part, this direction is trivial, you're getting associated with Algebra, you get the first derivative, you always got a Poisson bracket. and the other direction it turns out it's highly non-trivial namely giving first order derivative whether you can go on to find all the high terms and in a sense I gave me stuff E0 and B1 fixed, can I go on to high to get a B2 B3 B1 and BN so that is associated in order to get a solution and it turns out it's a highly non-trivial question I've worked on this for years to years Because in the very beginning, people just look at the equation, which I looked down, BI, the very beginning of this equation, BI, BJ, this equation tried to find the homological interpretation of this equation to get a solution. This then got to come up with various solutions, so special case of Cotinian bound over M,

7:30 sympathetic manifold, start with sympathetic manifold, Cotinian bound over M, then you got to, for sympathetic, it's true, but still then, different cases, We solved this question, so basically, and that's the, also probably related to the next week's conference, basically says, conversely, the inverse is 2, so namely, any Poisson comes from this way, that's the consecutive series. So any pulsion structure on manifold is equal to this now. So this is an equals one. This is one direction, and then the consecutive theorem says this is both directions. Any pulsion manifold, it comes from deformations. And that's sort of the wrong story. So that's... Excuse me. I'm a little confused about what you said before, because at first you had FG minus GF, and you did that sort of... And you rapidly changed it to this one. I'm sorry, sorry. Yeah, I'm sorry. I think the right thing I should write in this F, D, D, I'm sorry. This G of F. This is the right form. Yeah. Without any assumptions. Because coming from what I did before, from this fact over here. Because FG and GF, they kill this symmetric part, they get a C symmetric part. And then, what? I think I have to divide it by 2. up to constant of 2 and this is 2 so but then so this is true for arbitrary deformations but I want to write this in a way like a derivative this is not derivative I want some functions minus f0 divided by h so in order to write this way I have to impose the condition of this which normal people assume. So then they always schismetric, B1. If it's schismetric, then I can take 0. And now the formula becomes which I wrote down. The model looks like a derivative. And just the way I try to... So this is the right formula. This is the right formula. And for arbitrary. But if I really want this as a derivative of something, I need a schismetric. That's what I'm saying. And then this is the... I try to sort of expect this point of view of Poisson is the first derivative of non-conference directors

10:00 which in some sense is always the guideline for the study of Poisson geometries. Look at this order of non-conference geometries. Sorry about that. Is that okay? Yes? clarification on the on this last statement are you saying that there's given a possible structure there's actually a product on the function algebra which is a deformation and derivative is this or are you saying there's a formal power series formal power series so there's no variability of the power series I mean you can't make sense you can make sense but there's a definite version of deformation for instance deformation. And then this is, I see it's no longer policies, and then you have a, I forgot what is the street deformations in Johnson's list. And there the action is not for policies anymore, right? You have a... Right, yeah, that's much harder to achieve than a form of power series. So the solution, yeah, is that... Right, so there you have a different version of the deformations, but what I'm doing here is a sort of form of deformations. These are, see here, what I, this is what normally people call stopper, that's really formal deformation, this is really, sorry, I should say formal deformation. In the more, okay. Formal deformation. And, and, and then, then, then what it's considered just here, it says, any postmanifold comes from a formal deformation. A formal deformation always exists, okay, in the first order, you can always go on. So the algebraic obstructions are vanishing, it's not just stating that it converts to an algebraic Exactly, it's that initially people always try to look at this algebraic equation and then it turns out that always the third cohomology comes into the picture and no matter what they do with all the definition guidelines, the third cohomology obstructions you try to get a roundabout obstructions and then you've got to come up with various special cases to try to get rid of them but you cannot get rid of them forever So, but our care of concern is just a new idea, so then, that's the...

12:30 Yeah, so right on this ceiling pocket, any person's structure comes from a stoplight. Alright, so let's go back now. So this is another motivation where a person's structure comes from. So, now I'll go back to see, so let me talk about another thing, is the so-called sympathetic Well, as I just mentioned, porcel manifold is a single. You have single foliations. It's a higher-ease single. A simpatic manifold is very nice. So before this, I should say double serum is the same is an antisympathic manifold locally locally isomorphic to just what example I started and R2M instead of 2 with a canonical bracket. With the canonical bracket of PIQI, just like the bracket I gave in the beginning. And all the other zeros.

15:00 So it's very nice. Locally, there's no structure of symphatic manifolds. And a possible manifold locally, there's a frequency. I will get to this later on. But, so... What the... Historically, it's a foreign question. Basically, it started looking at it by Li. Between the 1890s, is what you call a function group. Actually, this is how the person's children comes into pictures. I guess around that time, this is the only bracket, only sort of the mechanics people are using this standard bracket of function as a partial pi, partial qi. So the question, Lee was looking at the following questions, you want to look at the functions which are functions of those colonic coordinates. But it's the bracket of those functions are closed in a sense. So those are the functions, this is a class shift of independent Functions, independent in the sense of independent functions, the differential, well, nowadays you can see what the e of those are independent. Functions of canonical coordinates. who's bracket under the standard bracket over here who's under the standard

17:30 It's close. Namely, so what this is, is basically you're looking at a bunch of functions, say phi 1, find out those are the coordinates functions here it's a bracket using the standard bracket and this is a standard bracket here, I'm putting standard as a function R2N and you want this to be closed in the sense of, closed under the composition functions so you want this guy is something big F ij and phi1 phi1 with r coordinates so that's what the Li looked at called function groups And, well, now into the modern language, exactly what it's looking at is now it's clear, so you want to look at this function, the bracket is still the composition of the functions. And because we have the Jaguvi identity, so this is in the modern language is exactly what this is really looking at is, first of all, seeing as a FIJ, this guy is really defined a personal structure on, so what are the consequences? The fact is that this one just means the following two facts, the meaning. So I'm just translating these meanings now. this guy is a person, defines a person bracket on the r-dimension space that's because all those Jakubi identities, right? So you can all those generate, but then there's all the EFIJ.

20:00 And the second thing, the second meaning of this is basically is that I'm putting the standard of 2n and just putting this map of phi, which are just each of those functions of phi 1 and phi r and this function to r r, right? I'm just putting all the functions together. And this bracket just means this map here is a Poisson map. So I, sorry, I didn't define what a person's map is, but you can sort of imagine what a person's map is between two person's manifolds. You just pull back the function because of the bracket. Okay, so basically this is what you want to interpret what function group is. is basically you are looking at a person structure on R, RR, such that you have the canonical symphatic structure to this person structure over here, and this is the person map. In In other words, you want to realize this Poisson structure to the big, relating to the canonical one here as a subalgebra. And in this sense it's very, very natural, you want to see. So this guy, so this guy is a Poisson structure which has singularities, but you want to embed this Poisson algebra to a sympathetic one, which is a non-symbol here. So, so, but after this review, the other way, how, how, how, how you get, I think, how we found the non-trivial Poisson structure by looking at this thing. we got a non-trivial processing structure. So, so what is sympaticalization is, now, let me define sympaticalization.

22:30 is a Sympathia realization of oposathan. is a sympathetic manifold a subjective, with a subjective sub-motion x cubed and j, which is what's going to happen? So you want to realize, so in a way you may want to look at the level of functions, basically you want to realize the algebra, Poisson's algebra as a subalgebra, a Poisson's subalgebra syntactic manifold. to put this one, so that's sympatialization. Any x and omegas are allowed to... No, I mean, that's definition. So the sympatialization of Poisson-Manifold consists of, I should say this, consists of a syntactic manifold with a suggested sub-motion such that that J is a Poisson map. I can take that entity. If it's identity, this M is not symphatic. It's just Poisson. You're blowing up all the symbols. So, I need X to be symphatic. So, you want to basically, you're giving Poisson manifold and you want to find something on the top of the symphatic which is Poisson map.

25:00 algebra level that I'm saying, you want to look at a function, the algebra of the prosome manifold, and to be a sub-algebra of something which is not single. It's a synthetic manifold. So that's what a syntactualization is, and what I hear is what is the function loop is. So, of course, you want to ask whether the question is is a very natural question, is there anything you want to see if it exists or not? If yes, what is unique? These two questions are very natural to ask. Well, So, let me just give a partially, let me just give an answer to this and then probably that's all for today. So I guess enough. So, the existence, the existing local existence, namely for arbitrary point if we look at the arbitrary small enough, do it now. But it's basically out there. And this is a regular. This is already B2L. Basically, that's what you look at. It already exists. For general, the regular, and I think the Weinstein, who was looking at around 80 degrees, this local structure of Pursu Manifold, and proved its existence. And Goldberg's existence is proved by and the idea is to try to prove locally and then try to patch them together that's to the next things which I'm going to answer, whether it exists or not, or the uniqueness.

27:30 Attribute also to uniqueness is no, because anti-sympathicularizations, for instance, anti-sympathicularizations here, and you can add another synthetic manifold and just take the same map and forget about this one, it would be some penalizations. So it's not unique. And, but on the other hand, a theorem of a calcium from the one stem, basically they proved, is the following. Maybe you want to prove it's not unique, but there is distinction one, there is one which which has a nice property. So, first is sync categorization exists. globally exists for any full to manifold. globally always exists. And secondly, it's not a unique but there is the extinguished one. There is always for any for some there is a distinguished and then this one is a unique this is a unique Distinguishing Sympathic realization

30:00 such that x is a local root point. And so there's a unique one which is locally, has a group-point structure, which acts in the midst of a local group-point structure, I should say, in the midst of a compatible. A compatible. I'm not going through the definition today. A compatible, it may not be globally group-point, but it's a local group-point in the sense define our neighborhood identity unit. Good question. Compatible means compatible with syntactic structure. So this is what is the, what is this call is the so-called syntactic groupoid. in this sense it's unique. So there's no, now not unique realization, but there is one distinct one, which has additional structure, Goupoi structure, which is compatible with sympathetic structure. And this sense is unique. The proof is very highly non-trivial. And you sort of choose, you prove this locally, you have something almost unique. This is kind of a symbolization. And then, because it's unique, you can touch them together. You've got to go over it in some sense. Somehow, part of my course is to try to give a proof from a different point of view of this cell, from a totally different point of view, why this is a true cell. But at least I have convinced you, I hope, there's a good point that does come into the picture from this point of

32:30 Poisson geometry here naturally if you want to realize Poisson as something sympathetic and have a good question. So next time I will continue. How I'm going to do it will depend on the stable audience later on and which part I'm going to do better. More or less I will just go through actions and that's all. It's very hard to survive French's system.