Lie Groupoids and Poisson geometry

0:00 For graduate students, it's a graduate course. So I'm going to start at the very beginning. It's elementary today. Also, I do not know who a former student of this class is supposedly a graduate. So, and I, later on, maybe after a couple of lectures, probably get more ideas, getting more better ideas, who are the permanent participants of this course, then probably can adjust to whatever level I need to go over. and so today I probably would do is something basic introduction what a course of manifolds are course on geometry and hopefully getting to the group points deliver motivations again sort of introduction what I'm going to do the rest of the semester of this course so yeah I have to admit that today I'm going to be very elementary, so people, some people will get this part of the law in those issues. Alright, so, I always start with something very, very standard for For people who are doing the first year of mechanics and the first year of physics, well probably out of six that would be probably the right. and you have consist of three positions without space and then the velocity is momentum, the the six coordinates and the functions on here is normally called harmonic functions. And this is very standard. And they messed up the sign somewhere. And this is a very standard of postal brackets, which appear in mechanics.

2:30 And a little bit of calculus, you can compute it before we talk this. For instance, this one's appearing in a lot of basic mechanics, right? this is the first introduced by Poisson you can see this is schismetric and also if it takes three functions, it takes secret connotations, and this does two stables. This is what you call Jacobi identities. and the bracket of two functions and this is what I call Lapanee's rules Well, it's very easy to check this properties. And anything that satisfies those is what you normally want to call the Poisson Algebra. which is what it's called some algebra, is just the associative associative commutative algebra with a bracket here, which satisfies those 1, 2, 3 properties. And that's basically, this is just . And so we're just abstract from, try to abstract from this standard .

5:00 And in particular, if you want to look at those algebra which comes from functions on a manifold. And that's just the Poisson manifold. So the Poisson Manifold is a Poisson Manifold such that this loose function on the manifold, which is economically associated with cognitive algebra, has its Poisson algebra. So it's a very simple definition and basically it all comes from this ax over here. Alright, so let me show you some examples where Poisson algebra is, Poisson manifolds are. And before I'm doing this let's see. So examples, well, I'm going to just concentrate on the load, suppose m is just out of n, with coordinates, x1, x1. And now, well basically from one is three, there's two axioms over here, this is symmetric and this is Lefnitz rule. This is what you want to call Lefnitz rule. From this two axioms, you can see that they're giving two functions, F and G, and this implies these two axioms, with the calculus, you can see the bracket of two functions, and you can really write in terms of coordinates. It's the same thing as

7:30 that's very easy x-axis you can check. Just two axioms tells you whenever you want to do the bracket you can always get a bracket from the bracket of accordion functions. and where they call it functions let's call x, i, x, d and this is species magic this would be species magic because it requires this rapid species magic the first examples And now, before I got an example, well, so, actually I'm just equivalent to this, honestly. And now you want to look at the second one. The second one is Jacobi identity over here. This is called Jacobi identity. If you want to look at Jacobi identity using this bracket over here, and you will end up it's the same thing it's equivalent to the Jacobi identity for the quadring functions that's also very simple to see you see this formula over here you look at Jacobi identity you can end up the expansion and then you just look by the hand you just require the quadring function as the Jacobi identity and therefore So what is this equation? So this one, well if you write down this equation, this guy is just pi ij, and now you use this formula over here again, so you end up with this equation, is the end of a partial pi ij, partial xl, and pi lk, right? I'm using this Einstein summation of the use, and plus signal communication equals the arbitrary ij. So in other words, if I want to just look at the Rn, what is Poisson manifolds on Rn?

10:00 proven to a symmetric matrix pi ij and by a matrix which satisfy this nonlinear PDEs so this looks very complicated around linear PDEs equations but it turns out it looks complicated if you look at the special cases it becomes very trivial so let's look at the examples so this condition together with this condition will give you the Poisson structure And now look at the first case will be, let's say, pi ij is constant, which is independent by x. Well, this condition is still required, and that just means you need to, in this condition, basically automatic. automatic. This condition is automatic because the partial derivative is zero, it's automatic. So it just says, well, it just any, this just says any species of matrix and define constant Poisson structure. Well, the second, you want to look at, let's say, another, a little bit more complicated than constant Poisson, would be a linear, right? The next level you want to look at is just linear, so let's say, what about pi ij is a linear, linear function.

12:30 Well, and now, this condition here, let's say star and this two stars, basically have two conditions. The first condition basically says you want the Cijk is minus Cjik. and second condition this condition here and you use this equation here in taking partial derivative and this cancels out and you get C, I, J, K and you have to manipulate the equations and basically you get a first order just the equation in terms of x and you look at the conditions to be 0 and you end up with this equation C, J, K, H, C, H, I, P, and plus the, so here's the single permutation of I, J, K, I, J, K, this equals here for R, G, P. It's pretty simple, you can just plug into the equations and just separate them. And now immediately you can recognize what this means. This equation means things. And this is just nothing but the structure constant of the Lie algebra. exactly what is Li algebra is given by this constant so this just says, equivalent says Cijk is the structure constant of Li algebra

15:00 So, now I'm getting something a little more interesting here, it's basically the first proposition which is trivial, but nevertheless it's the first non-trivial examples of a Poisson, namely, n-dimensional linear Poisson structure. Bilinear just makes brackets linear according to Poisson structure. Rn is bi-jection with n-dimensional real. So, a linear porcelain is the same thing just for the algebra, and for this reason those kind of porcelain structures, in fact this is the first discovered by Leeds around the 1890s, and this is normally called the linear porcelains. It depends on the background, where you are from. Different people call different names, called the Poisson, and then people from Russia, it's called Kielov Poisson, and sometimes called Constant, or Kielov Constant Poisson, and the French people might call surreal. So you have different names with this person's job. Very simple fact. And apparently people discovered in six days, six days, seven days, I guess, this person, NAD algebra come up with a canonical person's job. And I'm trying to put this local coordinate. It's a so simple fact. You can do it according to free later on. We can do it according to free, but today I just it's very simple. You can just look at this equation. That's how you can be out. All right, so, well, next level you might want to ask, what about quadratic? Well, it's become complicated. The answers I don't know. So, in the sense that I don't really have classification of quadratic Poisson structures, is sort of still open, except for, I think n-dimension equals three, there's some kind of results you can get in, classified, but in general, if a higher order is, it's extremely complicated, it looks like very trigger questions, but it's not completely clear what a person's structure you can write down, the quadrality.

17:30 All right. So, now let me just go back to using the coordinate free notations, and what I wrote down here is just basically using local coordinates. It turns out, if you look at a pi ij, and if you take a local chart of a Poisson structure, and as I say, to define Poisson structure, all you have is this matrix pi ij. But if you put that together here, and it turns out this is a tensor, which namely it's independent of the coordinates you choose. So this is the sections tension bundle. it's global sections. And therefore, that takes care of the first two coordinates. Then the Jacobian identity, which is the equation which I wrote down here in terms of pi ij. or this compensated equation, non-linear equation and it turns out you can write down a very simple way in terms of this bi-bactor, this normally equal bi-bactor u. And this one is equivalent to just so-called pi pi bracket equals zero, the scotton bracket

20:00 And do I, sure, recall what a scotting bracket is? Sure, sure, just briefly recall what this is. Okay. Let me, let me do this very quickly. The other thing I forgot to say at the beginning of this course is that I have a node in my web page. Anyone remember it? Anyone of you remember the site? I know the web page, but after my web page there's a slash of the... Well, I'll let you know. It's out of my webpage, but I forgot. Maybe Books. Books, DAW, HTM. See when it works. I'll find out. I just arrived today because I didn't really get time to look at it. I forgot exactly the website, the after the slash. And I have notes, which is what I started to write three or four years ago. And I'm teaching graduate course at Penn State, and then I taught again last semester. I tried to update this, but I don't know. It's probably ongoing project, which will take forever to finish. but nevertheless you can see some of the notes which I'm talking about I'll let you know next time where is the exact thing after this there is no right condition on pi so pi could be 0 right? exactly there is a condition pi equals 0 is a condition but pi is 0 is the set by this equation so the trivial bracket structure is okay exactly In this sense, you're right. And that's something, a point of view, which I learned from Alan Weinstein's Society of the Hallways. He always keeps telling me that manifolds is a special case of Poisson manifolds, instead of the other way around it.

22:30 You think a Poisson manifold has a different structure, but you don't want to think that it's a manifold. It is a special case of Poisson with their bracket. Exactly. And in general, you really want to think something, just like you really want to think some quantum mechanics, caustic mechanics, caustic is a special case of quantum or something, from that point of view. All right, let me say a few words about the Scarton bracket. What is Scarton bracket? It turns out that people, a lot of people, yeah, it's not surprising, I taught this in Penn State of graduate course, it looks like for first year, after first year of graduate students, people know the differential forms, and calculus on differential forms, but very, very few graduate courses, they cover this, all the multi-vector fields, the squirt and brackets, but it's getting popular because of those deformation quantizations. It turns out itself, this is really parallel to the differential forms. So, it's standard on the manifold. So, as a manifold, on vector field, you can have a bracket here, which is this bracket, right? That would be my standing point. So, you have x, y vector field, and you can take two vector fields, you can take a bracket, you still get a vector field. That's my statement. I'm not going to recall what this is, but let's assume you can take a bracket with this. And, and now, I can sort of take a bracket between vector field and functions, to a function. How do you do it? where the most natural way of doing this is x putting a function here and you let this to be just a vector-to-exon function. And now you force this to be schismetric and then extend it to the polyvector field or multivector field. So let's say XKM is a polybacter field or a multibacter field, depending on what you want to call it.

25:00 It's a wedge of K with the assumption, convention, 0M is a function. And then let's start the bracket. It is a bracket which is defined on all the multivactor fields, polyvactor fields. Basically, it is a bracket which extends this bracket over here by the, how you do the bracket of a vector field and a bracket between vector field and function and you want to extend it. how to extend it, is you want to enforce the Slap-Needs rules and in the sense, so first of all you want to be symmetric, so here's the AB is minus So, I put in slash here, it just means a degree of A. Okay, so, so whenever you take the bracket, and you want a single degree of this is shifted by minus 1, so, shifted by 1, so, so you, you want this member in this gradient. commutative so and we have the so-called greater jacopis and for greater jacopi I let me see if I got the right sign so I want to get the right sign for the greater jacopis Thank you.

27:30 again here is just shifted by the result and then you take secret permutation with ABC equals 0 this is called this two condition basically says this is a gradient Li algebra, it's Li algebra, gradient Li algebra and then the last condition would be Lapini's rule Thank you. No, no, it's not an algorithmic question. It's a history question. Are these close to Verasoro algebras? What? Is this like a Verasoro algebras? Well, you have to remind me what Verasoro algebras is. That's what you're asking. Verasoro algebras is an extension, but not the algebras. It may be, yeah, I'll look and see. I forgot what Verasoro algebras is. bracket but if you're going to stay soon for the next week for the workshop you will see a lot of this but as they can put this way this is what now people call

30:00 Gerstahopper any any structure which has this part they'll be here basically algebra is associative algebra associative algebra commutative associative algebra which has two structures with associative commutative algebra with a bracket so you have two algebra structures and the bracket is compatible with the multiplication of the multiplications and this is the and as a reason called Gerstenhaber algebra in fact this was a, this bracket is before Gerstenhaber and the Scotland bracket is before the Gerstenhaber the connection with Gerstenhaber is the following if people are interested in the geometry it's basically just if A is algebra and if you look at a Hochschild homology with coefficients on A itself, and it turns out there is a bracket over here which is actually having K, and this is what the Gersenhaber found in his very famous paper this is the screen and the mass two cables when you look at defamation by algebra and it turns out the Hockshaw homology have this bracket over here and which has this same part of this here this part of here and then what it connects is with this exactly if if you look at a is a function and and then That's the polybacter field. And then you can be reduced to the Scotland bracket. So in this sense, it's a Gerson-Harvard bracket, it is a generation Scotland bracket, it would be the other way around, but historically that's the least choice, but nowadays it's to ascend the people. Gustav-Alger is more popular than Scott and Bracket.

32:30 All right, so and how to get this bracket, the basic I already explained, you need to try to just using this extensively. I think that's one way, or you can have a global formula to define this, but there's a lot of ways of doing this, as I say, this is also one way of doing this. All right. Well, so then let me just write down propositions to summarize what I say. So what is a Poisson structure on a manifold is now, is basically a Poisson structure on a manifold So it's equivalent to a biobacter field, and satisfyingly this equation, Scott and Bracken equals it. And this definition is the first due to which noise, since it's around 78. The surprising definition is not that as old as you can imagine these previous definitions at least. Probably it's around there already, but I found it from literature, at first there's 78 of these JDD papers, which you know is defined as a bracket in Poisson's practice. And so far this is the Poisson's structure. and go to the definition. Now, a few more notions which I'm going to use in the future. And let's go back, as I say, locally. What is the... So, giving this pi... I think I probably couldn't go over it.

35:00 Giving a pi vector field. And you can define a bundle map from cotangin bundle to tangent bundle. and therefore you can talk around this well basically the local coordinates and it's just a matrix so you can talk around the matrix So, rank of pi at the point of x, it just defines the rank of this fundamental method, this map, this is the symmetric quantum map, this is just defined the rank of this fundamental map over here. same thing is if you look in terms of local coordinates which is rank of pi ij s of this matrix and by n matrix if you're taking local coordinates it's the same thing and therefore because it's a skee-symmetrical map this is skee-symmetric and therefore if a skee-symmetric matrix is the rank has to be even so this must be even The length of a Poisson structure at a point there must be evens and in particular if it's a full rank it's non-degenerate and we have a name of it, it's called Sympathetic Manifold. And people hear this much more often than Poisson for sure. But I'm doing the other way around, instead of thinking, talking Sympathetic person, Poisson is a Sympathetic. special cases just say this is just person manifold with the full rank so this is just person pi i j, pi in full rank.

37:30 It's not in this case, in other words, this is non-degenerative. This boundary map is isomorphism, is isomorphism. In this case, it's syntactic. And therefore, this pi sharp is isomorphism, whenever isomorphism, you can take an inverse of the isomorphism, so let's try a sharp inverse, is just again a schismetric map again, is from PN to P star, it's a skee-symmetric map skee-symmetric boundary map take an inverse of this, you get a skee-symmetric boundary map and a skee-symmetric boundary map is the same thing as two forms so in other words Because if it's a symphatic, automatically you will get a two-form. Such that my pi-sharp inverse is the same thing as this two-form defined by this math. Normally people using B, this means a bundle map from T-end to T-star-end using B-contractions. You see the two forms, B-contractions. Again, one can very easy to do a simple exercise. So far, here I'm just using this pi as a species matching, it's a bi-vector field, it's a bundle map. And remember, if I want to pose one, I have another condition, it's gotten rapid to be zero and we chance to play this in terms of two form, it exactly means this two form to be closed. So, so pi, pi bracket equals zero. In this case, you will do the calculations and you can see this is equivalent to this equal to zero.

40:00 So, just let me summarize what I'm saying here, which is not just the definition of synthetic manifold. So from this point of view, a manifold itself is synthetic, is equivalent to this non-degenerative the first two forms which isn't always the definition of Sympathian Manifold, I'm just define Sympathian Manifold as non-genre Poisson which become propositions, it's more or less the same thing so that's first thing, the rank, the rank of Poisson And one thing. If you have a manifold, say a sphere or something, you can always get this, if it's a symplectic manifold, are there other possible structures that have smaller and smaller rank on it? Sure, zero. taught me in the beginning. That's two. Well, yeah, there should be more than So that's part of the theory? I don't know if this is part of the theory, but I think you can just manage a constructor. The Poisson structures are very, very, very flexible, so it's unlike the sympathetic structures. Well, you might have some topology on it, but normally Poisson structures are very flexible. Yeah, well, I can't say there's no topology, but I think there's no topology, so more or less you can sort of manually define this part of the person and try to sort of connect them. Is there a maximum rank? Yeah. Could it be a maximum rank of the person structure?

42:30 what do you mean maximum rank the dimension of space I mean if I have a compact manifold and I think can I find a Poisson structure it's not simple necessarily so there may be some rank less than but you could the thing is the rank is not constant I should say so rank is a concept which is already what is maximum not each point. I'm talking about rank at a particular point over there. So, normally rank is not a constant. I will give you an example. So, you could get almost everywhere still sympathetic except to one point of a portion. For instance, a very simple example is an instance R2. xy equal to x squared plus y squared. this is a two-dimensional so and if any relation will give you possum because it puts them rapidly with the is you need a student max Scott and practice 3 so this any function here we put it you can put any function here to be possum and but the ranks one matrix here so what is the rank over here is a parameter of this pi is 2 everywhere except for zeros. At 0, 0 is zeros, and then this is everywhere else. Oh, because I'm just using coordinates. See, when I say this is a well, you want to fix all the, and then the other way is you just be symmetric. if you want you can write yx you can do minus but normally you only want to fix the wrong orders so in general for two dimensional space you can pick up arbitrary functions and you can define and then the zero point of those functions the point of those functions will be the everywhere is a sympathetic you see it's an anti-journal except the zero point That's one of the examples. So another thing which I want to say is that

45:00 let's go back to pi-sharp. and you want to look at image of this pi-sharp, let's call it D. this is a you don't want to call sub-bundles whenever you call sub-bundle it means constant rank, the thing is it's not a constant rank but it's not as bad it's not that bad, basically In fact, all the integrability conditions for the H is satisfied in the sense that any small section of a D and you take the bracket, it's still involuted. You're taking any section of a D, so it's just the integral. This is what you want to call integral. This is still which is not in the usual sense because this one is not constant. Nevertheless, this is in the sense of the Susanas Susanas I don't know how to spell it. It's in my notes. So it's in the general sense of Falkinian series. So in other words, this defined affiliation. So in other words, this is an integral distribution in the general sense. Therefore, you can have a foliation. So this defines a single foliation. And whenever you have single foliation, and now you look at this bundle map, restrict to this, basically this foliation is an image of this. So you look at this distribution on this restrictor, this D, and this map is not degenerate. This pi is not degenerate there. And therefore this just says this foliation basically is each leave of this

47:30 is synthetic. So it's a single foliation whose leaves are synthetic. and this is what we normally want in the literature called Sympathic foliations this is called Sympathic foliations which is simpler but it's not that bad So in some sense, I'll give you examples, for the R3, the Li-Poisson of SU2, the algebra SU2 or SU3. And basically this is a three dimensional algebra with three generators, the x1, e1, e2, this is e3 in the bracket. So you just use the same relation to define the bracket. So basically you have three quantities, x, y, z, and now the bracket is x, y, z, because that's just like the relation, then you take a single permutation. so Y is equal to X and so that's a Poisson Poisson structure and then you look at the foliation of this basically you look at the matrix of this and look at the distribution of this and you want you will see what you're getting is just a sphere. All the spheres and except for this point. Except for this point over here. So here you have a distribution. You have the single foliations

50:00 by sphere plus this single point of one point. So as a consequence, each sphere is a symphatic structure, so here's each sphere is a symphatic structure. In general, let me just state the fact that this is a... I will leave this exam, I'm going to leave a lot of excesses, I don't know, but when I told them this day, I guess nobody really did it in my excesses, I guess. So, I don't know, I don't know the French system, the American system is an anti-register course, automatic, you get an A's. and for the topic not French I was told I have to get exams but in case someone registered I think I was told getting zero registration and I don't have to make up his exams otherwise I have to make up exams so anyway so in that sense people don't have to do their homework so for the case of the Poisson m, m is just the Poisson and the algebra the Poisson structure. And then the syntactic foliation which I'm just talking about using the using this matrix, you look at the foliation, and it's the same thing as given by the quadrioring orbits, the sympathetic is just the, it's a sympathetic leaf, I should say, more precise. Sympathic leaves are just quadrioring orbit or adjoined orbit. The thing is I haven't got that far, I only defined using the coordinates, so that but I will later on if I want to give an intrinsic formula I need a g star maybe I should say

52:30 let me give you an intrinsic formula because I already did a global formula for the linear space as a manifold there's no difference between g and g star anyway but if you want an intrinsic formula Poisson bracket there, then you have to look at dual-lealgebra in terms of the algebra. So g is the algebra, and what I'm saying is what I defined, in fact it lives on dual-lealgebra. And then what is the bracket function is now two functions here. And the bracket at the particular point of u, u is g star. And now what you're having is df, dg, looking at the particular point of u. So what is VF, VG? Whenever I have a function, VF at a particular point of view is a co-vector on G-star, at a particular point of view, right, co-vector is G-star. And now the co-vectors, because I pick it up G-star, co-vector becomes the algebra, the G. If I take the algebra, I get G-star, so that's why I pick up the G-star and then take the co-vector and we get the algebra. So this guy is element of Li-algebra. This element of Li-algebra. So two element of Li-algebra you can take a bracket. This is Li-algebra. And now this element of Li-algebra, I can take pairing with u. It's element of u of Li-algebra, get the functions, get the numbers. And now it turns out this is intrinsic formula. So this is the same formula which I gave to you using the structural constant, C-I-J-K. but this is according to the free function. So, this is a Poisson bracket. So, the Li-Poisson is live, Li-Poisson lives on the view of Li-O to interest the group. And that's the Poisson bracket. So, and then the sympathetic leaves, now, is a co-journal orbit. The same thing as cojoined orbits. So as the covariate is any

55:00 cojoined orbit sympathetic. Colonically have sympathetic structure. And it turns out this is very important in the representation series of orbit method, the Q of constant orbit method is basically based on this what is the standard rule in France you take a break I think we probably take a break especially if you give grades it's better than we only got 15 minutes no they don't we have degrees of my students sometimes I have to invite them for lunch. Things change to get good valuations. And my department had to look at my valuations to determine my next year's salaries. This is useful to say and putting the quotation mark because it's not really is basically just a single foliation could be single foliation with synthetic structures on each leaf which, so that you can patch them together smoothly. So, with symphatic structure, I should leave, with it together smoothly in some sense, but what does really make sense is that, as you see, the five-activity is smooth. So, that's, but intuitively that's a good way to think of a Poisson manifold.

57:30 So another, let me give you another point of view where Poisson manifold comes in naturally is from the so-called deformation point of views. And so here's another viewpoint which also somehow in some sense is related to the next week's lectures, well, next week's workshop. Well, if you look at the, basically, you want the somehow the deformation series, deformation. You want the algebra, smooth function on the associative algebra, and former you would attach it to the policy of the H, so this is the H. And now you want some associative algebra here. and well this is just a form of series if you want associative algebra structure you know what which is the linear with respect to the edge so and to do this what you really need is that it is just for arbitrary function two functions and you define what those is and so-called deformation is you want this one is a formal poster is the first order is the same thing it's just the commutative algebra where you start with the functions and but the second order, the first order of F pulsers and so on. You want to write them as pulsers with those and

1:00:00 and what the, now you want to look at associativities and associativities and now you just write out this equation by using this power series and you will end up with this equation of fi plus j on the right hand side is equal to therefore you look at the fixed order of k you got a bunch of equations just i plus j equal to k of vj v i So, a deformation is equivalent to a solution with infinite equations, but if you look at

1:02:30 we're getting h squared orders h orders, sorry, I think you don't get h orders k plus 1, is that right?