 Yeah, yeah, no problem. Okay. So I start again. So okay thank you for the invitation. I'm in the basic notion seminar. And the title is super linear structures and conformal geometry I'll talk on both subjects which are very basic simple concepts more or less from. And the beginners of linear algebra, both of them. I'm in the context where combined is super structure with conformal group gives so called super conform groups and not many of them. The one with the richest structure is connected to four dimensional space and, and the main part of the talk will be, what is the representation theory of the super conformal group and what are the consequences of physics. There are consequences, which I think are not up to my knowledge not observed in physics. And the problem there we will come to that point is the representation theory has the problem that the problem of non semi simplicity of representation categories which looks like a disadvantage but it may be something which is interesting. And that's the point I'm going to and that's the whole motivation for my own considerations of the subject is it. This is basically motivated from physical theories. There's this standard model I do not want to go into details but most of you will know what it is. There are several forces internal forces, which are gauge series from the groups you want as you to as you three. There are several generations not so important but if you do it in Minkowski space you have of course the relativistic geometry is governed by a non compact group which is the provident group learning school or the universal cover the essay to, and the representations of these groups occur, especially of the universal cover is the, the spin or representation to dimensional. It's a two dimensional spin representation which plays a role in the standard one. It's not the four dimensional one it's the left spin or it's the half spin or representation to dimension standard representation of this group. There's a lot of, there's a long history the standard model. The problem is we see many groups and there's a lot of activity of the last 50 years to unify these forces into bigger groups is grand unification, and also to combine the relativistic symmetry with the internal symmetries. There's a bunch of trans unification embedded into the eight, eight, and, or have a whole series of such internal forces. Also in string theory you consider higher dimensions, most famous ones are the 10 dimensional string theory exciting instead of four dimensional string theory, and the so-called entity citizen five dimension also appearing in string theory. But the point is, there is, I think 50 years approximately ago, a theorem of Coleman and Mandela which says that under a reasonable assumption, which are technical assumptions on scattering up to its holomorphic city of the city of the scattering makers. It is shown that you cannot combine the compact groups and the dynamic groups like as you want for your covering group into one big classic leak group, unless it's a direct product. You cannot really have a unification where these are pieces of a larger group which is a simple. And this is theorem, then, later on, inspire a lot of activity in terraristic physics super gravity theories and so on forth, which tried to avoid this by not considering super gravity groups but super leak groups. Instead, because fermions play a fundamental role in physics. And so the thing I will focus on is this semi simple representations versus non semi simple representation, because the point is, once you allow super groups or super brass and their representations, their representations are genuinely non semi simple, whatever you do operations TensorFlow, you get representations you're definitely not semi simple, completely different to the classical picture, which you have either for compact groups, you know, by Peter violence, everything's dire some even the hip hop space setting and also in the setting for dynamic groups like as a to see, they are non compact leak groups but the representations here in physics is, is tied to unitary representations but the indefinite dimensions but even if you leave that you have non semi simplest. But that's not the point here. So, the two structures which are, I assume, to be relevant is this super structure is coming from family and both statistics, you have super symmetry is building on this ambiguity you have and then the conformal structure because if you look at the standard model it's, it's invariant it the, the, the, the homotages operate on the conscience so you have enlarged symmetry beyond the classical symmetry, at least in the most important case, where you look at super symmetry of a certain crew where you have for firm and decrease of freedom, then you have scalar invariance and that's the most important model so that's the physical background. I'm not a physicist. So, but that's what inspired me to think about this structure so first part of my talk. It's a mess now so it's just I want to remind you, simple reminder on what is a super representation, and then I will embed it at the end. And to what is the structure very abstract structure cancer category, but my focus will be on semi simple cancer. So let's not say we seem to have a shortly discussed. So, let's start with you see that this is not the piece of chalk somewhere here. So, it's better to write on what you, if, if you take a field, let's say carries to zero, a nice field, then you can consider an algebra which is commutative but by extracting a square root out of an element of pain. It's commutative algebra. Alpha it's commutative algebra. If Alpha zero it's freshman algebra of dimension two. It's commutative algebra. But it's naturally set to grade it. Because this is even if this is odd, if you multiply, even an even it's odd if you multiply odd and odd it's even because you land back in the ground field. Even times odd is odd, odd times, odd. So this is a two graded algebra commutative. And it's well known if you have two commutative algebras, which are set to graded and a tensor product. Again, it's commutative algebra set to create it. But this requires that you agree that you multiply in the usual way. You do it in commutative algebra by reordering. So this underlying. You have an identification between cancer products over the PK by mapping x 10s or y. Well, I will choose a sign factor because we come to the super case in a second. So here's the sign is not there. You just have this underlying convention for identifying different tensor products. And if you make this sign. Identity one, you get this multiplication law because you apply this if you move this here, it's a tensor product identity where you have been a alpha between a beta and alpha prime here, and you just exchange the two orders. And you use such an underlying identification. And if you this convention, you get a commutative algebra, but you have a set to creating so you can also believe this case we can also use the sign rules of course you will say okay, where I have to define the decrease. What is the degree so I write this as well I have to indexes. This is the even part and the decrease zero. And you have the odd part which is this, the decrease one so if I write this it just means a raise the bar, and this is set to create it. And I don't write it otherwise so it's the usual rule for differential calculus. If you use this, what happens. But it's new rule. This is the super pencil rule. You have again if you do this that. If you go back and forth. It's the identity. And in fact, these isomorphisms are functorial. If you have linear maps between if you think of this as vector spaces. But what you get in this case is. Well, it's the same tensor product. As before, as spaces, but the rule to define the algebra. Let's start as a different one. You have introduced an epsilon. Yeah, and by abuse of notation just remind you I put the apps on here. You have now a different convention for exchanging I'm right apps for the non trivial one. Then this is a clip at algebra. It's a non commutative. Too great. None. In fact, if alpha. Beta, I K star. It's a division algebra, but on an algebra. And this algebra splits. For instance, you put this. And it splits. It's as a market to do by two matrices over your. But what you get is an additional set to creating. This is not necessary depends on alpha. Take. Yeah, in this case, it's a Quartan algebra simple. If you're in K star. This is a simple algebra, but put alpha beta equal zero. It's a grassman algebra. So it's not simple. Can be grassman algebra. Grassman. Dimension for sorts. This sense not simple super commutative in this case but the point is, you have a set to creating so and this set to creating looks like this. The matrices of this type are even. Even part and matrices of this type are on. Usual rules, even times even or times odd is even, and the remaining products are out. But it's a set to create the algebra and this is what the super linear group is. Now you can take arbitrary. Take two integers and then you can do block matrices to take n by n m by n. And then you can define the, I better write it like this. You get the. Well, I like it like this. It comes to break it. But the same rule. Yeah, it's obvious because. You multiply the block matrix of this size again, block matrix of this size. So this is a set to create it non commutative algebra. But the point is, it's well known for everyone who knows what he has price and I suppose everyone knows what it is. If you take the user commentator makes this into a the algebra. However, we have a so called super commentator. If you have an ally in creating the super commutator makes it, you know, sleepy. As for super super algebra. But what is a super commentator I don't explain. It's too complicated here and to boring the sign rule, you can rewrite this definition of a super algebra in more convenient way. So super the edge was an ordinary leave as a vice the even part as a vector space it's a direct some even odd. The even part is a lead algebra is a user commentator of the bracket. She won is a she zero module. In the sense of the algebra module. And then you have a bilinear map from the odd part to the even part, which is symmetric if you exchange these two variables. We call the joint repair representation make this is a zero module itself. This was by definition, a module, and so this would be in fact this encodes the super commentator in an elegant way. The structure of us tells you what the super commentator is on on this with itself this with this, and the odd super commentator, the super commentator it's an even pair it's a symmetric map from the odd part. Yeah, to the even part. And then you have the Jacobi identity is built in this. The only thing where they try to call the identity has to perform with it as if you start with x, the odd part, then you this frozen in even part you can apply the even part on this module. And it should be zero for all inputs. That's a super legal. Okay. So you look at it. But that's the most convenient way to put it in terms you can really work with. So it's nothing bizarre it's very easy. Now, but now a little bit abstraction. So one can define a so called tensor category. What is a tensor category I will explain in a second. I don't want to try to make this precise as precise as possible after being a little bit sloppy, but a tensor categories as an obedient category, like a lot of vector spaces. You have exact sequence is carols corkens and so on axioms on a building category. They wanted to be K linear with the homo me homomorphism spaces, where our vector spaces. So the homes are also K vector spaces. So this is my category. I come to this in a second and so if the objects are created is finite dimensional spaces here and here. I find that the dimension vector spaces. Usually one writes, if zero is can x one is cam. This is the third way of notation for a super space homo for morphisms are required to preserve the grading. But it's linear homo morphisms in the category of vector spaces preserving the creating the dual is just the component wise tool. Of course you have to dimension which is the sum of the dimension but you have now a new dimension so called super dimension which is the difference is to the important concept. And the reason is this concept here is an invariant of the tensor category where whereas the dimension is not. So if you have the symmetry constraint and here I require this epsilon. And we already saw. So in this category what is a polynomial. It's just a polynomial in the first variables, and the craftsman polynomial the second words, because of this symmetry rules. If you write what an algebra is you can do it by diagrams, but a commutative is can do it by diagrams in the category. And this you've got the polynomials are symmetric powers, but by this sign rule. If you really write it out it's symmetric in the bisonic variables and anti symmetric variables. Okay, fine. And this is an example of a so called tensor category or field tensor category of a field. And this is a, I said it in a building category homomorphism are killing our vector spaces yes or yes. All axioms of an a building category have a monoidal structure. It's a tensor product. And you have a unit element, which is an object object. And then nothing happens but you have to certain axioms of compatibility. And then you have to submit. What is going on here. You have this symmetry constraint. Sigma, which I saw two examples. Sigma is the just exchange gives you some kind of vector spaces and which is twisted a causal epsilon it gives curry. And then if you want to talk about an algebraic tensor category require that all homes are finite dimensional. There are vector spaces and you require that have a dual vector space, a dual object, which is the Tanaka dual explained in a second certain axioms for the duals. And then you have to find the spaces in terms of category see a joint function, but essentially the identity. We all know from linear algebra is this one. So if all these properties are satisfied. And have an algebraic tensor category over K. And typical examples of vector spaces and his super vector spaces, but also representations of a super group. Which are finite dimensional over K. But it's super representation so it means that it said to create a representation. And I will not explain what a super representation is because it's not important for this talk. I think everyone can figure it out in a second if he has a minute time, but the relevant point is I will explain you what a super group is. So it wasn't what a superly algebra is. So a super group. Here. It's lucky to believe I should buy the triple. We had a classical leal. Here we have a classical algebraic group over the field K. But this is an algebraic group like chill and whatever. This is a. The one as before and the BS before, if you erase this and replace it by a slia and call it fraction g zero. So a super group is nothing but a classical lea group, and you forget the group structure, you require that this lea group acts on the G one. Instead of its lea and automatically if you replace. It's a group action here, but then replaces by lea by lea acts or by definition. algebraic lead super group has a leader by trust replacing the group in the even part by its leader. But it's nothing mysterious, and what a representation is is easy to be figured out. So if you have a representation object if you omitted in terms of diagrams in this tensor category of finite dimension super vector space. You can what an algebra is what a new algebra is what the representation is can be written in terms of diagrams. If you do it, you get what the representation is. And this category of representation is a tensor category. So it makes sense to if you have two representations to define their chance abroad. Keep in mind, the representations by definition of finite dimensional representation so answer for the, again, the finite dimensional representation. And this categories. Yeah, the problem is these categories. The finite dimension. And as hopefully I didn't forget it. Yeah, super representations. I don't write bold face here. All my super groups are both face without both face. This is a algebraic tensor category. Okay. However, as an abelian category. And it's not semi simple, almost never. Yeah, except for triple reasons. For instance, if the super group is an ordinary group, meaning that this is zero, of course, it's the group is reductive, you have, you have the classical complexity of the killing. So anyone is natural to ask. I write here the limit. So the main statement is a tonic statement is that every algebraic tensor category. At least if it's finally generated. So generate by tensor part of finitely elements is a representation category of some super group. And it is a finite dimension, the super group. But you can get rid of this finally generated because every tensor category as a union and inductive limit are finally generated subcategory so in general the classification theory, you have to realize by protective limit of algebraic super groups. And there is this mu this mu is just a homomorphism into the classical group itself, which says, which indicates its eigenvalues of the non trivial element here. So if you have a representation of Chi, you get a map into the endomorphism of the representations, the minus element here is, I can spaces, give you the even and the odd part of the representations when you encodes the, the superstructure of the representations. So this is a complete set of data. So every super group. Because of the mu gives an algebraic tensor category and vice versa. The question is, when is it semi simple. Well, hard to say. But if it's semi simple, if and only if. Let's assume it's finite generator to easier grasp the concept but it's only taken assumption. So the super group has a finite unremovified covering, which is only in the classical part make sense of a classical algebraic reductive group, and a certain type of super groups which are so called autogonal groups. This is very exceptional groups whose representation theory is indeed sameness. But if you have that all super dimensions. Now I have to define what the super dimension is. Now I have to own half because of the classification theory. These are representations in super spaces so it makes sense for the super dimension of representation is. You can do it a priori but a posteriori by this theorem. You know what the super dimension is, if the super dimensions of all objects are greater than zero this is an old theorem. This is corrected by the lead that in this case it's just if the group is classically reductive of you is true. So this is our tonic and statements and the essential point here and that there's a key point for this lecture is, as I said, for super groups the representation users are usually are not same simple, but there is a so called tensor function. This is a semi simplification. Then we simply see them on same is simply the patient. It's called Omega. And this is a semi simple. Something of this time. It almost looks classic in the sense you can really say what it is. It could be anything, for instance, an average way as you buy cancer category. So, something of this type. Super Gucci and certainly the semi simplification is exists, but it's very complicated to describe. Sometimes crazy properties. So, this constructions were studied by Andre and Khan, 20 years ago, but goes back to study of motives and was introduced first I think by Uwe Janssen, who constructed a category of motives of finite fields. But this is a very nice construction I don't go into the details. It is unique if it exists. It's a tensor functor. The tensor fun does it's a fun of a billion categories. It's an exact fun to it's a tensor fun to it commutes with tensor brother, and so on and it's unique. Yeah. And yeah. Okay. And we will study this later for. And then the GL or SL. And then the algebra I introduced before, but what the classical group behind it is of course the general linear group so that she, you know, classical. And the lead algebra of it she won. It's just matrices. As mentioned before, which are all in them was and matrices. And the map B is just the ordinary matrix anti commentator I don't like. Yeah, it's the most natural representation. You have. For these groups I will study this in the simplification. And I give you an example, just to show you what is going to happen. If I take G to be the group. She'll to cross she one one is boring. Yeah, I don't write it on the board, but the first interesting cases is she. She'll to slash to which means four by four matrices, and you have a G to cross to something interesting happens if you take this tensor category in this case. And you take the tensor subcategory, which is means you take here all irreducible objects in T so it's the tensor subcategory generated by the irreducible object in this category. The tensor products all direct summons of this add to them take new tensor pros and so on and forth. You'll go to this same simplification. If I don't restrict to this it becomes too complicated to write down. I don't want. So I restrict this omega here and the image full image is a tensor category, which is simply simple sitting in here. It's very strange happens. This body's theorem is equivalent to ordinary representations of a classical group so these factors don't occur. That's the reason why I was restricted to the category generated by irreducible objects. The tensor category generated by a group T to true classic group but it's not a classical group it's a projective limit of groups that I said, because you take all irreducible objects. It would have taken finitely many. I would get a classical group reductive. But here at all which is a countable infinite number. You'll get something which is quite large. This group is the following. It's a group. I don't give it any take it as a homo to the multiplicative group. And the kernel is more easy to describe the product from zero to infinity of the classical as a to see, or overseas, your case. Or any algebraic those field of characteristics. So this group. You start with representations or even the sub category representations of a finite dimensional supergroup. It looks harmless it looks like chill to cross chill to plus something up. But then the semi simplification kills many many objects. So there are much fewer objects in here, then in here and of course much fewer objects here. But the representation of this. The underlying tunnel. Not finite. It's not that type I should break open a limit that is very large. You see the complicatedness of this super representation is really already at this example, showing up for the first time. Okay. So now come to the second part which I will do rather quickly. This is one of the simple things here. So conform a groups I will be very brief. On that. I think it's not so important, except for one point. I made some body. The interesting points are this here. Let me start. So what is a conform a group. I think everybody knows the classical things or you have vector space a non-determinary quadratic form over the wheels. And let me write it in the board. This is more convenient you have R to the end, but that is for signature r, s. And then you have to use the auto group. RS. And these are conformable with respect to the generalized metric and then you have translations. These are definitely isometrics here. Then you have scalar laps. So we scale, which are conformal. So if you take these together, they generate a group, which are called P is a generalized punk array group, but not the reason for P. This would be the punk array type group. But this is a parabolic group in the super in the in the, in the, in the conform group. conform a group. She conform the is generated by P. And an evolution star. As usually in auto group is isomorphic to all r plus one s plus one, but the evolution you can define as follows, you take any sigma. This is not so important, but in a minute if in fact is important. You just take the reflection on the on the on the on the unit ball. You take the Euclidean metric is just x divided by absolute value of x squared. This is a conformal web. And you know in dimensions more than three. These are the only conform elements here and their products. So this is this group. And of course, these are only densely defined so you have to define only defined on the dense open subset of our end because you have to see the zero locus of this clinic has to be removed but you can still define it as sometimes some maromorphic, so to speak group, but in fact the lead algebra. It's easy to see that act by in polynomial. So it conform a group is a nice group acting by vector fields, but not well defined production is a well known thing. And you have one possibility to repair this. And this is rather interesting now for my talk. So if you make a reflection in a classy case let's say as to our tool, you reflect at the unit circle. The point zero moves to infinity. That's the only annoying thing so if you add a point at infinity replies are to buy the, by the probability that you get the usual merguez transformation everything works by fine and the same you can do an average situation you just, you have here dimension and plus to remove the point zero, take the project real of space, and take in this the zero locus which is defined because this is a homogenous equation of this extended quadratic form here. And then this is a quadratic of the same dimension as the RN, and you can easily show that are any bets, and it's equivalent embedding pose you have natural action on the conform a group on this credit so on the space. And this is an open dense subset. This is the Minkowski space or this RN, in this case, and the vector fields active active active. So this is the conformal compactification. I do not want to comment further on this thing, except that I want to say that the same construction of conformal groups should work, should work for the super case. Let's say our for this is the space that's most interesting in physics, the Minkowski space with the Lawrence metric should be able to add odd pieces. I have written in red and construct a conformal group or supergroup or a conform superly algebra. But in fact, this is possible in four dimensions. It's not like certain natural assumptions but not in higher dimensions. It's not possible. So in fact if you require that the conformally algebra should look in a very specific way namely it's the leadership of the original group. And some group, which we've opened it should be anything. But you require that the odd part is the Spino representation of the Minkowski group so that's the essential impression is looks like it would be essential requirement that this part acts here as a spring representation. We make this requirement, which is usually what the physicists do. Then the only case is for dimension nothing beyond. So there are no really interesting supergroups conformed supergroups which after a nice property that the even part is really generated by the image of this map from the symmetric map called. We're looking for now super relations is this things like that. We call the one symmetric map a covariate is zero. And now you assume that if you take the image of this map and the generated subspace of this image, you require the, you know, even require that it's she's zero but you require it's containing this is an assumption. So we're defining the as you are a classic group. If you do this, almost no go beyond that dimension, and both is not very difficult concerning the Spino assumption here, which are made. This is quite natural assumption because the weights here are very small to the joint representation. And you have a symmetric map, it's not that on a on a vector, and you can see it's not that on the highest weight vector. So the highest weight work cannot have weight beyond one so it must be weight one half so you have Spino representation must occur but the assumption is the only type of representations are multiples of Spina half Spina plus or months. So then it's quite restricted, but this is the usual assumption made in physics you have spinos which should generalize the direct direct fields. So there is almost nothing so what remains is this case as you 31 or 13 is the remaining case. The question is, is there super conformal group, or you can also say SL to see which is the same they are isomorphic covering group, covering group. So in this case is there such a conformal super group but the answer is yes. And it's something an exotic exception structure. It exists for a stupid reason which except these are exceptional isomorphisms in the whole dimension. So carefully analyze the proof why it exists in the mental poor. You see the proof is exactly the same why the lead group eight exists. And the, as except the largest exception group, the proof is almost similar. There's a proof by wit, which I like very much and if you look at which proof for the existence. And while there is no construction beyond, you find the proof. Can you use it carried over to the, to the supersymmetric case so you have this super linear proof which is already quite complicated. I cannot write down it it's formulas. It's too long, it's too tedious. But the principle idea is very simple. Sorry, it's written here, you have to find a super involution and then we define this the complexification of this four dimensional conformal group. First for the complexification and then the descent to the real form by polynomial super vector fields on the super space, which means polynomials in this variables and craftsman. No, the polynomials are polynomials here and craftsman polynomials here, but active fields are derivatives with respect to the Bosonic variables on our farm, usually derivatives, and you have also super derivatives. You derive, let's say craftsman variable. Just kick it out of an alternative polynomial can make derivatives also in the super polynomial so you can decrease the decrease or vector field is something where the coefficient coefficients of the vector fields are super polynomials, and the derivatives are with respect to Bosonic and or family on each variable. This is a graded algebra. Once you find an evolution which extends the involution we get here. You can make the following definition. You can find these polynomial vector fields on this super clean in space, such that the convoluted differential operator you apply first the involution on the polynomials on the on the functions, then act with this vector field and then switch back. And if you require that the convoluted differential operates again polynomial coefficient operator. You can see that this is a graded vector space and must be finite dimension because of the creating. So that's very simply. Once you have such an evolution. And the point is you don't find an evolution in higher dimensions, but in dimension for you find, because you can identify our four with the emission two by two matrices as usually done in physics. And theta is two odd variables such just think of the first case where m is one, then is theta one and two or two craftsman variables. The craftsman polynomials constant plus constant times to the one plus constants times to the two plus constants times to the one times to the two. And the two point almost the chill to group acts because this is a C to complex and emission matrix can be inverted on a dense opens up that. Of course you have poles this is like in the classic case, and the inverted my matrix also acts on the odd part, because two by two matrices act on vectors in C to whether I even ordered. You don't, doesn't care. You see it's an evolution, it's densely defined, and you make this definition, and what pops out is exactly the conformal supergroup. You're looking forward to decide properties. So this group exists, and key point is so I don't comment on the decision strings here I think it's only interesting for physicists. Physicists very often considered the under the sitar space which appears in 10 different string theory, but the anti the sitar space is a double cover of this conformal compactification I explained before. We call we have this conformal compactification by this quadratic. It's not like manifold oriented, but it's a total and unremifed covering is the boundary of this under the sitar. And you know this, what is called under the sitar safety correspondence, the fields on this boundary correspond to supergravity under the sitar. So this feels on this boundary or on this combatification is what I'm having in mind for applications because if such a series called for superconformally invariant the representation theory of our group comes in. So when I come to the last part of the lecture, which is I cannot go into details here very much. So up here, you can replace it this is exceptionally group business, like here, the lead algebra here the conformal and also the super algebra. You have an exceptional isomorphism which identified identifies the complexification of our is as you to to is the same as the conformal the algebra. I'm not super. But if you do this, I put here an M, we get an SL for, but if you do fermionic variables here by this bizarre construction would you end up the group becomes very simple it's the SL we saw before. So it's well known and easy to get hands on. And that's the reason why I can prove something with the SL for what is an exceptional thing only existing dimension for most of the computation here really depending on this low dimension. But this category here is quite complicated. At least if MS, it's really super case MS not zero there's a classical as it forward and tensor brothers are not understood irreducible representations are trust the irreducible representations of the even part of the group so this completely well known. Well, the irreducibles are very easy but they're infinity of many of them non isomorphic ones. Handsome products are completely non understood, whereas in the classical case you have the little bit Richard's rules would calculate tensor products, all simple groups. And the category is wild, which means if MS bigger than two, which means that in the composable objects cannot be classified. It's very hard to understand if you're not a representation theorist, but it means precisely but it's very complicated categories. Then the next point is, you can hear this is literally filtration. If the category is not semi simple, you can objects are finite dimension you can filter by irreducible so take the maximum semi simple. The maximum semi simple sub module, you divide it out. The question to take the next maximum semi simple. The soccer again and you get a filtration. This is so called low, lower. The refiltration, the length of this filtration is bounded. And in our case is here the conformal group at most by nine, you have the most nine steps of filtration, but my nine is very much. You will see it in a minute why this is quite awkward. But the whole category looks a bit weird. So it looks at least completely different than what you know from standard one. Everything is tends to be semi simple in representation sharing what you can have there is continuous spectrum. Yeah, but this integrate but it's nevertheless you have a decomposition you don't have anything like that here. Okay. And what is the main tier and what what do we know we already explained this semi simplification functor. And the point is what does it do. If you put hands down on it. This functor kills certain irreducible objects and everything that goes with it. So, if you're an irreducible object L was super dimension. It's zero. Then omega of L goes to zero. It kills all irreducible objects with super dimension is zero. Okay, that doesn't look so bad because super dimension zero might be a tiny subset. But when super groups came first up, Victor cuts classified them and also try to understand their representations. And he coined the notion of typical and typical representations. He could manage the typical representations. In fact, irreducible ones. These are exactly one which are projective so same simple. And for them money far he found a character form like in the classical case, and all the rest, he called it typical. The thing is, if you saw a coin into the set of irreducible representations so to speak, the coin but almost with 999,999% hit a typical representation representation was super dimension zero which is projective, but goes away. The other two remains are the atypical representation but even not all you have a whole bunch of atypical representations in the, in the case here we study we have zero atypical which is typical one atypical two atypical three atypical typical and four atypical. This is maximal atypical. And the only ones that survive. In this condition are the maximal atypical representations. And this is a very thin subset which is combinatorial is not easy to describe, and not really understood. And it looks very bad. And what the nice point is, the kernel of this map can exactly describe an element goes to zero if and only if it's a direct summer. So this is the sum of in the composable objects, whose super dimension. This is the kernel of this function. Objects that go to zero. Super dimension zero. So all the objects which are direct some of in the composites, because it's not semi simple you can only decompose in in the composable objects, all those which are the finite sums of in the composable. All those that have super dimension zero. This is the exact kind of objects. Which got to see every single simplification. So only maximal atypical to survive this was a conjecture of of cuts and it is proven by now so you could also say the surviving representations are the maximal atypical representations. And I do not go into the details here so you see the coin here's maximal atypical these are the most complicated guys, and these are exactly the guys which make the category. Non semi simple. This is same feature, because the whole category decomposes in blocks or the direct sum of our building categories that are no extension between the blocks, and the maximal atypical are one block, and that's a black body highest very high arises. And if, and it's different from them, you can easily obfuscate the tunnel car groups and reduce it, which is the most interesting and the most difficult case. So I come on end. So there's many things to say. I want to say something on the tunnel car group I explained it in this special case. The same picture happens in the case one is considering here and the physical application but the conformal group is dimension four. It's the minkowski space is because of this exceptional isomorphism between. Two comma four with SF four or two, or with a group which contains it. And as I said, conformal invariance, the most interesting places before, because you have an invariant measure here it's DX one up to DX for the theta one up to the theta eight recall we have our to the four times. And I mentioned super space tends to m, if m is four, you have eight variables here they scale with square root of these variables. The odd variables scale with a square root so there's an invariant measure, but in this case, the most interesting case. Yeah, this I showed you what the result is here result is more complicated. I skip all these things. You can complete all this interesting invariance here which I think it's not so important. What I want to show that these are typically representations are classified. By certain diagrams. So the irreducible switch survive in this functor fall into 10 families are to be precise 14 families, but you see they are indexed by trees. So we have four points which comes from the fact that we study here. G44 of people would look at chill and and they would have n knots, the trees so we are here in the four dimensional case so we have trees with four knots. But we look at forests. There's a forest with one tree. Maximum height as a forest with tree with two branches. There's no tree with three branches. There are two trees in the forest. There's three trees in the forest for trees, young trees. And there are trees with one single route you say, especially these, these, these, these. These, these, these, these have one note and the others have more. And I wrote certain groups below. And these groups play exactly the role that the group as a tool did here. What is the sum. The sum is over all representations in two families they have two families here. The one family consists of me with zero. This is the triple representation. I belongs in this case to the analog of this tree. But only with two knots. This is equal to zero this is this tree. If any of the tools it looks like this, and all the others from one to infinity. They correspond to a forest with two young trees. The point is, this is a family because you see if there is more than one route. You can in a forest they can have a distance. Yeah, these are trees for you for them and you can have certain distances is a variable. And so, for each distance, a possible distance you have an area to maximum a typical representation. And in this case, here, the new, more or less, if you have this this is a new one. If you have one point distance, you will do so. And so on and forth, you see, there's two families, but they all contribute. Oh, sorry. No, I really know these all from able to zero one. This means distance is zero sorry distance is one. Excuse me. And this one comes from this representation. And I switch between GL and SL. Forgive me, this has a reason conformable groups have SL, but I compute with chill. And the chill adds a certain determinant like in the classical theory, which is so called Paris in the term which is important only from a more generalist point of view which I come to in a second. So, these are infinite. These are infinite many members, but you see instead of the group as a tool which appears for this family. You have certain classical groups which depend for every representation in this family, you have to add in a product. As 24 here for every independent of the distance and SP, and each of these are used to generate an independent copy of a group. Like this, you start with some finite dimension supergroup but to get a countable number. They become independent and that is much reminding of physics situation. If you look at the standard model. The groups don't have anything to do their direct products, but you have generations. We call that three are many copies separate is three. That's the point and you see, if you do the smaller case here, very often is three from me only got two from me from you and various, the kind of maximalities is smaller. In this case, you have only this for families. And in this case, it's, it's much like chill to us to have only these two guys. And if you look at these two guys and freeze these two rules you see, it says, and these two. It's exactly what we had here. It's an inductive structure. And all most of the proofs use this inductive structure that in chronological focus. So, I come to the last point for a finish, I should say that each of these families has a unique member and namely, this representations have dimensions. They have super dimensions. You see the super dimensions of their representations which are yes is one, two, six, six, 122434812 that's the super dimension. But they have a dimension and the dimension depends on the spacing. The distance is zero. This is the unique basic member. It's that smallest dimension in this whole family of representation. What you see in this group is only controlled by the super dimensions always as a tool. This tool is a super dimension of all these irreducible representations. And, but you have dimensions and you take the unique one which is where you shrink this distance to zero there's a basic representation, just give you an example. And this is the largest case. In this case, you find the super dimensions 24, but the dimension of this is 11 million and something as the smallest dimension you have in this family is a very large, but it's the most interesting guy. You can describe it and construct it you construct cuts model is something like a verma module. It's a representation of power of two it's like spin or it's the only representation which is this funny project. And if you restrict it to the Lawrence group. So, I should say the super group contains the classical conformal group, the conformal group contains the group you start with the autogal group, or, as. And in the four dimension case this is the covering of the Lawrence groups the SL two. Yeah, and you can restrict to this Lawrence group and look at the spins that occur if you restrict this gigantic representation, it decomposes, but only representations of spin lesser than. And that is what physicists are interested because you know spin series Kayla fields one half is steric fields, one is vector fields, three half is gravity knows and gravitation was spin tools or their potential candidates for field theories. The groups that occur here. I rewrote it from, from this eternal thing on this tensor quotient. This is the tensor quotient which simplifies these groups are in four dimensions. These groups and what you see is the groups that appear in the standard model. But these groups only appear if you go to the large families. There might be a connection and the question is, would it be reasonable that it's really connection is done. I have no idea I'm not a physicist, but there's a simple impression one could have. If you are physicists and make any experiment. If you find knowledge you get a number where you get a number you, you compute something in a fine money diagram, I'm an integrals. And what is a fine money integral. You, I should write it down here. A fine money integral is something that is not quite well defined but doesn't matter, really, because it's an integral or something that is not well defined, but you put in here, a finite number of fields. And here's some term e to the minus, which is defined, but the integration is not defined because it's integration or the space of distributions which is only intuitively defined from my point of view, I've got the impression that it doesn't count because these entries can be made sense out of what are the fields that are fields, or even the distribution. But you should have better look at this to take the net finish functions on your manifold minkowski space but I take this compactifications let's take areas five boundary or this projective compactification of minkowski space you don't need compact support, you have values in a certain F, which is. That's the, and then you have distributions on that on the space so something where you can he or fee and so on. They depend on variables and so on back and forth. I don't explain what a farmer integrals but it's an object, which has values it's in the classic case of a direct field. You have here are for not the compactification of our for and here, for instance, you have the spin off field to see to left in the case of the standard model or see for in the case of classical Derek equation left and right combined. The point is on this space group act, it's Lawrence group in this classical case he has the conformal group. And on this vector space, if the conformal group act so you have a representation finite dimension, you can write something like the field with values in a representation. And the point is, can you write a lacrangian here, here's an integral or lacrangian density, which is not well defined but intuitively makes sense and gives you rules how to define it. Indeed. If you take one of the representations in this list I had. Yeah, for instance this gigantic one, or the smaller ones is a representation of the super group you have the super group acting on here and on here. So, what does it help. Well, it helps because you can define a lacrangian. Anyway, as you write the Derek operator you have kind of Derek operator in the setting, because this high symmetry. You can write a location, usually in the locations for Derek, yes, comes like this. Yeah, which means you have a bar linear form and an operator to operate. The point is you have operator, which brings back functions itself. And usually if you have a connection, you get out of this space, you get, like you take a connection or something like that, you'll end up in have something. If you are the algebra as he has something. And what does the Derek operator do you introduce the Clifford algebra and then can multiply back. It's a similar situation because if you consider so called cartoon geometry, which is just this situation you have here you take this conforming compactification. I need to see the boundary or whatsoever. You can write it because the super conforming group is this also super super situation, the super group extensively. So this is a parabolic group. So this is a client in geometry. There's a super parabolic. And so, in other words, and bars. Homogeneous space, super space to be precise of this type and you can do geometry like this, instead of looking at fields here you look at it. So P a query to use a convention is, you'll let it act from the right side on G. And the quotient is here. The space you are interested in originally. And then what you do is you look at a set of together their operator, you'll get the one form, a, which is what the physics also called a, if they have a field gauge field. It's, it's a one form. Where is a one form on G with values in the lead algebra of G. Well, this looks a little bit weird at the first side, but they have a natural derivative, which maps this. Or you can also. It is as well as in G. You have a natural derivative, which maps, you can also look at this. Which maps one forms with valued in an arbitrary representation of G you see a representation of G is a representation of P, and you can descend it to a vector bundle here. And instead of looking at vector bundles on this space with valued in F, you can again view it here as an equivalent object. And then you can write it like this. So, the rift is this maps to do that as a usual over and differentiation on G. You have the potential bundle. But now if you have a map. It goes from here back. You're done. The composite is something like a direct. If you do this together you have this. And if you add this, you can also apply. If you have the connection here, you can also write this. We can also do this same same procedure once you have a map. Equivarian thing from here to here, but you have because you don't need a cliff at algebra. You have a representation. And what is a representation. It's a lead group homomorphism from the lead group of representation F super representation is a leak group map from the super leak group. The algebra G into F. Let's write it and what is and it's the dual. And so in this tensor category. Haha. What does it help if you do allies this map. This map. If you do allies it. What is it. It's a map you bring this cheat to the one to this side. You move the G to this side. It becomes a G dual. You move this F to this side. It becomes a map of wait a minute of F. We leave this we move it to this side because we do allies. No, it doesn't depend. It's a map from here to here. This is correct. No, she moved here. Sorry. It's a map here from here to you. That doesn't look nice because we want to go the other way around. You can do allies this again, and replace F by after all this. It doesn't matter you replace F by after all, you get a map from, from here to here in this direction. And this is the map you're looking for you have to replace F by F tool. And you have to keep in mind that the lead algebra has an invariant form so as a representation itself to I wrote it in the more geometric way but this is cheese. It's kind of their operator and together with connection you get all the tools you need for, let's say classical age series. So it might be that in this series. If you apply this field. If you make now such integrals. You have fields with various in F. And I do these fields are self tools I wrote down. I, and write down tools or F and so on. And you have this is an invariant function, and you have an invariant. What a measurement or many you have many copies of M in this. But these are tensor contractions, the tensor rules of this categories determine what what the outcome is. And what you have with this is a functional of a tensor big tensor grid of spaces. And these fields, yeah, tensor. And this integral is a tensor for representation. And this is homomorphism this integral is a number that is complicated to see the one dimensional representation. So it's a contraction operator it's an equivalent map of tensor products of representations to the one dimensional representation. And now comes the point. You can compose it, at least on this level here without the dynamic variables next. This map has to respect the tensor product rules. So the tensor products to do in here are highly complicated. I cannot compute a single example of this of this tensor in dimension for I never succeeded. But if you press it down here. The tensor rules become these of classical groups. And these are the groups I listed you want as you to as you see. So, if was a miracle. These final integrals have the property that summons, which are in the current of this property have lower contribution asymptotically, depending on energy range. And this is suppressed at least to first order. You might come up with the impression that these groups rule the final. The truth is more complicated, of course, but an asymptotic expansion could happen. And this is a typical thing. This is something we might produce cancellations. I have no idea what this happens. This integrals. Yeah, yeah. Yeah, the quantities, all the terms which when you turn so out in the composites all the quantities if those that are in the composer and have super dimension zero if these are suppressed. In the integral in the sense that the asymptotically doesn't contribute to the leading term. You might not see it in certain energy ranges that we are testing at the moment. It's extremely complicated. I did computer computations. I did not succeed. I could not even resolve one representation. I have a question, namely, I was interested just to be to be honest, this is complicated is 11 million dimensional representation F if I take the tensor product of this representation. It says approximately f square times 10 to the what is 612 dimension is approximately this. At least 1000. You're done her the constituents. I do not know what you're done. Components are I do not know what the levy filtration is. And the interesting thing that we represented expected to have several levels, seven layers. Seven pieces this soccer. The maximum semi simple part on the top. What I'm thinking is this is the middle. But the interesting point is also here, because it's not simply simple. If you have a map to something simple, you must have a one dimensional representation on the top, because only the top survives if you map to a number so in these interviews. Yeah, so that's that's part of this philosophy is horrible, but it might be helpful but you cannot compute anything I do not know what the answers. I tried on a computer, but non-simple simple objects there's no good common logic algebra on that. You can extensions you can do, but I have no idea how to have adequate methods from mathematics to deal with it. I don't have a computer but my running on a computer broad problems because I always put data on a on a on an external drive and back and forth and back and forth the system hangs up after 10 hours or so, and I couldn't do this. And I didn't even do a final integral in any complicated case. There is the simplest case with the SU two, I started computations and, but I didn't do all computations just started this and there might be indeed the problem is you have to find the connection and this contribution is complicated. And you have to do, you have to really know what you, what you use as a final integral you have many degrees of freedom to to define the location you can add terms you have to curvature here as usually the curvature of this carton connection. Hey, they have a curvature term in light is there with the twice curvature which usually appears, but the truth is, you shouldn't even consider usually connection but a connection in the sort of in the sense of quilling which is, it's not the one form it's you can add some odd forms and higher degree and so they have to go a little bit more complicated words perhaps. And so I'm not even sure what the right fine and trust me have to be this is just a speculation but let me end with one sentence independently of this is nonsense or not. In fact, once you accept the conformist super group of minkowski phase is important you cannot avoid this problems of this category. So whatever you do. It must play essential roles in computation of woman. Whether it's whether it's this or not I have no idea but I think that's a matter of fact that's mathematics which rules behind the tensor products. The, the, the, that's a simplification even that is complicated. And you cannot avoid. So that's that's my final comment on that. Yeah. You mentioned several times from the setting that for was very important. Yeah. So, I, what, how does it be the fact that things happening in in real life are connected to things being manifested in mathematics. Yeah, that's an old question physics I have no idea to air is modern. I have a certain name argued that only light propagation can only happen in certain dimension four or three plus one and so on. You can make many of these speculations. I wouldn't never dare to make a speculation like this I'm not a physicist have no intuition for that. So this is a mathematical obstruction, if conformality as a conformal invents of the language is important. And if supersymmetry is important, then you cannot avoid this and the rest is mathematics and you have to think what are the conclusions what they are, I have no idea, but in fact if you believe in both dimension four is something which is extremely. 10 dimensional super space is what you do you don't have a fully super. What you do is you play tricks and I can tell you what what they do because it's always difficult to say what is like this and that you. So in 10 dimensions are 110 the conformal space would be what that okay 12 dimensions are the SO to 10. And you don't have a conformal group with a nice property that it looks likes. So if you look naively at something that looks like as oh. And plus something which is called our group the physicists and so that is the even part. And the, and this G one which is spin type, whatever it is spin means you can have many copies doesn't mean if you look at this it's not possible as I said but what they do is, and that's a nice substitute is this is, if you look at this part it's easier to understand, then you have, let's go to the complex situation is easier. Yeah, I did as well complex. In this case, you have the spin representation. It's due to the six dimensional it covers in two pieces the spin. The composites as plus plus as Minos, which is dimension to the five, which is 32. So you have a map from there to there and then the self dual representation, and the self dual pairing is on. What you get is an embedding or a faithful representation of this group into the symplectic group. This is because of this oddity of the self duality of this spin representation in 22. 32 dimensions and what you can look at here is an extension, where you replace this by this SPO we had it before in this classification theorem. This one. Now, again, our complex numbers. This is odd 32 dimensional and even one dimensional form. This is super. And, simply, the odd in classical sense this group is a nice superlier group or simple, see algebra. That's exist. And, of course, as an module under this group, this is a group here, you can decompose this into SO 12. It's auto complex, which is something. And then you can pull back the odd part and push it there. But then you're killing form. What does it do. So the odd part is exactly G minus spin representation we did it like this spinning, but but then you get the spin here. But if you take the anti commutative of two spinors. Well, what does it put out it puts out a sum of an element here. It's a component x from here, small x or let's see. From here, plus a component from the auto complement. And this is some practical. This is some, this is some complement and if this is easier for a remember correctly. If you would decompose the symbolic group group classical one into the minute SP 32 is as a module of the smaller group as well as a 12. As, as a 12 modules. I think you get here the alternating fifth power of the standard representation. Both dimensions standard presentation and this gives you a 12, a five form, it can be interpreted as a five form alternative in five form. This is the definition of this bundle five form on our job. And what physicists do this is this is not a supergear to buy anymore is that some time which doesn't fit in. And you can even not put it into an hour because of the direct sum. It's only existing in the sublactic group not in the autograble. But what they do is they invert as five, five sphere, and this I RDS until the city space. And here, they restrict section of this bundle, or this Lee, I just want to ask you, and integrate our as five. The, the pullback and that's a star of this form, whatever the form is differential form. And by this integration, you kill this. And up is, is the artist five, which is again, the S O, who stole symmetry, which acts on. So that's how they cannot answer such a question because this is very natural, you know, string theory in 10 dimensions is the only realistic body, no ghost theorem, no ghost theorem, you have to have full string theory, some higher but that's the way they and some of their models. Work around this problem now. Okay. Yeah. Yeah. Yeah. Yeah. This topology, these are algebraic groups. And that's important. These groups are fine group schemes, each single one. And it means what an algebraic group is. It's not a group in it's in truth it's a hop culture by dualized the category of algebraic groups is dual to the category of hop algebras, and the hop algebras this projective limit becomes an inductive limit. And it's a fin hop algebras and inductive limit of a fin hop algebras isn't a fin hop algeba, but over an infinite dimensional ring. And that's, in fact, an algebraic group scheme which is not a finite hyper algebraic super group seem also it looks like a product there's no topology. And in terms of algebraic geometry algebraic super group it's a hop algebra, which is a union inductive limit of sub hop algebras, which are very nice finite dimensional. And that's that's the way you should look at it something purely algebraic. There's no topology there.