 Fourth and final talk and finally I should bring the homological Hall algebra as in the title into the game and So that's the last opportunity. Okay, so where are we in our logic? So we started With the slogan we just want to count river representations and as a device for counting we Defined this motivic ring or this localized rotating ring of varieties and so this led us to Writing down this zeta function like a partition function like motivic generating function in the motivic Quantum space of the quiver and then we have seen three different ways of factoring Just write factorization one factorization two and factorization three three different ways of Factoring aq in a purely formal sense. So this was kind of a logarithmic q derivative and it led to motivic generating function of Hilbert schemes of Associated to this quiver the factorization two that was the wall crossing formula and it led us to look at these modular spaces of See me stable representations and their motives and then finally the third factorization brought us to DT invariance and to the Intersection core model G of these modular spaces Okay, so just from this purely formal factoring this motivic generating series lots of interesting geometry popped up And the slogan for today is we want to categorify Everything so categorify aq and the various factorizations and What does it mean to categorify this thing? So this means Find an algebra So really an interesting algebraic object in this case a graded associative algebra whose Poincare series is Precisely this motivic generating series So find an algebra appropriately graded whose Poincare series is aq so and In the first three talks I I really wanted to go into all the details of these shift factors Yeah, all these ugly shifts by minus squared of the left sheds motive This were really important for me because it is important in all this theory But just for for time reasons I will not manage to do all the details today So at some point I just so for example In the verification that this algebra which I will now write down has this property that is Poincare series is this I Have to skip a few details and work a bit suggestively Because otherwise we're just doing a 20-minute calculation, which at the end is almost trivial Okay, so and this algebra is the so-called homological Hall algebra or A candidate for this the only known candidate in general. This is the co-homological Hall algebra of q so that's the algebra I will I will now introduce as Precise as possible and then try to give you a bit of a feeling of the of the features of this and So what we do is the following so first recall Well, the the key to all the geometry of quiver representations I'll let me write this down here the key to all geometry of quiver representations was this idea that quiver representations our fixed dimension are Correspond to the points of some affine space on which you have a base change group and The stack of quiver rips into is a sack of iso classes of quiver representations It's just the quotient stack of this affine space by this group. Yeah, so this will be central again so recall This group action and for simplicity since I will write down many many of these spaces are D Let me now omit the Q Because our work over fixed quiver. Yeah, so I should really write rd of q, but let me omit it in the notation All right, so here's the candidate for the co-homological Hall algebra, which will just write as curly H of q Take the direct sum over all dimension vectors What like in the generate in a motivic generating function? We took the sum over all dimension vectors of the motive of the quotient stack and here we take the Aquavariant core moloji of the representation space say with the rational coefficients Yeah, so the direct sum of all aquavariant core moloji of these spaces this is the space the underlying space and We will see that this has more or less nothing to do with the quiver The fact where the quiver comes into the game is only in the multiplication, which we will now define So define a multiplication by multiplication on H of q by and now let me first define the the multiplication by a one-line slogan and then give the exact definition and the slogan is by Convolution along The stack of the stack of short exact sequences Should be along the stack of short exact sequences. So short exact sequences Define kind of a hacker correspondence namely from a short exact sequence You can either project to the middle term or to the two outer terms the two outer terms are representations of dimension vector d and e say and the middle term is a representation of dimension vector d plus e and this and if you somewhat convolve along this hacker correspondence, then you get the multiplication, but let's make this precise because ultimately, I want to convince you that all we are doing here is Basic linear algebra with a lot of conceptual overbau, but at the end is just linear algebra and also in the definition of this Convolution product. It's the same. So let me write down the following diagram of spaces and groups So we want to have a Aquavarine Cormology class here and produce an Aquavarine Cormology class here out of it So inside here, we take the closed subset of Now this is kind of a symbolic notation upper triangular block matrices Okay, so what does this mean? Well rd plus e consists of tuples of linear maps Linear maps from a di plus e i dimensional space to a dj plus ej dimensional space Okay, since we have this this decomposition of the dimension anyway We can split these matrices into two by two block matrices And we just look at the closed subset of two by two block matrices with a lower left block being zero Okay, that's the natural inclusion iota and we also have a natural projection map Namely this projection means forget this extra datum and only take care of the diagonal and What you then get is here a representation of dimension vector d and here representation of dimension vector e Okay, that's basically the Hacker correspondence Which we will use to define the convolution product, but we have to decorate everything with group actions So here we have a natural Do we need a name for this? Yes, we need a name for this story. Let's call this zd So here we have a natural action of gd cross g e here We have the natural base change action of gd plus e and now we also need Need a group which somehow Mediates between these two groups and what we take is the natural parabolic group PDE this is also a group of upper triangular base change matrices That's a maximum parabolic product of maximum parabolic Yeah, and from this parabolic group You can project to the groups on the diagonal This is the the levy Levy quotient and it also embeds into this group here and it next acts here because upper triangular block matrices Act on upper triangular block matrices So that's a simple idea and this actually gives you a very nice operation in acrobarian homology, so let's start with acrobarian homology With respect to the group gd of rd tensor acrobarian homology with respect to the group g e of r e and we want to end up in Acrobarian homology with respect to gd plus e of rd plus e No, and that's the geometry we can use for this Okay, so let's first take the goodness morphism in acrobarian homology to get to the gd cross g e acrobarian homology of rd cross g e Okay This map P is a trivial vector bundle, so we can easily change to the gd cross g e Acrobarian homology of the total space of this vector bundle instead Okay, then we can change the group because well here We are just taking the factor by a unipotent group and Acrobarian homology is insensitive to unipotent groups, so we can identify this to the P de Acrobarian homology of Z de Then there is a general induction Isomorphism in acrobarian homology It allows you to replace this by gd plus e acrobarian homology of the associated fiber product gd plus e over P de of Z de and From there you have a canonical multiplication morphism to the whole thing induced by This embedding and you take the composition of all these quite natural maps in acrobarian homology, and that's it okay Yeah, so every single step is a very natural operation in acrobarian homology. No mysteries there and Okay, then you have to verify associativity of this product that's lots of fun and you have to write down huge diagrams but Nothing really Spectacular happens in there. Yeah, so associativity Mainly has the reason that you can view the space of three by three block matrices in two different ways You can either view it as a two by two block extended by a one by one block or one by one block extended by a two by two block That's associativity Basically Okay So Wonderful So fact this map here, which I will now call multiplication star star Defines Unital associative But in general non commutative Nq zero graded Algebra structure the common logical whole algebra of q well even without the reference to Categorification of of Motivic generating invariance this looks like a fun object to study Yeah It's it's one of these convolution type algebras Which you see regularly in geometric representation theory. It's definitely something which deserves study But of course, we don't want to study it only because it looks natural But because it really should categorify a of q so I have to convince you that It's Poincare series is well at least closely related to a q That's the next thing. All right, so Now let me do this here. So what is this? What is the common logical whole algebra as? Graded or in fact, we will see in a minute by graded vector space Okay, so for this we have to study this equivariant homology and Now that's a huge surprise or better disappointment this group Acts on this vector space Linearly yeah, that's the this linear conjugation action, which we have seen many times as The topological space this vector space is contractible. You can easily contract it to a point Namely to zero since the action of gd is linear you can compatibly contract Our d as a gd variety to zero with the trivial gd action So just by being contractible you see that this is canonically Isomorphic to the gd aquarium homology of a point and this is Really a big disappointment Since the the structure of the quiver is not reflected at all in this yeah all the arrows of the quiver are gone The only thing we remember is well how large the group is how many vertices the quiver has Okay, but yeah, let's do it anyway the equivariant homology of the point is by definition of equivariant homology the same as the Usual homology of the classifying space gd is a product of general linear groups and you know how the classifying spaces look like They are infinite grass manians, so you can really compute this homology and it is nothing else then tensor product over the vertices of the quiver and for every single vertex you get equivalent homology with respect to gl di of a point and that is rational function symmetric polynomials in Generators x i1 to x i di So you have two sets of gender and you have generators are indexed by two indices one index for the vertex of the quiver and One index running from one to di and Then you take symmetric polynomials in these Yeah, and concerning homological degree the degree of this x i K is To K is K Sorry I'm not taking the elementary symmetric functions. It's just K. It's it okay This you can further identify Sorry for that with now you take the elementary symmetric functions in these and This is now in degree 2 di these elementary symmetric functions, and then you write down the Poincare series of HQ in this realization is a priori Some overall dimension vectors product over all vertices Yeah, and then we have a polynomial ring and the generators are in degree 2 to di and So it's one minus one minus q squared one minus q to the 2 di Mm-hmm recall how we wrote the motivic generating function This we wrote as some over all D minus L to the one half minus Euler form of DD divided by Modified Pochama symbol Okay Aha, just one quick question from a participant. Can you briefly recall what the park at a series of a graded algebra? Okay Yes, thanks Yeah Okay, and there's a missing t to the D. Okay, so so this I define as the sum over all D sum over all K dimension of the case a covariant homology D and then Q to the K T to the D Yeah, so the generating function for the dimension of the of the homogeneous parts and here a priori We have two gradings. We have the grading by the dimension vector and we have the homological grading Which we have to modify in a minute, but at the moment. Let's just take like Lexis. Okay That's the Jackson integral or Mellon Barnes presentation of a Q shot natural in this complex If only I knew what this is So could the person making asking this question, please send an email to me with some details I'll be happy to have a look at this. No idea Okay So so that's the the Pochery series of that algebra a priori we have the the dimension vector grading and the homological grading and And that is what we get from this elementary calculation and Well, we see a little bit of the features of the motivic generating function at least the denominator denominators look similar Here if you replace Q by L inverse then that's fine, but the numerator The Euler form is not yet incorporated Of course, it isn't because this guy as a vector space doesn't depend on the quiver on the arrow structure at all So what's going wrong? Well the arrow structure is encoded in the multiplication. Namely what I haven't told you in writing down the multiplication is While you see these dots here. So this was really hiding a problem namely This multiplication does not Um It's not compatible with Comological degree if you start in comological degrees K and L there you don't add up in end up in degree K plus L but So let me add something to this diagram if you start in degree K here and Degree L here then you end up in Comological degree K plus L minus the Euler form of the dimension vectors D and E Okay, so there's a hint that really the multiplication of the quiver plays a role Yeah, so but this means in particular that The Comological Hall algebra a priori is only a graded algebra Integrating by dimension vector and not by Comological degree because Comological degree is messed up There's one case where you can fix this if The Euler form is Symmetric globally symmetric then this shift here this minus DE you can rewrite as minus ED and then you can regrade the algebra we can renormalize the Comological grading by a term minus one-half Euler form DD and then this suddenly this multiplication Becomes compatible with the Comological grading. Yeah, you add a minus one-half DD here You add a minus one-half DD to the Comological grading here a minus one-half EE to the Comological grading here and then here you arrive at minus one-half DD minus one-half EE and This you can rewrite as minus one-half DE plus minus one-half ED so this whole thing is then nothing else than minus one-half D plus E D plus E Okay, that was the calculation, which I told you at the beginning. I will not show you so I showed it anyway. Okay, well Okay The problem is here So here you are taking the embedding of ZDE in RDE and if you push forward Then you get a degree shift and you also get another degree shift from from the change of the group here Okay, but this is a calculation. I really don't want to show now. Sorry Yeah, okay, so if the Euler form is symmetric you can renormalize the Comological grading by this and then You get that the whole algebra Comological Hall algebra is actually Nq zero cross Z graded You get another hidden grading and Then I guess you believe me that the relation to the motivic generating function becomes much stronger because well It's precisely here. This term minus one-half DD Yes Yeah, yeah, you're right. I should better Yeah, yeah, you're right if I have an odd number of loops at a vertex, then I then I have to allow half integers Thank you. Thank you Yeah, one-half Z. Okay Okay, and if you do this little renormalization by the Comological in the Comological grading then you certainly believe that in the numerator this additional term pops up minus one-half of Euler form DD and so That is roughly the reason why the Comological Hall algebra categorifies a of q Okay, so we categorified a of q Actually, yes, it is yes, this is only for special quivers, but this is interesting enough Is this some sort of a colloquial condition? No It's simple minded answer, but no, I don't think so, okay Okay Well globally for this Euler form to be symmetric You need that the quiver is symmetric the number of arrows from i to j is the same as from j to i Globally, this might be very special, but we already have seen this condition locally Yeah, that the restriction to a certain slope has this property this will of course resurface now Okay, I should briefly mention something But for this I will not write down the formula because we will not use it The fun with the Comological Hall algebra is that in working with it you can argue geometrically Yeah, really compute homology of say certain strata in these are D But you can also work algebraically because there's an algebraic description of it. So using a covariant localization you get an explicit algebraic description as a shuffle algebra with with kernel explicit description of the multiplication as Shuffle product with kernel. I don't want to write down the formula now Because it is really long and I have to explain all the terms, but the idea is somehow Quite natural So as a vector space we have identified the co-har with well tuples of symmetric polynomials Now what is the multiplication in terms of these symmetric polynomials and one can really make this explicit Using standard techniques a career in localization. You really take these tuples of symmetric polynomials You perform something called a shuffle product But in the shuffle product there's an additional term the so-called kernel which reflects again the arrow structure of the quiver Yeah, the formula is long and we don't really need it in the following because we will stick to our geometric intuition, but the fun in co-har is Combining the geometric approach and this purely algebraic approach Yeah, so sometimes you just do a few pages of algebraic calculations for shuffle product All right, so we accomplish the first thing for today namely we Categorified a queue We realize that as a punk area series of an algebra Which is naturally associated to the quiver it reflects the The geometry of the quiver the geometry of this variety of representations and it reflects the the category behind the quivers It reflects the category because well what we are doing is we are somehow Involving all short exact sequences in the multiplication. So the whole category structure is somewhere reflected. So Categorify a queue and now the question is What are then the categorification of these various factorization identities, which we found in the last three days and This is what I will now show Okay, so we got three factorizations and each of these factorizations has admits an algebraic analog for the Comological Hall algebra and I want to formulate this and get to the point where you see that That these algebraic facts, which I will write down are really Categorifications of these factorization identities Okay, to do this we need one more ingredient So if mu is again Final ingredient So if you not only have the the quiver but also a stability on the quiver in the form of such a slope function as before Then you can define a local version of the koa HQ x see we stable So for any real slope x and the slope local version is Well, we have a slope local version of the motivic generating function so you can almost guess the definition This is defined as direct sum of Aquivariant Cormology But now the direct sum is only over those dimension vectors Which are of slope x and This time I'm ignoring the problem with d equals 0 you can guess what it is and Then we don't take the Cormology of The whole variety but just of the open part Of semi stable points Okay, so in complete analogy with this slope local motivic generating function you have the slope local Cormological Hall algebra Set of semi stable points is GD. Aquivariant. That's why this is well defined and Concerning the convolution along short exact sequences if you have semi stable Substantations of a fixed slope form an Abelian subcategory. That's a completely general statement Yeah, in particular if you have two semi stables of the same slope take an arbitrary extension is again a semi stable of the same slope So that means our convolution product along short exact sequences preserve semi stability and we really get this local algebra This guy is much more serious than the the global co-op the global co-op admits a very simple algebraic description Because the space whose Aquivariant Cormology we take is just a vector space and Here it is just some strange Sirisky open subset. Yeah, so this guy you can not treat algebraically you can only treat it Geometrically all right, so um, let me write down categorified Facturization one so we have to remind ourselves. What was this factorization one? I took this motivic generating function aq with a slightly shifted Variable t and divided it by another shift of the variable t and then what we got was the motivic generating series of Herbert schemes the algebraic analog of this is if if you take the direct sum over all d of all the non-aquivariant usual Cormology of these hermit schemes Then the Cormological Hall algebra acts on this is a module for the Cormological Hall algebra for any n and in fact one which is even cyclic and If you take the limit for a large n then You get the whole coa In a sense. Okay. Let me try to make this this precise. So Let's let's take this this thing here and now let's change and make it bigger and bigger Well, this is a module which is cyclic. So you always have a projection from HQ to this Originally you have an action of HQ, but since it is a cyclic module. You have a projection Okay, in fact, they're even natural connecting maps between between these so We can take the limit over all n now I have to be careful it's the Inverse limit and in the inverse limit We have an isomorphism so that means the kernels of these projections get smaller and smaller At the end Yeah, so we can approximate the Cormological Hall algebra by all its modules and these modules are given by Cormology of Hill schemes And that is the categorification of the first factorization formula That's quite satisfying, right? We have an algebra and the Hilbert schemes give us natural modules Okay factorization two is something you can almost guess from the definitions Factorization two was the wall crossing formula that the motivic generating function aq is an ordered product over all the local aqx and So categorified factorization Two is the coha analog of the wall crossing formula and it just tells us that H of q is isomorphic to an ordered tensor product over the reals of These local guys these local coas Unfortunately, this is not an isomorphism of algebras, but only of by a graded vector of graded vector spaces q vector spaces Okay, but the map is some all quite natural So that's the way you Categorify wall crossing. Yeah, and now well, you can imagine if you take Pongary series of both sides in the right ring Then you get precisely Pongary series of the left is Ordered product of the Pongary series of these guys and since these Pongary series are the motivic generating functions This reproves the wall crossing formula or this is the categorification Okay, maybe we will have time for a concrete example where we will actually see this Yes Well the map again comes from the fact that our D is stratified Into hard on our Siman strata How did I call this D dot h n So you have these the stick composition into these locally close hard on our Siman strata and now you have to Look very closely what this induces in aqua wearing comology. Actually what we do is we first Pass to aqua variant Chow groups and show that the that the aqua in cycle map is an isomorphism Then we work with chow groups because well some things are just nicer in there Yeah, you don't have the long exact sequences of comology and there you can see easily that this decomposes the aqua rain Chow group into a career in chow groups of of these local guys But it's essentially the same trick just the hard on our Simon Does it also preserve the degree in the shifted grading of the ecology ha ha Yes, yes, if you start with a symmetric river and do the corresponding shift here and here then the this modified Comological grading is preserved Yeah categorified factorization three now That was the Donaldson Thomas invariance. Yeah What we did was we took the motivic generating function the local version and Roded as a plastic exponential and the coefficients Popping up there. We called the Donaldson Thomas invariance and then we saw indeed. They have geometric meaning This also works here. So and you can already guess what the assumption is assume again that the Euler form restricted to one slope is symmetric and Then as I explained globally Yeah, I only explained this for the global coha But you can do this the shift in chronological degree then also locally If this thing is globally symmetric, we can renormalize the chronological grading on the whole coha and under this local condition, we can renormalize the grading Locally for the local hall algebra So let's do this renormalize the cohomological grading on the local Hall algebra Then So we want to categorify this notion of factoring it into an exp to a plastic exponential and Well, this is really a key calculation, which one should do at some point If you do exp on the level of Pongery series Then what you are doing on the level of algebras is you're taking the symmetric algebra Yeah, the exp is just taking the symmetric algebra Well, if you have ever calculated Pongery series of symmetric algebras Well, you get a nice product factorization and these product factorizations are encoded in the exp so In this case we somewhat expect as Categorification the statement that this is related to some symmetric algebra. Well, this is not literally true, but You can filter this algebra and the associated graded is then a symmetric algebra then there exists a filtration on the slope local cohomological Hall algebra such that the associated graded is Isomorphic to a symmetric algebra of Some by graded space dt dot dot and a free variable z in degree zero two Okay, let's try to digest this Let's compare it with with the aq side. So that was there was the factorization for the aq Okay, so taking the associated graded with respect to a nice filtration does not change the The Pongery series at all yeah So the Pongery series doesn't know whether you took the associated graded or not, but it has to be there and and Then you get a symmetric algebra because on the level of Pongery series taking the symmetric algebra is the same as taking this Plastic exponential, but you have to be careful. We are working with even and odd degrees So you have to understand this as the graded symmetric algebra so you have to understand in the super sense the symmetric algebra over Even variables is a polynomial ring the symmetric algebra over odd variables is an exterior algebra We'll see this in the example Exterior algebras really appear this free variable that that corresponds to this obligatory term here which we have seen geometrically coming from the virtual motive of a trivially acting multiplicative group of the complex numbers Yeah, so this appears here as a free variable and this double graded dt space Well, that's the space whose Pongery series is this Yeah, so Going through this term by term you really see that this fits perfectly with what we wrote down yesterday Is this semi-stable koho a sort of root sabal jubah a root sabal jubah Yes, okay, so I will give you in a minute the example of the Konecker quiver then you can really see what happens Yeah So in general the problem is well, you have to find a stability Which is well adapted to the roots of the corresponding root system. This is not always works out well We already discussed this two days ago But in the case for example of the Konecker quiver it really fits nicely and we'll see this Okay Problem is this Nobody knows what this filtration is. This is the Bandavis and call it the cause of the perverse filtration. So it's really deeply buried in algebraic geometry in the weight in the weight structure of homology and Well, at least I Can't compute it in any single case where it is non-trivial Yeah, so the only case where this Grading I will see this an example. There's one case where this grading really actually appears and plays an important role and In this case, I can compute the associated this algebra algebraically with this approach by Aquavariant localization and shuffle products, but I can't really work with a perverse filtration and so this is so this theorem is really nice and really general but with this theorem, it's difficult to work out examples and This brings us to finally two examples Well, our standard now we will also categorify our standard examples, of course Yeah, our standard examples always were the trivial quiver the one loop quiver and Then with respect to the DT invariance. We saw yesterday two vertex quivers And so what we will do is trivial quiver one loop quiver and the Konecker quiver and the Konecker quiver case We will see the full power of these factorization two and three so Examples here with the trivial quiver Of course, it would be great to do this in all details, but No, I'm not suggesting that we do this in the question and answers because All these shuffle product calculations. I don't know if I can do them out of my head. Well, okay, we'll see In this case the coa is The symmetric algebra Yeah, without any good without any associated graded. It just is the symmetric algebra over a one dimensional DT space Oh, that's something we already know right? We know that for the quid a trivial quiver We have a DT invariant in degree one only which is one Aha, so this DT dot dot should be one dimensional space. Okay So it's just a one dimensional space That's the DT and we have this free variable that so we have a polynomial ring Forget the ring structure just polynomials with its usual countable basis and then you take the associated You take the symmetric algebra over it But the symmetric algebra you have to be careful about it's in the graded sense and in fact this is now placed in odd degree Yeah, so this Q is actually placed in degree one one one for the dimension vector and one for the Coma logical shifted Coma logical degree So the symmetric algebra is an exterior algebra So the coa is an exterior algebra in countably many generators Okay, for the one loop quiver Well, we have also seen that the DT invariant Lives only in one degree and it was minus L to the one half. So also only one dimensional aha So we have again just this But now in degree one zero in the shifted degree in even degree. So what you get is really a symmetric algebra Symmetric algebra in countably many variables okay, and seeing this The countably generated Exterior and countably generated symmetric algebra appearing this of course cries for both on fermion correspondence as we have Already discussed on Monday. Well, unfortunately, that is the only example of such a duality Yeah, so the duality is when you turn the number of arrows in the quiver to the negative and Shift on the diagonal and the only case where this is really related to quivers is this duality between the trivial and One-loop quiver and that's it. Yeah So unfortunately this duality doesn't doesn't generalize to other classes of quivers Okay, so and final example is The chronicle quiver and again we have to choose a stability. It's more or less arbitrary. Let me take D1 minus D2 divided by D1 plus D2 and And we have seen the root system yesterday. Yeah, we have the take the a1 till the Root system or it's positive part. These are the real roots. They are of the form one zero two one three two and so on and then you see the slopes are one divided by an odd number One or minus one divided by an odd number. So what you get is You get slope local whole algebras only in degree only in for slope x and for all the One divided plus or minus one divided by an odd number minus one third minus one fifth minus one seven and So on so these are the only slopes which appear so This is a set of slopes to make it precise If x is a real number which is not in this set of slopes then the local Koho is the trivial algebra. Okay The local all algebras for plus or minus one divided by something odd Corresponding to the real roots. They look like the Koho's for the trivial quiver. That's what always happens for real roots Yeah, they look like the trivial quiver so exterior algebra in countably many generators and Finally we have this Zero part for the imaginary root and that algebra is really complicated. It somehow Feels yanian so It is actually generated by two series of infinitely many generators and Then terribly many quadratic relations Which are not really in illuminating if I write them down But it really looks like a very degenerate form of a yang in algebra So that's where really interesting Comological whole algebras appear But in general these local whole algebras are if they're interesting They are almost impossible to calculate Okay So that's all I can say about these koho's and that's it No the the whole algebras they usually define these wringled whole algebras You define in a completely different theory so Wringled whole algebras are defined by a similar Or let me just say by the same convolution product, but In very different theories, so if we stick to to the complex numbers then what you would take is the You would take the the space of Constructible gd invariant functions on rd Constructible gd invariant Functions on rd This is what you would do over the complex numbers or in the finite field case. You would just work with arbitrary Gd invariant functions, so this is really functions. This is not polynomial functions Yeah, I'm not talking about invariant polynomial functions just Invariant constructible functions So this is what you do over fq. You would just take C valued q sorry q valued q where it is enough q valued and Also here q valued just arbitrary gd fq invariant functions on rd fq and I don't know of any reasonable of any well behaved map from say a career in homology to a current to invariant constructible functions, which induces a map in on the whole algebras on a deeper level on A very deep level a posterioria after many many computations Then you suddenly see that certain algebras can be realized both as How all algebras are functions and they also pop up as common logical whole algebras? but the experts for this are Here in the room, but it's not me. Yeah, but at a very deep level. Yes, but there's nothing obvious some all a priori The two-dimensional cause oh, yeah, okay the two-dimensional if you start with say pre-projective algebras Yeah Computational finite fields related to computations Yes, yes Yes, so I When I when I Showed you the ideas behind the proofs of these of these factorization formulas I always tried this try to do this as Geometrically as possible But in fact, it is also possible to prove these factorization identities algebraically in This comological in this this usual wringled whole algebra. So To briefly indicate what you do Yeah, let me work with finite fields So I take the direct sum over all D and now here I'm taking the GD invariant functions on on this over finite fields and then there is a map which map The whole algebra integral To our motivic ring. This is not really true. We have to We have to complete here. Let me take not the direct sum of the direct product There's a natural map from there to there namely um Essentially you map a function on our D to No, let me not formulate it. Sorry Let me just say this exists a Q algebra homomorphism and that means you can also formulate identities in the wringled whole algebra and you Find these identities somehow out of the representation theory of the crewers and integrate them to get identities here so representation theoretic identities in the left-hand side, which is the wringled whole algebra integrate to certain factorization identities in the motivic quantum space and Well, here is don't see the motives One feature of the specialization is that Q is mapped to left-shades motive Which is quite natural because well the number of of of FQ rational points in the affine line is Q And this should map to the left-shades motive okay, just as a As a vague identification of how one can prove these factorization identities representation theoretically which is an aspect I didn't want to elaborate on because I wanted to emphasize the the geometry and At the end the the koa categorification Yes, that's what yeah Well, so so so that's the pragmatic solution what one should really take here is The motivic whole algebra there also exists the motivic whole algebra and this would be the correct thing to write down there Namely the motivic whole algebra Would be that we're working this completed setup again you take the Not the good the group of varieties over See but a relative version got the group of varieties over the variety Okay over the quotient stack Now okay, I should do stacks. Okay So all this is in a nicely explained explained in Tom Richland's papers Yeah, you take the the Grotten D group the relative version of the Grotten D group of stacks over the quotient stack Rd by Gd and On this level you also have such a convolution product and from this You have an algebra homomorphism to this because Counting points over finite fields is a motivic measure Yeah, so the map here is just to the motive X you associate Well, the number of rational points over fq of X basically because this has this motivic behavior So that's yet another ingredient which we would just behind all this So questions on zoom Is there a double action of the coa on the co-mology of Hilbert's scheme a doubler. Okay No, so there are many attempts to to double this action. Yeah, so I Know the direction of this question. Yeah, so the HQ acts on on direct some of co-mology of of Herbert schemes But the fact that this is a cyclic module is a bit disappointing what you really want to have is you want to have something like a like a Drinfeld double of this for which this is well, maybe even an irreducer representation So the action of aq HQ is raising the degree Raising operators But you also want lowering operators you want to have this this should be kind of a fox base Yeah This was defined somehow in an ad hoc way in several ways, but there's no easy geometric way to do it Yeah, so Not clear how to do it in general Another question some people have answered it, but still maybe you want to answer it Are the shuffle algebras which were mentioned related to those which appear in the description of quantum toroidal algebra? I don't know I'm glad when other people answered great Again, it's two-dimensional. It's again this two-dimensional thing. Okay There was a question by Fabian Yeah, I wanted to ask for this integration. You don't need the assumption that the curve is symmetric No, no, that's that's at this point. It's completely general Yes, please So in this theorem about the associated great beam symmetric, yeah so can this DT Break it that can be equipped with some lee break it so that Palschers look looks tempting, right? I mean the the best known example of an algebra Which looks like a symmetric algebra up to taking the associated graded with respect to a Filtration is the enveloping algebra of a lee algebra because then you take the pbw filtration and the associated graded as a symmetric algebra That's the pbw theorem And of course tempting to look for a lee algebra structure. I think again the answer is in the 2d setup so for pre-projective algebras there you can expect expect this related to Well, the analog of DT invariance is then a cut poly norm yields and you can expect then interestingly algebra. I Doubt that there is such an interestingly algebra here in the the crew of setting. Yeah, the crew of setting is somehow to de-generate I guess in some appropriate Modification this still holds and was proven in full generality by Sven Meinhardt and then Davis but It's even less explicit And I don't know of any examples which you can calculate in the three CY setup. Oh Yes, sorry It's again by a convolution. Namely you you look at this basic convolution diagram Which we had ID plus E and Well, this this helped the end This we realized as a quotient Well, we only did this in the exercise session. Yeah, but we did You take a certain set of stable points in an extended representation variety and that does the trick So all you have to do is Extend this convolution product To these extended representation varieties, but you shouldn't extend everywhere. Yeah, so here you take the usual Unextended here take the extended here take the extended and some corresponding Hacker correspondence and then you have to think about that this everything is compatible with stability and Then this convolution operation gives you this action Yeah, and then to check that this is really a cyclic module you have to know about the fine structure of this convolution Thank you