 So, thank you very much for the invitation to speak here and the opportunity to present my work. So, and also to speak in front of Mazaki Kashiwara, whose work has influenced many of my current research. So, in this talk, my goal is to explain how classical results concerning representation of Lie algebras, if they are adapted in a suitable categorical framework, can be helpful to understand classical results of algebraic geometry. So, I will first present two parallel stories concerning the geometry of the diagonal of a smooth scheme. If all mistakes are of that sort, I would be happy. Okay, so the first one is the so-called Hochschild-Costen Rosenberg. Okay, isomorphism. And the second parallel story is the least structure on the shifted cotangent complex, complex or the coulis structure on the cotangent complex. So, now the modern point of view is that these two things are related by what's called the derived loop space, that's called LX, which is the derived fiber product of the diagonal with itself in the category of derived scheme. And so, L of X is a derived group scheme over X, if you fix a projection. So, it's Lie, algebra is exactly that stuff. This shifted tangent bundle with its least structure. And if you look L of X over X, it's also the total space of a linear vibration, which is omega X1 of 1, and this is the HKR isomorphism. So, this is a modern way of seeing these two things related, but I will do a bit of history about these two objects. So, originally goes back to these three authors for a regular associative algebra A over a field K of characteristic zero. So, HKR isomorphism computes the Hofschild homology ring of A. It's just the east differential over the base field K. And so, this statement has been generalized to geometric contact. So, let's say if X is smooth and projective over K, and if I look at the diagonal injection of X into X cross X, the geometric HKR isomorphism can be written as an isomorphism between the derived pullback of the structure sheath of the diagonal and the symmetric algebra of omega X1 shifted by 1 in the derived categories of sheaths on X. You mean non-coherent sheath, just sheath. As you wish, current sheaths also. You can put with current common g's. So, the first statement is smooth algebra over K. Regular. Regular. It doesn't require characteristics. Here, regular will work. So, in the first statement, HHI, you don't need characteristic zero. Are you smooth? No, but for regular, I'm not sure. Ah, no. Let us say are you smooth over K. In the second, you need characteristic zero. Yes. Yeah, yeah. Here also, yeah. Let's assume characteristic zero. And so, this has been done by many authors. One, Yekutyele, and I was aware of an unpublished proof of Kashiwara in a letter to Shapiro in 92 that works, in fact, for even for X complex, which relies on a completely different strategy that you don't try to sheafify the bar complex, but you produce an ad hoc resolution on the first formal neighborhood of the diagonal in X cross X. So, on the other side, the least structure on the shifted tangent complex on the commutative algebraic side, it has been discovered independently by André and Quillon, I think in the 70s. So, this least structure on the shifted tangent complex of a commutative algebra. And then, the make-breakthrough was done in a paper by Capranoff, and later on by Marcarion, 99. And perhaps, I'm not aware of the literature, but the most up-to-date paper on this thing in the Union, 2016, where he proved the structure for X is any derived art in stack. So, for instance, if X is BG for an algebraic group, the shifted tangent complex is G itself, and you recover the Lie algebra structure on the Lie group of G. So, the two stories were linked originally the link was to prove index CRMs. So, Riemann-Roch type CRMs. So, for Kashiwara, it was in the framework of D-modules, motivated by a conjecture of Shapira and Schneiders. For myself, it was a motivation to the standard Rotendick-Riemann-Roch theorem in complex geometry where there are some cases that are still open. And I would like, again, to recall a formula due to Kashiwara that was my main motivation to enter into this story, which was a conjecture in 92 that I solved in 2011, is that you take X, a complex manifold, and you look at the following maps. So, you take the Rotendick-Cycle class of the diagonal, say, of dimension n. This is just the Rotendick-Cycle class of delta in X cross X. And then I project here on the diagonal, and here it's a canonical bundle. And then I use a dual version of the HKR isomorphism. And so, let's write, I prefer, okay, the direct sum of H i X omega X i. And the conjecture is that the composite map here is the taut class of X. So, it looks exactly, you see how it's linked with index theorems. It's exactly what you consider when you want to apply left sheds formalism. You take the cycle class of the diagonal, and then you restrict it to the diagonal. So, in the topological case, it's just the Euler class of the diagonal. But in the complex or algebraic case, it's much more complicated, and the HKR isomorphism entered into the picture, even if there was no lee theory aspects, lee theoretic aspects in a cashier's approach. So then, there was some conceptual explanation, let's say, lee theoretic explanations on why the taut class is related to lee theory. So, this is due to, this was hinted by Fegin. When you wrote in slow formula, it stopped with shifts. No, it's one, sorry, H. So, it was hinted by Fegin, and then done by Markerian. And the reason is this. If you take G, a lee algebra, say, a finite dimension over the base field, then you can consider the Duflo element of G. It is the element, I write it this or the inverse, depends on the convention. So, I see it as a formal power series of invariant polynomials on G, and a conceptual way of looking at this element is as follows. If you have a lee group G, such as its lee algebra is G, you have two maximal differential forms in the neighborhood of the origin in the lee algebra G. You can transport the left invariant volume form on G, so you transport it by the exponential map, and you divide it by the natural top invariant form on the lee algebra. So, it gives you a function in the neighborhood of the origin, and it is exactly the Duflo element of G. And in fact, what Markerian proved is that, if you take, so for G, now, you take Tx shifted by minus 1. So, this leaves in the derived category B of x and it is a lee object endowed with capron of structure. So, I have to describe. So, D of x, symmetric monoidal category. So, what is the lee bracket? It's just the Atia class of Tx. And if we carry on this story in the categorical framework for this lee algebra, the Duflo element of Tx of minus 1 is the top class of x. Okay. So, for interested readers, there is a recent proof of Grotten-Decrimenrock using this formalism by Gates-Gorie and Konsevich. So, this is a classical story for the diagonal and it's not written yet. I mean, Gates-Gorie gave talks about this, but... Okay. Sorry, but... You prove Grotten-Decrimenrock also? I mean, it's a real issue for you. Yes. So, as a matter of fact, Gates-Gorie-Konsevich approach works. It's a two-categorical approach, so it works. I think it's perhaps the most intrinsic proof you can get in the categorical framework, but the problem is with traces. So, in the complex case, it doesn't work because the category of current sheaves has not enough good properties, such as compactly generated and things like that. So, I proved... So, for the Grotten-Decrimenrock theorem, the first proof, so let me say summarize, the first intrinsic proof, so the original proof in the complex case was the O'Brien-Toledo-Tung, which is used in the left-shed formalism, but it's not at all conceptual. And then, so there has been the proof by Mark Harian, and on the other side, the proof by Kashiwara, so the unpublished proof of Kashiwara in 92, modulo the conjecture, so the conjecture that I proved that completes the proof. And there is this Gates-Gorick-Konsevich thing that's perhaps even nicer, but works only if X is algebraic. Okay, so now, so I was really interested in this interesting formula expressing the Todd class as hidden in the HKR isomorphism, and there is some natural thing to do to try to generalize the story, not for the diagonals, but for arbitrary sub-schemes. So I'm looking at a closed sub-scheme or a closed complex of manifold X and Y, and the question is, when do we have the same property, so the derived self-intersection of OX isomorphic to its formal associated object. So, again, this unpublished work of Kashiwara was really helpful, so in 92, again, Kashiwara proved that it is true if the embedding of X into its first formal neighborhood splits. It's made that the normal sequence is globally split. And Arenkin and Kaldararu, so 20 years later, proved that it is true if and only if the conormal bundle extends to a locally free sheaf on the first formal neighborhood. Sorry? Because there are two different versions of the equation. It's isomorphism, both in shifts of X. In the derived category, I mean... No, no, because one can write isomorphism given by differential degrees. I don't know. It's the derived... It's X linear. X linear. So in the derived category of OX shifts, yeah, yeah. Otherwise, yeah, well, you know, it's... Yes. Yes. Yes, but it was for the film. Okay. Pardon? Ah, it was for the film. So what I will call a quantized cycle, it's a triplet, X, Y, Sigma, where X is a closed sub-scheme of Y and Sigma is a splitting of the first neighborhood if it exists. So it's just a splitting of the normal exact sequence. And so if you are in this situation, so we can decompose the restricted tangent bundle of Y and restrict it to X, it splits at the direct sum of the shifted tangent bundle of X and the normal bundle. So... That's the assumption. Yeah, it implies, yes. If you have this, it implies a splitting and the splitting depends on Sigma. And then it's the geometrization of something which is well known in the framework of Lee's theory. It's called reductive pairs. When you have H included in G, but G splits as a direct sum as H modules. So this guarantees that the couple Tx minus 1, Ty minus 1 restricted to X is a reductive pair in the categorical sense. But the secret factor is not necessarily the answer. No, that will be my point. But... Okay, so... Now, the question is what is this... How to describe the Lee's structure on that guy? So since the first formal neighborhood splits, the second formal neighborhood of X and Y is determined by two comological invariants. So the first one is a class Alpha that lives in X1 of the conormal bundle with values to the second symmetric power of the conormal bundle. And it is simply the extortion class of... So I denote by I the ideal sheaf of X and Y. So all the sheaves lives in the first formal neighborhood and I push them down on X by my retraction. So this is N star and this is S2N star. And the second class, which I call beta, that lives in X1 omega1 S2N star, this class is the obstruction to lift sigma to the second neighborhood. And these two classes completely encode this second neighborhood. Okay. And so the Lie algebra, it is Ty shifted by minus 1 restricted to X. It's computation gives that it is. So I have these two factors, Tx of minus 1. So here the action of Tx on N, so I told you it's a reductive pair, so I have no factor on Tx, so I have no factor on N. It's simply the action by the Acha class, so the tangent act on N. Okay. On Tx shifted by minus 1, it is a Lie subalgebra. So it's simply the Acha class of the tangent bundle. So here it's the same. And the last part, so what is the Lie product on the complement of the Acha subalgebra? So I have two parts. The first one is alpha from N cross N to N, if I return by duality. And the second one, which arrives with values in Tx is beta. Okay. Why do you want the T-class direct sum 0 or something? Sorry? Do you want the T-class direct sum 0? Yeah, yeah, because I put the first component with values in Tx, and the second with values in N. Here is the bracket of two elements in Tx. So I'm saying that this subalgebra is no component in N. So here is the 0 in... Okay, so this 0 means exactly that it's a reductive pair. That is the complement is a module. So this bracket, it's just... It's a rest... So I just... Here it's only on the derived categories. So I'm not looking at higher bracket. It's just the Acha-class in homology. Okay, so a priori there is no reason, as I say, that there is a least structure on the normal bundle. But returning to my original motivation, I define a cycle class for quantized cycles. So the definition is the same as Kashiwara's definition. You start with a relative dualizing bundle. So this is a Grotendick cycle class of X and Y. Then I project it again to X. And then I use HKR. So generalized HKR for quantized cycles. So it depends on sigma. And I arrived into the symmetric algebra over OX of the normal bundle shifted by minus 1. And so this gives me a class that leaves in the homology of the exterior product of the conomer bundle. And the question is what is how to compute this class? So for the diagonal, it's the tot class as we saw. There is also the fact that what is new is that if X is the zero set of a section of a vector bundle on Y that vanishes transversely, and if there is a compatibility with the sigma, meaning that E restricted to X to the first neighborhood is a pullback of the normal bundle, then this class is trivial. It has only a component in degree zero. So in a sense, it's an obstruction class to have a linear tubal neighborhood. But what it is, so a recent answer to this question was given by Shilin Yu in 2015. It's a surprising answer. And the answer is if the pullback of the normal conomer bundle, which is a local free sheaf on the first neighborhood, can extend to a locally free sheaf on the second neighborhood, then this class can be computed. I won't be precise here. It looks like the tot class of the normal bundle. So a generated series involving Bernoulli numbers related to the normal or conormal bundle doesn't make much. So what I want to explain is, so again, use approach made no use of Lee's theory whatsoever to this homological perturbation theory. And this condition, this extension of the second neighborhood, I want to give a Lee's theoretic explanation. So I don't know if I said at the beginning, but everything is joint work with Damien Calac from Montpellier. What is the definition of the original... It's this. The definition of X sigma is this. You take the Grottenlich cycle class of the diagonal, you restrict it to X, and then you use HKR. It gives you a class like this. But the Grottenlich cycle class should involve some differentials, no? Oh, I just see it in the X of the co-dimension. For me, it's here. Everything is smooth. So there is no higher terms. So we introduce, so let me give some terminology. So it's for general reductive pairs. So I take GH plus and a reductive pair. So there are two cases, two general cases where the quotient is naturally a Lee algebra. So the first condition is called the split condition. It's the most easy you can think of. If you take the bracket of two elements of N and you project on H, it is zero. Then, of course, G is a cross product and N is at least a algebra. There is no component. Since the bracket gives no component of H. But we introduce the second condition that's called Tame. So here N, it's even an ideal, Lee ideal in G. So the Tame condition is that if you take the bracket of two elements, you project in H, and then you take again the bracket with element of N, you get zero. Of course, it's weaker. But this condition implies that N is also a Lee algebra, so with the restriction of the bracket. Sorry, pi N, thanks. Because when you take the Jacobi identity, you only need these brackets to vanish, not the first component. So what we proved is this. So the first part. It's purely the algebra. Yeah, it's purely the algebra. So this condition really means it's the subspace element and desegregation also. No, no, it's for any three elements. Any three elements. Yeah, yeah, yeah. No, no, it's for any three elements. And then what we proved for the first theorem is that the pair, so we're again looking at the quantized cycle. So the pair Tx, so this pair of the algebra object in D of x is, so it's split if and only if beta equals zero. So the retraction extends to the second order. So this is the easy case. It is same if and only if use condition is verified, which is sigma star N extends to the second neighborhood. So to prove this, it's simply you know the explicit table of the algebra T y restricted to x and you have just to interpret. So since everything is categorical framework, these computations involve extension classes. So this condition is, lives in x2 of s2n tensor nn, vanishing of a class in here and you just have to identify it with the obstruction class to extend this bundle as a second order. And as soon as you have this, so if I take a reductive pair, which is tame, so there is G action on the universal enveloping algebra of n with its least structure. So the action of n is the multiplication and the action of h, since n is an algebra in the category of h modules is just the action by bracket. And yes, you can prove that, I will go back to this later, the invariant, the induced u of n is the induced representation of the trivial representation of h from h to n. I will come back to this point later. But so now coming back to the cycle class, second theorem, it's the induced representation is the enveloping algebra of n with its least structure. What do you mean? I mean, it's induced so from h to g, sorry. So everything is in geometric in the categorical framework, so it says that it's very far the universal property. Okay, so the second theorem is that so I give again x, y, sigma, which is tame, then the cycle class is the du-flow element of the shifted normal bundle. So this is u theorem in a conceptual form. So I should say that this least structure in our case is just given by alpha by definition, it's the projection of the bracket to n. Okay, so the last part and I will give a hint of proofs that we did is so fundamental work of Markarian and also Amados is to compute the universal enveloping algebra of tx of minus one. So this is related to its geometrization of classical problems in least theory such as the du-flow isomorphisms. And in the setting of the diagonal, it's simply given by the projection, so again it involves the hkr isomorphism. So it's the x-algebra of the diagonal and most important in this framework the Poincaré-Birke-Witt theorem for tx shifted by minus one becomes the hkr isomorphism in cosmology. This point of view allows to understand the algebra structure on this object with respect to the Yoneda product because the dual hkr isomorphism is only additive but its multiplicative structure is encoded by the constant structure of the enveloping algebra. So here it's an iterated bracket involving the Atia class of the tangent bundle. Tx minus one is the same as h or n. So here, no, for the diagonal, so if in this case it will be n but in the diagonal it's the same r. So both are the same. But here I'm looking at this at n. So I should say that Calac-Caldera-Rou and 2 give a very, the most general algebraic structure for the normal bundle shifted by minus one even if there is no quantization condition it's what they call the derived Lie algebra. So it's a much more complicated structure but they computed its universal enveloping algebra. It will be the r-home over o-y-o-x-o-x but only as a sheaf of modules over the base field because it's an algebraic. So here we'll do a refined version in the case of the normal bundle as a least structure. So again, so sigma is sub-tame. Then, so the first thing is that if we derive, we derive the respect to the first variable. So I put left here and not as a bifunctor. So it's naturally an algebra object in d of x. So this is not obvious because if you derive only with respect to the first variables you cannot compose a priori. So it's an algebra in d of x and is naturally isomorphic to u of universal enveloping algebra of the shifted normal bundle. It means that I derive, it means the derived functor, I should say r-home o-y. So I derive this functor and I apply it to o-x. So I don't derive it as a bifunctor. So here it is, everything is o-x linear. So this functor goes from o-x modules to o-x modules because of that. O-y module to o-x module, sorry, yes. And it is universal enveloping algebra of a ly. Of this ly object, so it's an object in d of x. So when you say ly object in d of x do you mean you have to say it's a... It's an object with a morphism in any... I mean I can make sense of a ly object in any symmetric monoidal category. You don't need a sub-enhancement. No, no, no, no. I'm doing it, you can do this L-infinity structure and do enhancement here. I do it only on the level of derived category. So the first bracket and I kill the higher motor piece. Okay, so now a few words of the proofs. So to compute the cycle class we introduce the notion of let's say L-torsion morphisms. So this is a strange property because it doesn't seem to happen often in classical ly algebras but nevertheless, so I take c, a symmetric monoidal category and I take an object p in c and I say that the morphism phi from p to... So I have to add some hypothesis, something like Karubian otherwise and infinite direct sum. It's countable direct sum. So I take a morphism from my object in... so I take g, a ly object in c into the filter part of the symmetric algebra of g. I say that it is a L-torsion morphism when I take the following composition. So I start from my g algebra, ly algebra times the initial object. So I do identity time phi. So I arrive in my filter part. So by the Poincaré-Bierck-Covet I send it into g tensor products u of g. Then I do the multiplication in u of g and then I apply pbw reverse. So I go back to S of g and I ask that it only contains terms of degree L plus 1. So all terms in degree less than or equal to L are killed. Factors through SL plus 1g. It's definition. So I say that such a morphism... So it involves all terms but if I multiply morally by element of degree 1 in u of j it kills everything. So only the product on the L plus 1 component remains. So implicitly you are in category 0. Yeah, yeah, yeah. So you are symmetric, monoidal. Category... yeah, k linear. k linear where k? k linear, yeah, everything. Yeah, otherwise with characteristic k. It's k... the true hypothesis has to be Karubian. So every idempotent splits. It has to be k linear of a field of characteristic 0 and you have to allow countable direct sums. And then, I mean, we proved that it was in the folklore but pbw holds. So you're almost as in the classical case. Deline wrote it for a billion categories but in fact his proof transposed without problems. He wrote it in this case to cover gradedly algebras but if you assume Karubian everything works the same. So morally an L-torsion element it's an element in the L's filter part of the universal enveloping algebra but if you multiply by degree 1 elements everything is killed. You just get the top product after pbw. And the theorem is that if phi is an L-torsion morphism then phi is entirely determined by its top component of degree L phi is in fact the du-flow element of G contracted with its component of degree L. So as soon as you have the top components you go all the other ones by contracted with this element and then in the geometrical case you can prove that the morphism from the dualizing complex to this r-home o-x-o-x is an L-torsion morphism. Okay, and then no time left just to say for the description of the universal enveloping algebras so here it will be the top of the co-dimension. So the reason is really easy. It's just, you can explain, it's just if you do this so if you go from o-x to the shifted normal bundle so you apply the morphism o-x to the conormal bundle which is the Atchya sequence this is zero because o-y you map it to o-x-y-1 and this is a distinguished triangle and this thing is the dual of the multiplication G times u of G goes to G, u of G if you write it properly so you get the L-torsion property on the nodes geometrically. Okay, and last part is how you describe this the algebra structure on that stuff so you have this isomorphism so this is by the HKR isomorphism isomorphic to the symmetric algebra of the shifted normal bundle and so what we prove is by hand that the algebra structure here the structure constants are exactly the structure constants of S of G via PBW by proving the induction relations on all the structure constants. Okay, I'll stop there, thank you very much. Question remarks? Here, this one. It's just, it's this one. So this is... How about you hold o-x, not o-x? This is o-x. On the middle? No, this one is a die. This, this, this. So it factors through o-y goes to o-x second at the first order and the composition of these three ones I say that this composition is zero this is because you go there. This is the conormal bundle. Ah, you just say the composition is zero, not this. No, no, the composition is zero because because I compose the extension. I compose this map with the extension map which goes there. This is too successive. No, no, yeah, I see. No, no, it's not a distinguished triangle. The composition of two successive arrows of a distinguished triangle. Sorry, I probably missed it. If you go back to Leith's theory, what's the statement about u of n that you kind of assume that it's tamed? So then what? So I wrote it as an induced representation but in the classical case, so I have g is h plus n. So u of n as a g module is this statement and it was believed but without proof that in the geometric case this guy would be that and so the algebra of invariance of this algebra h invariance of this algebra would be the x-th algebra and it is indeed the case as a corollary of what we did in the tamed case. So it is really interesting because there is a conjecture of du-flow related this algebra so the center of this algebra is with the Poisson center the symmetric algebra of g over h and so this conjecture is widely open and takes the opportunity to point that there is one other case among symmetric pairs in the classical case which is well studied which is the cause of symmetric pairs. For symmetric pairs h is just a fixed locus of an involution and this case is somehow completely disjoint from the tamed case because in the case of a symmetric pair there is a canonical splitting but the projection so there is a v-algebra structure but the projection is zero for eigenspace reasons. But in the tamed case do you prove that the floc engine? No, no, no. So in the tamed case it says that it relates the h invariance of u of n but I mean there is no mystery because in the diagonal case so if you embed g into g plus g if you take the splitting given by one of the projections it's tamed this is the case of global retraction you end up into the classical du-flow thing so there is no simple proof of that. Other questions? Let's sing the spear again.