# Séminaire d'Algèbre

### salle 001, rez de chaussée, 11 rue Pierre et Marie Curie - 75005 Paris

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Lundi 8 janvier 2018 à 14h00

Ryo FUJITA (Kyoto), Affine highest weight categories and quantum affine Schur-Weyl duality of Dynkin quiver types.
[The notion of affine highest weight category introduced by Kleshchev is an affine'' generalization of the notion of highest weight category and axiomatizes certain homological structures of some non-semisimple abelian categories of Lie theoretic origin. By comparing such structures, we can see that Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor associated to each Dynkin quiver $Q$ gives an equivalence of monoidal categories between the module category of the corresponding quiver Hecke algebra and Hernandez-Leclerc's monoidal subcategory $C_Q$ inside the module category of the quantum loop algebra.]

Lundi 15 janvier 2018 à 14h00

Christof GEISS (Mexico et Bonn), Crystal graphs and semicanonical functions for symmetrizable Cartan matrices.
[In joint work with B. Leclerc and J. Schröer we propose a 1-Gorenstein algebra $H$, defined over an arbitrary field $K$, associated to the datum of a symmetrizable Cartan Matrix $C$, a symmetrizer $D$ of $C$ and an orientation $\Omega$. The $H$-modules of finite projective dimension behave in many aspects like the modules over a hereditary algebra, and we can associate to $H$ a generalized preprojective algebra $\Pi$. If we look, for $K$ algebraically closed, at the varieties of representations of $\Pi$ which admit a filtration by generalized simples, we find that the components of maximal dimension provide a realization of the crystal $B_C(-\infty)$.
For K being the complex numbers we can construct, following ideas of Lusztig, an algebra of constructible functions which contains a family of semicanonical functions'', which are naturally parametrized by the above mentioned components of maximal dimensions.
Modulo a conjecture about the support of the functions in the Serre ideal'' those functions would yield a semicanonical basis of the enveloping algebra U(n) of the positive part of the Kac-Moody Lie algebra g(C).]

Lundi 22 janvier 2018 à 14h00

Bruno VALLETTE (Villetaneuse), Ubiquité du groupe de jauge homotopique.
[La théorie de déformation des algèbres à homotopie près est codée par une algèbre de Lie de convolution. Comme cette dernière provient d’une structure pré-Lie, on peut l’intégrer avec des formules plus simples que celle de Baker—Campbell—Hausdorff. Ceci permet de décrire efficacement le groupe de jauge homotopique dont on montrera que l’action donne “toutes” les constructions fonctorielles de l’algèbre homotopique : théorème de transfert homotopique, torsion des structures par des éléments de Maurer—Cartan, hiérarchie de Koszul, etc.]

Lundi 5 février 2018 à 14h00

Toshiki NAKASHIMA (Tokyo), Geometric crystals on cluster varietie.
[The notion of geometric crystal was initiated by A.Berenstein and D.Kazhdan to consider certain geometric analogue to the Kashiwara's crystal base theory. Their structures are described by rational maps and rational functions.
If all these rational maps are positive'', such geometric crystals are called positive'' and they can be tranfered to the Langlands dual crystal bases'' by tropicalization/ultra-discretization procedure.
V.Fock and A.Goncharov defined certain pair of varieties (A,X), called cluster ensemble'' which is obtained by glueing algebraic tori using the A-mutations and X-mutations'' respectively.
They gave the conjectures on tropical duality'' between cluster ensemble A-variety and X-variety (called Fock-Goncharov conjectures).
We shall define the positive geometric crystal structure on cluster varieties and then obtain the resulting tropicalized crystals, which will be a guide to understand the Fock-Goncharov conjectures in terms of crystal base theory.
Finally, we shall show some compatibility of geometric crystal structures on A-variety and X-variety in the classical type A case.]

Lundi 19 février 2018 à 14h00

Travis SCHEDLER (Imperial College London), Résolutions symplectiques des variétés de carquois et des caractères.
[Nous allons expliquer une classification des variétés de carquois (Nakajima) qui admettent une résolution symplectique, et similairement pour les variétés des caractères d’une surface de Riemann (espace moduli des systèmes locales). Nous expliquons également un résultat parallèle pour les espaces moduli de Higgs de A. Tirelli (doctorant). Ce travail est en collaboration avec G. Bellamy.]

Lundi 26 février 2018 à 14h00

Amnon YEKUTIELI (Be'er Sheva), The Derived Category of Sheaves of Commutative DG Rings.
[In modern algebraic geometry we encounter the notion of derived intersection of subschemes. This is a sophisticated way to encode what happens when two subschemes $Y_1$ and $Y_2$ of a given scheme X intersect non-transversely. The classical intersection multiplicity can be extracted from the derived intersection.
When the ambient scheme X is affine, it is not too hard to describe the derived intersection, by taking flat DG ring resolutions of the structure sheaves of the subschemes $Y_1$ or $Y_2$. This also works when the scheme X is quasi-projective. However, derived intersections in more general schemes X could only be treated using the very difficult homotopical methods of derived algebraic geometry.
Several months ago I discovered a cheap'' way to construct flat resolutions of sheaves of rings. The resolutions are by semi-pseudo-free sheaves of DG rings. The main advantage is that the geometry does not change: all the action takes place on the original topological space X.
Using semi-pseudo-free resolutions it is possible to produce derived intersections as above. It is also possible to get a direct presentation of the cotangent complex of a scheme (at least in characteristic 0). Presumably the derived adic completion of Shaul, so far studied only in the affine case, could be globalized using our our methods.
Lastly, the semi-pseudo-free resolutions give rise to a congruence on the category of sheaves of commutative DG rings on the space X, that we call relative quasi-homotopy. The functor from the homotopy category to the derived category turns out to be a faithful right Ore localization. This fact gives tight control on the derived category. It should be noted that in this situation there does not seem to exist a Quillen model structure, so the traditional approaches would fail.
In the talk I will explain the various ideas listed above. More details can be found in the eprint arxiv:1608.04265.]

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