Groupes et algébres de lie

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Reflection Groups And Coxeter Groups

These are notes for a minicourse on Hall algebras given at the ICTP in Trieste in January 2006. After giving the definition and first properties of Hall algebras, we study in some details the classical Hall algebra, the Hall algebra of quivers, and the Hall algebra of coherent sheaves on smooth projective curves. The last section deals with the Hall algebras in the context of derived categories. (The figures only seem to come out nicely in the .ps file)

/pdf/lectures-on-hall-algebras-4nkdeafypj.pdf

Lectures on hall algebras

We study support \(\tau \)-tilting modules over preprojective algebras of Dynkin type. We classify basic support \(\tau \)-tilting modules by giving a bijection with elements in the corresponding Weyl groups. Moreover we show that they are in bijection with the set of torsion classes, the set of torsion-free classes and many other important objects in representation theory. We also study \(g\)-matrices of support \(\tau \)-tilting modules, which show terms of minimal projective presentations of indecomposable direct summands. We give an explicit description of \(g\)-matrices and prove that cones given by \(g\)-matrices coincide with chambers of the associated root systems.

Classifying \(\tau \)-tilting modules over preprojective algebras of Dynkin type

The Hall algebra UE+ of the category of coherent sheaves on an elliptic curve E defined over a finite field has been explicitly described and shown to be a 2-parameter deformation of the ring of diagonal invariants R+=C[x1±1,…,y1,…]S∞ (in infinitely many variables). We study a geometric version of this Hall algebra, by considering a convolution algebra of perverse sheaves on the moduli spaces of coherent sheaves on E. This allows us to define a canonical basis B of UE+ in terms of intersection cohomology complexes. We also give a characterization of this basis in terms of an involution, a lattice, and a certain PBW-type basis.

http://www.mi.uni-koeln.de/~burban/BurbanSchiffmann.pdf

On the Hall algebra of an elliptic curve, II

Each integrable lowest weight representation of a symmetrizable Kac-Moody Lie algebra \(\mathfrak{g}\) has a crystal in the sense of Kashiwara, which describes its combinatorial properties. For a given \(\mathfrak{g}\), there is a limit crystal, usually denoted by B(−∞), which contains all the other crystals. When \(\mathfrak{g}\) is finite dimensional, a convex polytope, called the Mirkovic-Vilonen polytope, can be associated to each element in B(−∞). This polytope sits in the dual space of a Cartan subalgebra of \(\mathfrak{g}\), and its edges are parallel to the roots of \(\mathfrak{g}\). In this paper, we generalize this construction to the case where \(\mathfrak{g}\) is a symmetric affine Kac-Moody algebra. The datum of the polytope must however be complemented by partitions attached to the edges parallel to the imaginary root δ. We prove that these decorated polytopes are characterized by conditions on their normal fans and on their 2-faces. In addition, we discuss how our polytopes provide an analog of the notion of Lusztig datum for affine Kac-Moody algebras. Our main tool is an algebro-geometric model for B(−∞) constructed by Lusztig and by Kashiwara and Saito, based on representations of the completed preprojective algebra Λ of the same type as \(\mathfrak{g}\). The underlying polytopes in our construction are described with the help of Buan, Iyama, Reiten and Scott’s tilting theory for the category \(\Lambda \text {\upshape -}\mathrm {mod}\). The partitions we need come from studying the category of semistable Λ-modules of dimension-vector a multiple of δ.

http://www.numdam.org/item/10.1007/s10240-013-0057-y.pdf

Affine Mirković-Vilonen polytopes

To an abelian category A of homological dimension one satisfying certain finiteness conditions, one can associate an algebra, called the Hall algebra. Kapranov studied this algebra when A is the category of coherent sheaves over a smooth projective curve defined over a finite field, and observed analogies with quantum affine algebras. We recover here in an elementary way his results in the case when the curve is the projective line.

https://www-irma.u-strasbg.fr/%7Ebaumann/kapra.pdf

The Hall algebra of the category of coherent sheaves on the projective line

The purpose of this paper is to apply the theory of MV polytopes to the study of components of Lusztig’s nilpotent varieties. Along the way, we introduce reflection functors for modules over the non-deformed preprojective algebra of a quiver.

/pdf/preprojective-algebras-and-mv-polytopes-261exkffws.pdf

Preprojective algebras and MV polytopes

We propose an analogue of Solomon's descent theory for the case of a wreath product G/G n , where G is a finite abelian group. Our construction mixes a number of ingredients: Mantaci-Reutenauer algebras, Specht's theory for the representations of wreath products, Okada's extension to wreath products of the Robinson-Schensted correspondence, and Poirier's quasisymmetric functions. We insist on the functorial aspect of our definitions and explain the relation of our results with previous work concerning the hyperoctaedral group.

/pdf/a-solomon-descent-theory-for-the-wreath-products-2b8o1n33j3.pdf

A Solomon descent theory for the wreath products ≀_{}

Let G be a complex connected reductive group and let G be its Langlands dual. Let us choose a triangular decomposition n ⊕ h ⊕ n of the Lie algebra of G. Braverman, Finkelberg and Gaitsgory show that the set of all Mirkovic-Vilonen cycles in the affine Grassmannian G ( C((t)) ) /G ( C[[t]] ) is a crystal isomorphic to the crystal of the canonical basis of U(n). Starting from the string parameter of an element of the canonical basis, we give an explicit description of a dense subset of the associated MV cycle. As a corollary, we show that the varieties involved in Lusztig’s algebraic-geometric parametrization of the canonical basis are closely related to MV cycles. In addition, we prove that the bijection between LS paths and MV cycles constructed by Gaussent and Littelmann is an isomorphism of crystals.

/pdf/on-mirkovic-vilonen-cycles-and-crystal-combinatorics-4lt9l37ois.pdf

Pierre Baumann

Papers

Affine Mirković-Vilonen polytopes

The Hall algebra of the category of coherent sheaves on the projective line

Preprojective algebras and MV polytopes

A Solomon descent theory for the wreath products ≀_{}

On Mirković-Vilonen cycles and crystal combinatorics