Tag - Associative rings

Raphael Steiner: Taking the Hecke algebra to its limits

12th September, 2019

We parametrize elements in the full Hecke algebra in a way such that the parametrization represents a generic automorphic form. By convolving, we then arrive at pre-trace formulas which are modular in three variables. From here, various identities for higher moments may be derived. We give applications to the sup-norm and fourth-norm of holomorphic Hecke eigenforms as well as Hecke-Maass forms on Γ \ ℍ and furthermore outline future work on higher moments of periods and quantum variance. This is joint work with Ilya Khayutin.

Carl Wang-Erickson: Derived structures controlling representations

28th February, 2019

The point of this talk is to give three examples of derived structures influencing representations that have connections with number theory. These structures arise from the differential graded algebra of group cochains valued in the endomorphism ring of a representation.

Two examples have to do with representations of a Galois group. One of these realizes a number theoretic criterion for the modulo p multiplicity one condition for Jacobians of modular curves at an Eisenstein maximal ideal of a Hecke algebra; this is joint work with Preston Wake. Another furnishes a realization as a derived Galois deformation ring of an exterior algebra considered in works of Galatius-Venkatesh, Hansen-Thorne, and Venkatesh. The third example features smooth modulo p representations of a p-adic Lie group, answering some questions of Sorensen about the relationship between its Iwasawa algebra and the associated derived Hecke algebra.

Shiquan Ruan: Hall polynomials for tame type

18th August, 2016

In this talk we will show that Hall polynomial exists for each triple of decomposition sequences which parameterize isomorphism classes of coherent sheaves of a domestic weighted projective line X over finite fields. These polynomials are then used to define the generic Ringel–Hall algebra of X as well as its Drinfeld double. Combining this construction with a result of Cramer, we show that Hall polynomials exist for tame quivers, which not only refines a result of Hubery, but also confirms a conjecture of Berenstein and Greenstein.

Edward Green: Brauer Configuration Algebras and Multiserial Algebras

18th August, 2016

In joint work with Sibylle Schroll (Univ. of Leicester), we introduce a generalization of Brauer graph algebras which we call Brauer configuration algebras. These will be defined in the talk. Brauer graph algebras are the symmetric special biserial algebras and are currently under active investigation. Defining an algebra KQ/I to be special multiserial if, for each arrow a in the quiver, there is at most one arrow one arrow b such that ab ∉ I and at most one arrow c such that ca ∉ I, we show that KQ/I is a symmetric multiserial algebra if and only if it is a Brauer configuration algebra.

An algebra is called multiserial if the Jacobson radical as a left and as a right module is a Σi Ui of uniserial modules Ui such that the intersection of any two is either (0) or a simple module. We will present a number of results, including the following:
(1) A special multiserial algebra is multiserial.
(2) The trivial extension of an almost gentle algebra by its dual is a Brauer configuration
algebra.
(3) Every symmetric radical cubed zero algebra is a Brauer configuration algebra.
(4) Every special multiserial algebra is the quotient of a Brauer configuration algebra.

We say a module M is multiserial if rad(M) is a sum Σi Ui of uniserial modules Ui such that the intersection of any two is either (0) or a simple module. Although special multiserial algebras are usually of wild representation type, we have the following surprising result which indicates that although wild, the representation theory is worth studying.

Theorem If Λ is a special multiserial algebra and M is a finitely generated Λ-module, then M is a multiserial module.

William Crawley-Boevey: Quiver Grassmannians and orbit closures for representation-finite algebras

18th August, 2016

We show that Auslander algebras have a unique tilting and cotilting module which is generated and cogenerated by a projective-injective; its endomorphism ring is called the projective quotient algebra. For any representation-finite algebra, we use the projective quotient algebra to construct desingularizations of quiver Grassmannians, orbit closures in representation varieties, and their desingularizations. This generalizes results of Cerulli Irelli, Feigin and Reineke.

Kyungyong Lee: Positivity for cluster algebras

16th August, 2016

Cluster algebras were first introduced by Fomin and Zelevinsky to design an algebraic framework for understanding total positivity and canonical bases for quantum groups. A cluster algebra is a subring of a rational function field generated by a distinguished set of Laurent polynomials called cluster variables. The Positivity Conjecture, which is now a theorem, asserts that the coefficients in any cluster variable are positive. One proof was given by Schiffler and the speaker, and another proof was obtained by Gross, Hacking, Keel and Kontsevich. We outline the idea of our proof.

Hugh Thomas: Semistable subcategories for preprojective algebras

15th August, 2016

A linear form φ on the Grothendieck group of an algebra determines a abelian, extension-closed subcategory of its finite length modules: the φ-semistable subcategory (in the sense of King). This subcategory is abelian and extension-closed. As φ varies, the subcategories picked out exhibit a wall-and-chamber structure. If the algebra is hereditary and finite type, we recover the combinatorics of Igusa-Orr-Weyman-Todorov pictures, or, equivalently, of the cluster complex. It turns out that for finite-type preprojective algebras, we obtain combinatorics described by Nathan Reading's 'shards' (originally introduced by Reading to study the combinatorics of weak order on the associated Coxeter group). Shards provide a beautiful picture from which we can recover the combinatorics for any quotient of the preprojective algebra, including the hereditary cases. Time permitting, I will also say something about affine type. This project is joint work with David Speyer, and also draws on previous joint work with Osamu Iyama, Nathan Reading, and Idun Reiten.

Rosanna Laking: Indecomposable objects in the homotopy category of a derived-discrete algebra

15th August, 2016

In this talk I will present joint work with K. Arnesen, D. Pauksztello and M. Prest. We classify the indecomposable pure-injective complexes in the homotopy category of projective modules K(ProjΛ) over a derived-discrete algebra Λ. The set of indecomposable pure-injective complexes are the points of a topological space known as the Ziegler spectrum. We give a complete description of the Ziegler topology and, making use of the interactions between this space and categories of functors, we prove that every indecomposable object in K(ProjΛ) is pure-injective.

Claire Amiot: Cluster categorification and applications to tilting theory

11th August, 2016

This series of talks is based on joint works with Oppermann, Grimeland, Labardini and Plamondon. Cluster categories are triangulated categories where quiver mutation appears as a natural operation. A first class of example is given by cluster categories associated with surfaces with marked points. A second class is constructed using the derived category of finite-dimensional algebras of global dimension 2. Mixing both constructions, one may consider surface cut algebras, that are algebras of global dimension 2 constructed from a surface and show how cluster combinatorics permits to deduce information on their derived category.

Bethany Marsh: Dimer models and cluster categories of Grassmannians

10th August, 2016

The homogeneous coordinate ring of the Grassmannian Gr(k,n) has a beautiful structure as a cluster algebra, by a result of J. Scott. Central to this description is a collection of clusters containing only Plücker coordinates, which are described by certain diagrams in a disk, known as Postnikov diagrams or alternating strand diagrams. Recent work of B. Jensen, A. King and X. Su has shown that the Frobenius category of Cohen-Macaulay modules over a certain algebra, B, can be used to categorify this structure.

In joint work with Karin Baur and Alastair King, we associate a dimer algebra A(D) to a Postnikov diagram D, by interpreting D as a dimer model with boundary. We show that A(D) is isomorphic to the endomorphism algebra of a corresponding Cohen-Macaulay cluster-tilting B-module, i.e. that it is a cluster-tilted algebra in this context. The proof uses the consistency of the dimer model in an essential way.

It follows that B can be realised as the boundary algebra of A, that is, the subalgebra eAe for an idempotent e corresponding to the boundary of the disk. The general surface case can also be considered, and we compute boundary algebras associated to the annulus.