Program

Titles and abstracts

Raymundo Bautista. Tame-Wild dichotomy for the category of filtered by standard modules for a quasi-hereditary algebra.

Let A be a quasi-hereditary algebra over an algebraically closed field k, with D = D₁,...,D_n the standard modules. We consider the category F(D) consisting of the finitely A-modulesM for which there is a filtration 0<M₁ <M₂ <...<M_l =M, such that M_i+1/M_i is in D. We prove that F(D) is either tame or wild but not both. Moreover we prove that in case F(D) is tame for any dimension d, for almost all indecomposable N in F(d) having k-dimension d, there is an almost split sequence of the form 0 → N → E → N → 0.

Zajj Daugherty. Representations of the periplectic Lie superalgebra.

The periplectic Lie superalgebra p(n) is one of the two so-called "strange" Lie superalgebras, in that it is not one of the analogs of the simple Lie algebras, nor is it of Cartan type. The representation theory of p(n) has posed a particular challenge —the category of finite dimensional representations has been shown to be a highest weight cate- gory, but otherwise early attempts at applying standard methods for classification had been unsuccessful. In this talk, we will explore some of the newly understood combinatorial structure of this category.

This is joint work with Martina Balagovic, Inna Entova-Aizenbud, Iva Halacheva, Johanna Hennig, Mee Seong Im, Gail Letzter, Emily Norton, Vera Serganova, and Catharina Stroppel.

José Antonio de la Peña. The exponential of Coxeter matrices.

Let A be a finite dimensional k-algebra. It is well-known that the Coxeter matrix Φ is an integral matrix encoding a lot of information on the representation theory of A. It is the object of study of many papers, but no attention has been paid to e^ΦA , the exponential of the Coxeter matrix. In this talk we consider some basic properties of the exponential matrix in general, as well as properties of e^ΦA arising from the algebra A or the matrix Φ_A. We give examples of calculation of this matrix. This is joint work with Jesús Jiménez.

Matthew Hogancamp. Categorification of Young idempotents.

I will discuss the construction of a family of complexes of Soergel bimodules in type A which categorify the classical Young idempotents. The construction uses the theory of categorical diagonalization of Ben Elias and myself. This is joint work with Ben Elias.

Jesús Jiménez-González. Quivers and their incidence quadratic forms.

In the early 2000’s Michael Barot described the combinatorial structure of a positive integral quadratic unit form q of type A_n or D_n, and used it to determine a finite set of relations for a Lie algebra with Killing form q.

In the talk we give a transparent approach to the case A_n (through forms associated to quivers) that generalizes to the non-negative case. This route may be thought of as a categorification of such forms.

Ryan Kinser. Decomposing moduli spaces of representations of algebras.

This talk is based on joint work with Calin Chindris applying methods of Geometric Invariant Theory (GIT) to study representations of finite-dimensional algebras. I will present a Krull-Schmidt type decomposition for moduli spaces of semi-stable representations for arbitrary such algebras. If time permits, I will discuss some applications of this decomposition result.

Ellen Kirkman. Reflection Hopf Algebras.

The Shephard-Todd-Chevalley Theorem states that when a finite group G acts linearly on a commutative polynomial ring A = k[x₁, . . . , x_n] over a field k of characteristic zero, the invariant subring A^G is a commutative polynomial ring if and only if G is generated by reflections. More generally, let H be a finite dimensional semi-simple Hopf algebra that acts on an Artin-Schelter regular algebra A so that A is an H-module algebra, the grading on A is preserved, and the action of H on A is inner faithful. When A^H is Artin-Schelter regular, we call H a reflection Hopf algebra for A. We present some examples of such pairs (A,H).

Ivan Losev. Bernstein inequalities and holonomic modules.

Bernstein inequality is an inequality relating a GK-dimension of a module over a semisimple Lie algebra to that of the quotient of the universal enveloping algebra by the annihilator. Holonomic modules are particularly nice D-modules. We will discuss generalizations of the inequality and of the notion of holonomic modules to more general algebras: those obtained by quantizing Poisson algebraic varieties with finitely many symplectic leaves.

Tim Magee. GHK mirror symmetry and the Knutson-Tao hive cone.

In 1998, Allen Knutson and Terry Tao introduced a rational polyhedral cone with some amazing combinatorial properties in their proof of the saturation conjecture. The “Knutson-Tao hive cone” encodes the number of copies of a given irreducible representation of GL_n appearing in the tensor product of two others– so it tells us how to rewrite a tensor product as a direct sum. Choosing the representations of interest slices the cone to give a bounded polytope, and counting the integral points in this bounded polytope gives the number we’re looking for. This cone (and plenty of others with the same wonderful combinatorial properties) can actually be obtained by completely general mirror symmetry considerations, without any representation theory at all. We’ll get much more than the cone too. In this setting, integral points are elements of a canonical basis, and the combinatorial data is just the cardinality this basis. Moreover, this is a construction that in theory applies whenever you have a space equipped with the right sort of volume form, so the Knutson-Tao hive cone is part of a very broad framework when viewed in this way. I’ll give an overview of how this all works.

Andy Manion. Higher representations and Heegaard Floer homology.

I will discuss joint work in progress with Raphaël Rouquier on a relationship between tensor products for 2-representations of U_q(gl(1|1)) and some constructions appearing in Douglas-Manolescu’s cornered Heegaard Floer homology, a type of twice-extended TQFT framework for Heegaard Floer homology, as well as connections with other work and potential applications as time permits.

Jacob Mostovoy. Racks, multiplicative graphs and cubical products.

Augmented racks (or crossed G-sets) are algebraic structures that appear in knot theory, in the classification problem of Hopf algebras and in the study of the Yang-Baxter equation, among other subjects. Here we show that they can be thought of as 1-germs of cubical multiplications. This gives a new interpretation for the rack (co)homology and leads to a definition of several Hopf algebras associated with augmented racks.

Alfredo Nájera-Chávez. Nakajima categories and universal cluster algebras.

Nakajima categories were introduced by Keller and Scherotzke in order to study quiver varieties using Gorenstein homological algebra. In this talk I will explain how to categorify a large class of finite-type cluster algebras with coefficients using completed orbit categories associated to Nakajima categories. This class includes all finite-type skew-symmetric cluster algebras with universal coefficients and all finite-type Grassmannian cluster algebras.

Valente Santiago. Wide subcategories of finitely generated Λ-modules.

This is joint work with E. Marcos, O. Mendoza and C. Sáenz. We explore some properties of wide subcategories of the category mod(Λ) of finitely generated left Λ-modules, for some artin algebra Λ. In particular we look at wide finitely generated subcategories and give a connection with the class of standard modules and standardly stratified algebras. Furthermore, for a wide class X in mod(Λ), we give necessary and sufficient conditions to see that X = Press(P), for some projective Λ-module P; and finally, a connection with ring epimorphisms is given.

Gordana Todorov. Cluster Categories coming from Cyclic Posets.

This is joint work with Kiyoshi Igusa. Cyclic posets are generalizations of cyclically ordered sets. We show that any cyclic poset gives rise to a Frobenius category over any discrete valuation ring R. The stable category of a Frobenius category is always triangulated and has a cluster structure in many cases. The continuous cluster categories, the infinity-gon, and the m-cluster category of type A_n, (m ≥ 3) are some of the examples of this construction. Some of these new cluster categories were used by several people and most recently by Jorgensen and Yakimov in their work relating cluster categories to Borel subalgebras of sl_∞.

Yadira Valdivieso-Díaz. Computing homologies of algebras from surfaces.

One possible way to fully understand the representation theory of algebras is to understand their cohomology. For example, the n-Calabi-Yau property, cluster, and (support) tau-tilting, the newly emerging n-representation theory and homological mirror symmetry are all based on the understanding of extension spaces.

In this talk, we show how to compute four different homologies of algebras from surfaces, which are finite dimensional path algebras constructed from closed Riemann surfaces with marked points, using the combinatoric and topological data of the triangulated surface.

Weiqiang Wang. Canonical bases arising from quantum symmetric pairs.

A quantum symmetric pairs (QSP) consists of (U,Uⁱ), where U is a quantum group and Uⁱ is a coideal subalgebra corresponding to a Lie subalgebra fixed by an involution. (The classification of QSP’s of finite type corresponds to the classification of real simple Lie algebras.) We shall present an i-canonical basis theory on the modified coideal subalgebras of finite type and the tensor product U-modules. In a special case when U is of type A, the i-canonical bases admit positivity properties as well as application to super Kazhdan-Lusztig theory. We will give examples of i-divided powers, which exhibit rich q-combinatorics. This is joint work with Huanchen Bao (Maryland).

Harold Williams. Affine cluster monomials are generalized minors.

One interpretation of cluster algebras is that they organize homological relations among generating functions which count the subrepresentations of a given quiver representation. On the other hand, the theory originates in the study of coordinate rings of varieties attached to a Lie group. The cluster bases in these coordinate rings generally include a small number of simple functions called generalized minors, which in type A are just restrictions of ordinary matrix minors, but for the most part their elements lack a priori any elementary Lie theoretic meaning. However, the cluster algebra of any acyclic quiver can be realized via a specific double Bruhat of the Kac-Moody group with the corresponding Dynkin diagram, and we show that this realization has a remarkable property: infinitely many (all, for affine and finite types) cluster monomials are in fact gene- ralized minors. This says in particular that the structure theory of rigid representations of affine quivers is completely and explicitly encoded in the representation theory of loop groups, in a way that illuminates certain parallel classifications in the two subjects. This is joint work with Dylan Rupel and Salvatore Stella.

Yaping Yang. From homotopy theory to representation theory.

We will talk about a construction of affine quantum groups using cohomological Hall algebras in the setting of a generalized cohomology theory. The representations are given by the corresponding cohomology of Nakajima quiver varieties. In my talk, I will explain in detail two examples.

1. We use the Morava K-theory to construct a family of new quantum groups parametrized by a prime number and a positive integer. Those quantum groups are expected to be related to Lusztig's 2015 reformulation of his conjecture from 1979 on character formulas for algebraic groups over a field of positive characteristic.

2. We use the equivariant elliptic cohomology to establish a sheafified elliptic quantum group for any symmetric Kac-Moody Lie algebra. The rational sections give the algebra of elliptic R-matrix. I will also explain the relation between the sheafified elliptic quantum group and a global loop Grassman- nian over an elliptic curve.

This talk is based on my joint work with Gufang Zhao.