Generalized Associahedra and Newton polytopes of F-polynomials.

Generalized Associahedra and Newton polytopes of F-polynomials.

Abstract: Generalized associahedron is a polytope whose outer normal fan is the g-vector fan. Considerable attention has been given to combinatorics of such polytopes since their relation to cluster algebra. In this talk, we will discuss constructing generalized associahedra based on quiver representations for simply laced Dynkin quivers. Our inspiration comes from a paper by Arkani-Hamed, Bai, He, Yan (2017) on scattering forms on the kinematic space in physics. Their construction can be viewed as giving an associahedron associated to the linearly oriented type A quiver. Our approach generalizes this associahedron to all simply-laced Dynkin types. Furthermore, we show that our construction can also be used to realize the Newton polytopes of the F-polynomials associated to the cluster variables. This is a joint work with Véronique Bazier-Matte, Guillaume Douville, Kaveh Mousavand, Hugh Thomas.
 

Representation Theory of Special Biserial Algebras.

Representation Theory of Special Biserial Algebras.

Abstract:

Due to the seminal work of P. Gabriel in 1970’s, if k is an algebraically closed field, each finite dimensional k-algebra A has a presentation of the form kQ/I, where Q is a finite directed graph and I is an ideal in the path algebra kQ. Although classification of all indecomposable modules over A is, a priori, a hard problem, assuming some combinatorial constraints on Q and I results in the description of indecomposable A-modules and their homological interaction in terms of some concrete diagrams.

In this talk, after reviewing the rudiments of representation theory of quivers, we focus on an interesting family of finite dimensional algebras, called special biserial. We observe that this family of algebras is closed under quotient and we use this to reduce the study of τ-tilting finiteness of special biserial algebras to that of a tractable subfamily, called minimal representation-infinite algebras. Using this subfamily, we show that τ-tilting theory, introduced by Adachi, Iyama and Reiten in 2014, can be studied via the classical concept of tilting theory and we obtain new criteria for τ-tilting finiteness of special biserial algebras.

 

Bracket width of simple Lie algebras.
Special Algebra Seminar – begins 4:00 p.m. in KED B005

Bracket width of simple Lie algebras.

Abstract:

For an element of a group G representable as a product of commutators, one can define its commutator length as the smallest number of commutators needed for such a representation, by definition the other elements are of infinite length. The commutator width of G is defined as supremum of the lengths of its elements. Recently it was proven that all finite simple groups have commutator width one. On the other hand, there are examples of infinite simple groups of arbitrary finite width and of infinite width.

In a similar manner, one can define the bracket width of a Lie algebra. It is known that all finite-dimensional simple Lie algebras over an algebraically closed field have bracket width one. Our goal is to present first examples of simple Lie algebras of bracket width greater than one. The simplest example relies on a recent work by Yu. Billig and V. Futorny on Lie algebras of vector fields on smooth affine varieties.

This talk is based on a work in progress, joint with A. Regeta.

 

Del Pezzo orders with no worse than canonical singularities.

Del Pezzo orders with no worse than canonical singularities.

Abstract: We classify del Pezzo non-commutative surfaces that are finite over their centres with terminal and canonical singularities. Using the minimal model program, we introduce the minimal model of such surfaces. We first classify the minimal models and then give the classification of these surfaces in general. This presents a complementary result and method to the classification given by Chan and Kulkarni in 2003.
 

Restricting Supercuspidal Representations to the Derived Subgroup
Special Algebra Seminar – begins 2:30 p.m. in HP 4325

Restricting Supercuspidal Representations to the Derived Subgroup

Abstract: The representation theory of reductive groups over p-adic fields reduces to the study of supercuspidal representations. In 2001, J.K. Yu described a construction that allows us to obtain supercuspidal representations of any positive depth. It was later proved by Kim (2007) that these constructions exhaust all supercuspidal representations for large enough p. To construct a supercuspidal representation of a reductive group G over a p-adic field F, J.K. Yu uses what he calls a G-datum. In this talk, we will be interested in the derived subgroup of G, which we will denote by G_der. We will discuss how we can obtain various G_der-data from a G-datum. We will then explore the relationships between the supercuspidal arising from the G-datum and the supercuspidals arising from the various G_der-data. In particular, we would like to know how the supercuspidals arising from the G_der-data appear in the restriction to G_der of the supercuspidal arising from the G-datum.
 

Motives, Periods, Algebraic cycles, and graphs.

Motives, Periods, Algebraic cycles, and graphs.

Abstract: In this talk, we will attempt to give a gentle introduction to the art of constructing categories of Mixed Motives and their (Tannakian) Galois groups. We will lightly touch on the connection to special values of zeta functions and L-functions via a focus on periods and regulators. We will discuss one particular construction of such categories built out of algebraic cycles. If time permits, we will then explore a graphical interpretation of the subcategory of mixed Tate motives, and muse on whether this makes working with algebraic cycles more tractable.
 

The orbit philosophy for Spin groups.

The orbit philosophy for Spin groups.

Abstract: Let G be a semisimple Lie group with Lie algebra g and maximal compact subgroup K. The philosophy of coadjoint orbits suggests a way to study unitary representations of G by their close relations to the coadjoint G-orbits on g^*. In this talk, we study a special part of the orbit philosophy. We provide a comparison between the K-structure of unipotent representations and regular functions of bundles on nilpotent orbits for complex and real groups of type D. More precisely, we provide a list of genuine unipotent representations for a Spin group; separately we compute the K-spectra of the regular functions on certain small nilpotent orbits, and then match them with the K-types of the genuine unipotent representations. This is joint work with Dan Barbasch.
 

Unfurlings and tensor product categories.

Unfurlings and tensor product categories.

Abstract: The categorification of quantum groups by Khovanov-Lauda and Rouquier (KLR) was a significant advancement in representation theory. However, the KLR categorification faces a major obstacle outside of finite-type A: because the morphism spaces are defined by generators and relations, it is difficult to tell whether unexpected relations would cause them to be much smaller than expected. To get over this difficulty, Webster studied how representations of the categorified quantum group behaved under deformations of their spectra. These deformed representations sometimes carry the action of a larger quantum group called an unfurling; by studying the unfurled representation, which is somewhat simpler, it is possible to bound the dimension of the original categorified quantum group because of the upper semicontinuity of dimension under deformation. In this talk, we will explain this technique in the context of categorified quantum groups, and then discuss joint work with Christopher Leonard (Virginia) in which we apply it to compute the trace decategorification of Webster's tensor product categories.
 

Chiral algebras, factorization algebras, and Borcherds' "singular commutative rings" approach to vertex algebras.

Chiral algebras, factorization algebras, and Borcherds' "singular commutative rings" approach to vertex algebras.

Abstract: : In the late 1990s, Borcherds gave an alternate definition of some vertex algebras as "singular commutative rings" in a category of functors depending on some input data (A,H,S). He proved that for a certain choice of A, H, and S, the singular commutative rings he defines are indeed examples of vertex algebras. In this talk I will explain how we can vary this input data to produce categories of chiral algebras and factorization algebras (in the sense of Beilinson--Drinfeld) over certain complex curves X. We'll discuss the failure of these constructions to give equivalences of categories, and obstructions to extending this approach to more general varieties X. I will not assume prior knowledge of vertex algebras, chiral algebras, or factorization algebras.
 

Integrable representations of weight-graded Lie algebras.

Integrable representations of weight-graded Lie algebras.

Abstract: A weight-graded Lie algebra is a Lie algebra L, which is a weight module with respect to the adjoint action of a finite-dimensional semisimple subalgebra g. Several types of Lie algebras recently studied can be viewed as weight graded Lie algebras. We will present results on the structure of integrable representations of L whose weights are bounded by a dominant weight of g. We will link the category of such representations to the module category of an associative, not necessarily commutative algebra, which we will describe for some special cases of L. The talk is based on my joint paper with Manning and Salmasian.
 

Noncommutative Resolutions and Annihilation of Cohomology.

Noncommutative Resolutions and Annihilation of Cohomology.

Abstract: Being smooth for a reduced affine variety is equivalent to having a coordinate ring of finite global dimension. In other words, when there are singularities, the (bi)functor Ext^n(-,-) is nonzero for any positive integer n. This is a classical result which intersects algebraic geometry, commutative algebra and homological algebra due to several mathematicians including Zariski, Serre, Auslander and Buchsbaum. We say that a ring element is a cohomology annihilator if it annihilates Ext^n(-,-) for arbitrarily large n. In this talk, we will discuss cohomology annihilators for Gorenstein rings. Firstly, we will look at the situation in dimension 1 and give a complete answer. Then, we will look at the relationship between noncommutative resolutions and the cohomology annihilator ideal.
 

Around the Riemann-Roch Theorem for Abelian varieties.
Special date and location : Friday at 2:30 p.m. in Macphail room (HP 4351)

Around the Riemann-Roch Theorem for Abelian varieties.

Abstract: In this talk, I will explain how the Riemann-Roch Theorem for divisors on an Abelian variety A is related to the reduced norms of the Wedderburn components of its endomorphism algebra. Motivated by this result, I will also mention more recent observations, building on work of Atiyah, Brion, Mukai and others, which pertain to Severi-Brauer varieties over A. For example, the Brauer group of A can be interpreted through the concept of theta groups.